Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Reservoir turbidity modelling using artificial neural networks and the estimation of performance indicators Chan-Yan, Deborah A. 2000

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2000-0177.pdf [ 5.29MB ]
Metadata
JSON: 831-1.0064040.json
JSON-LD: 831-1.0064040-ld.json
RDF/XML (Pretty): 831-1.0064040-rdf.xml
RDF/JSON: 831-1.0064040-rdf.json
Turtle: 831-1.0064040-turtle.txt
N-Triples: 831-1.0064040-rdf-ntriples.txt
Original Record: 831-1.0064040-source.json
Full Text
831-1.0064040-fulltext.txt
Citation
831-1.0064040.ris

Full Text

R E S E R V O I R TURBID ITY M O D E L L I N G US ING ARTIFICIAL N E U R A L N E T W O R K S A N D T H E EST IMATION O F P E R F O R M A N C E IND ICATORS by Deborah A. Chan-Yan B.Sc.Eng., Queen's University, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2000 © Deborah A. Chan-Yan, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of G i ^ l l ^ ^ j ^ A f l& JU The University of British Columbia Vancouver, Canada Date Aali ^ ,70CO DE-6 (2/88) ABSTRACT Ensuring proper suspended sediment and turbidity control requires an understanding of water body response to turbidity-causing events. This thesis describes an approach which evaluates turbidity control using risk-based performance indicators, and which is suitable for application in a wide range of water resource problems. In particular, the performance of a water supply reservoir under major drawdown conditions is evaluated in terms of meeting water quality turbidity objectives. The case study presented is based on the Capilano Reservoir located in North Vancouver, British Columbia. Response to turbidity-causing events is simulated using artificial neural networks, and turbidity levels are estimated for both the annual and wet season periods. In addition, artificial neural network models are developed for short-term forecasting of turbidity levels at the reservoir outlet. Reservoir reliability and resilience are then estimated using FORM. Reliability and resilience estimates are derived for various scenarios of interest and, as a result, can be used to evaluate tradeoffs between meeting proposed drawdown objectives and the elevated turbidity levels that may occur within the water body. The reservoir is shown to exhibit poorer performance during the wet season, particularly in the October to December period, compared to its overall performance on an annual basis. ii TABLE OF CONTENTS A B S T R A C T II T A B L E O F C O N T E N T S HI LIST O F T A B L E S VII LIST O F F I G U R E S VIII A C K N O W L E D G E M E N T S IX 1 INTRODUCT ION 1 2 L I T E R A T U R E R E V I E W 5 2.1 Artificial Neural Networks 5 2.2 Risk-Based Performance Indicators for Water Resource Systems 8 2.3 First-Order Reliability Method (FORM) Applications 9 3 ARTIFIC IAL N E U R A L N E T W O R K S (ANNS) 11 3.1 Neural Network Description 12 3.2 Network Training 14 3.2.1 Genera l izat ion and Overtraining 14 3.2.2 Error Back-Propagat ion 16 3.2.3 Epoch S i ze 16 3.2.4 Stopping Criter ia and Network Val idat ion 17 3.3 Network Architecture and Internal Parameters 19 3.3.1 Number of Layers 19 iii 3.3.2 Number of Nodes 20 3.3.3 Transfer or Activation Functions 20 3.3.4 Learning Rate and Momentum Coefficient 21 3.3.5 Initial Connection Weights 23 4 T H E F I R S T - O R D E R RELIABIL ITY M E T H O D (FORM) A N D R I S K - B A S E D P E R F O R M A N C E IND ICATORS 24 4.1 Risk-Based Performance Indicators 24 4.1.1 Reliability 24 3.1.1 Resilience 24 4.2 First-Order Reliability Method (FORM) Theory 25 4.3 FORM Based Estimates of Reliability and Resilience 30 4.3.1 Reliability 30 4.3.2 Resilience 31 4.3.3 Reliability and Resilience Estimates for Turbidity 33 4.4 FORM and Monte Carlo Simulation (MCS) 34 5 C A P I L A N O R E S E R V O I R C A S E S T U D Y 36 5.1 Background 36 5.2 Turbidity 37 5.2.1 Turbidity as a Water Quality Indicator 38 5.2.2 Elevated Turbidity Levels within the Capilano Reservoir 39 5.2.3 Historical Reservoir Drawdowns 40 5.3 Existing Turbidity Model for Capilano Reservoir 40 5.3.1 Estimation of Sediment Loads 41 5.3.2 Surface Water Quality and Hydrodynamic Modelling 42 5.4 Case Study Scenarios 43 iv 6 ARTIFICIAL N E U R A L N E T W O R K D E V E L O P M E N T A N D E S T I M A T E S O F P E R F O R M A N C E IND ICATORS F O R C A P I L A N O R E S E R V O I R 45 6.1 Turbidity Data 45 6.2 Artificial Neural Network (ANN) Development 46 6.2.1 Network Inputs 47 6.2.2 Network Architecture 55 6.2.3 Network Training 57 6.2.4 Turbidity Forecast ing 59 6.3 Performance Indicator Estimates 60 6.3.1 RELiabi l i ty A N a l y s i s ( R E L A N ) Software 6 0 6.3.2 R E L A N Inputs - R a n d o m Var iable Character ist ics and Corre lat ions 61 6.3.3 R isk-Based Per formance Indicator Calcu lat ions using R E L A N and F O R M 64 7 R E S U L T S A N D D I SCUSS ION 67 7.1 Artificial Neural Network Turbidity Simulation and Forecasting 67 7.1.1 Turbidity S imulat ion 67 7.1.2 Turbidity Forecas ts 75 7.2 Evaluation of Reservoir Performance Indicators 78 7.2.1 R a n d o m Var iab le Generat ion and Ass ignment 78 7.2.2 Annua l Reliabil ity and Res i l i ence 81 7.2.3 Wet S e a s o n Reliability and Res i l ience 86 7.2.4 Reliability and Res i l ience for the Wet S e a s o n Months 89 7.2.5 R a n d o m Var iable Sensitivit ies and Dominant A N N Inputs 92 7.2.6 Appl icat ion of Importance Sampl ing (IS) to R e d u c e Computat iona l Burden 96 8 C O N C L U S I O N S A N D F U T U R E W O R K 98 R E F E R E N C E S 100 A P P E N D I X I A N N TURBID ITY S U B R O U T I N E S F O R R E L A N 107 v A P P E N D I X II C O R R E L A T I O N M A T R I C E S F O R R E L A N C A L C U L A T I O N S 118 A P P E N D I X III R E L A N R E S U L T S 125 LIST OF TABLES Table 6-1 ANN Dominant Input Results from Fuzzy-Neural Algorithm 53 Table 6-2 ANN Input Sets - Annual 54 Table 6-3 ANN Input Sets - Wet Season 55 Table 6-4 Network Architectures - Annual 56 Table 6-5 Network Architectures - Wet Season 56 Table 6-6 Data Set Characteristics for Turbidity (NTU) - Annual 58 Table 6-7 Data Set Characteristics for Turbidity (NTU) - Wet Season 59 Table 6-8 Distribution Descriptors for RELAN 62 Table 6-9 Random Variable Characteristics - Annual and Wet Season 62 Table 6-10 Random Variable Characteristics - Wet Season Months 63 Table 7-1 ANN Performance Results - Annual 68 Table 7-2 ANN Performance Results - Wet Season 68 Table 7-3 Selected Turbidity Models - Annual and Wet Season 69 Table 7-4 ANN Performance - Turbidity Forecasting 76 Table 7-5 Random Variables for RELAN - Annual 79 Table 7-6 Random Variables for RELAN - Wet Season 80 Table 7-7 Effect of Probability Accuracy on Resilience Estimates - 8 NTU 85 Table 7-8 Effect of Probability Accuracy on Resilience Estimates - 14 NTU 86 Table 7-9 Computational Time for Resilience Estimates - Wet Season 97 LIST OF FIGURES Figure 3-1 ANN Schematic - Three Layer Network 13 Figure 3-2 Good Generalization vs. Overtraining 15 Figure 3-3 ANN Transfer Functions 22 Figure 4-1 Failure Surface in Two-Dimensional Random Variable Space 27 Figure 4-2 Transformed Failure Surface in Standard Normal Space 29 Figure 6-1 Historical Daily Average Inflow Record for Capilano Reservoir showing two extreme values (363 and 472 m3/s) 51 Figure 7-1 ANN Validation for the Annual Scenario (1) 71 Figure 7-2 ANN Validation for the Annual Scenario (2) 72 Figure 7-3 ANN Validation for the Wet Season (1) 73 Figure 7-4 ANN Validation for the Wet Season (2) 74 Figure 7-5 ANN Turbidity Forecast Performance 77 Figure 7-6 Reliability for the Annual Scenario using MCS, FORM, and IS-FORM... 82 Figure 7-7 Failure Surface and Under-Estimation of Reliability by the FORM Approximation 83 Figure 7-8 Resilience for the Annual Scenario using MCS, FORM, and IS-FORM.. 84 Figure 7-9 Reliability for the Annual and Wet Season Scenarios using MCS and IS-FORM 87 Figure 7-10 Resilience for the Annual and Wet Season Scenarios using MCS and IS-FORM 88 Figure 7-11 Reliability for the Wet Season Months using IS-FORM 90 Figure 7-12 Resilience for the Wet Season Months using IS-FORM 91 Figure 7-13 Sensitivity of Reliability Estimates to Random Variables for the Annual Scenario and 10 NTU Turbidity Standard 94 Figure 7-14 Sensitivity of Reliability Estimates to Random Variables for the Wet Season Scenario and 10 NTU Turbidity Standard 95 viii ACKNOWLEDGEMENTS The author would like to acknowledge the National Science and Engineering Research Council (NSERC) graduate fellowship that funded this research. Information made available by Klohn-Crippen Consultants Ltd., and comments and data provided by the Greater Vancouver Regional District are also greatly appreciated. A special thank you is extended to Dr. Barbara J. Lence for her continued support, and to Dr. Holger R. Maier, J. Graham Lang, and Bryan A. Tolson for helping make this work possible. ix 1 INTRODUCTION Suspended sediment and turbidity control is an important aspect of overall water resource management. Elevated turbidity levels degrade water quality and adversely affect its usefulness in terms of water supply, fisheries, and agricultural and industrial purposes. As such, understanding reservoir response to turbidity events is important for both managerial and operational decision-making that relate to water quality concerns. The goal of this research is to investigate the application of artificial neural networks (ANNs) to estimate reservoir turbidity levels and risk-based indicators that describe system performance in terms of meeting water quality standards. The application of ANNs is investigated for both simulation and short-term forecasting, and a method that combines an ANN and a first-order reliability approach is developed for estimating system performance. For numerous reasons, the application of ANNs for water quality and turbidity modelling is advantageous compared to standard dynamic two- or three-dimensional simulation modelling. While standard dynamic models rely on complete time-series input for simulation, ANN models are fundamentally different and only require single-point-in-time inputs which are then related to single output values. Significant time, savings can result with the use of ANNs since their processing speed is fast and their computational burden is generally less than that associated with simulation models (see Flood and Kartam 1994a). In some instances, ANNs can also be employed as simplified versions of more complex simulation models where they replicate and achieve comparative results (Johnson and Rogers 1995; Yang et al. 1996). Another advantage of ANNs is that networks can be developed, using a similar approach, for both simulation and forecasting applications. The ability to forecast turbidity allows for advanced planning of such operations as raw water treatment. 1 The use of ANNs in the area of water resource modelling and management has gained increased popularity over the past ten years (for a recent review see Maier and Dandy 2000). ANNs consist of numerous computing elements that are interconnected and arranged in layers. They are computational mechanisms that are able to acquire, represent, and compute a mapping from one multivariate space to another (Garrett 1994). The name "Artificial Neural Network" and the theory behind ANNs is derived from their biological counterpart where, as in humans, previous "learning" is key to the successful processing of new information. In a similar but much simpler way, ANNs work by pattern recognition where patterns are "learned" from examples presented to the network during a training phase. After training, the ANN can then be used to process new information and provide output results characteristic of the mapping obtained. Many ANN models are in fact similar or identical to popular statistical techniques and can be interpreted as nonlinear regression functions (Cheng and Titterington 1994; Sarle 1994). As with the statistical techniques, ANNs are also "calibrated" or "trained" by minimizing a measure of error associated with a set of calibration or training data. The usefulness of performance indicators in the management of a water supply reservoir warrants the development of a method that easily quantifies a reservoir's ability to meet water quality objectives. Simulation models, in general, are useful tools. However, in order to adequately reflect the complexities of such systems, the effect of model and parameter uncertainties and natural variations in system inputs (see Vicens et al. 1975) need to be considered. Turbidity modelling is no exception as there are numerous stochastic and random inputs that directly impact turbidity levels within the water column. These inputs include precipitation, runoff, streamflow, and meteorological data, among others. In an effort to assess the performance of water resource systems under naturally varying conditions, Hashimoto et al. (1982) introduce the risk-based performance indicators of reliability, resilience, and vulnerability. These indicators represent the probability, expected length, and likely magnitude of a water resource failure such as a shortfall in water supply or the violation of a water quality standard. 2 Maier et al. (1999) develop an approach to estimate reliability, resilience, and vulnerability using FORM (Madsen et al. 1986). Reliability analysis is based on the premise that engineered systems function by resisting the loads to which they are subjected. As long as a system's capacity exceeds the demand imposed on it, the system is stable; however, when the demand or load exceeds the resisting capacity, failure occurs. This is the case, as mentioned previously, when a measured water quality indicator (load) surpasses the particular standard (capacity) associated with it. FORM estimates the probability of failure (and hence the reliability) of a particular system given the mean values and other statistical parameters of the random variables involved. In this research, ANN modelling for turbidity simulation and forecasting is illustrated for a case study based on the Capilano Reservoir, North Vancouver, British Columbia. The technique developed by Maier et al. (1999) is then applied to estimate both the reliability and the resilience of the system in terms of meeting various turbidity standards. The Capilano Reservoir case study assumes major drawdown conditions that are maintained indefinitely. This represents a hypothetical situation, but was chosen in order to examine the potential water quality threat posed by exposed sediment beds that are subject to erosion and reintrainment in the water column. The remainder of this thesis is organized as follows: Chapter 2 provides a background literature review of the use of ANNs in water resource applications, the evaluation of risk-based performance indicators, and FORM. Chapter 3 discusses ANNs in detail and Chapter 4 describes FORM theory and how it is applied to estimate system reliability and resilience. A description of the Capilano Reservoir case study is provided in Chapter 5, and Chapter 6 presents the ANN turbidity modelling for Capilano Reservoir and the evaluation of reservoir performance. Results are presented and discussed in Chapter 7 and the final chapter, Chapter 8, 3 provides a brief summary along with relevant conclusions and recommendations for future work. 4 2 LITERATURE REVIEW 2.1 Artif icial Neural Networks Although the concept of artificial neurons was first introduced in 1943 (McCulloch and Pitts 1943), significant advances in their application have only been possible since 1986 with the introduction of the back-propagation training algorithm (Rumelhart et al. 1986) and advances in computing capabilities. Since then, and particularly within the last decade, researchers have begun to investigate the potential of ANNs as a tool for supporting the modelling of engineering systems (Garrett 1994). Flood and Kartam (1994a; 1994b) discuss the use of ANNs in civil engineering and provide a description of system applications. Ample information is available in the literature regarding ANN theory and development (for example, see Battiti 1992; Bebis and Georgiopoulos 1994; Hegazy et al. 1994). Comparisons from a statistical perspective can also be found (Cheng and Titterington 1994; Sarle 1994). ANNs have been applied to a wide range of water resource problems. For example, ANNs have been used in the optimization of reservoir operating policy (Jain et al. 1998; Raman and Chandramouli 1996), the long-term operation of hydroelectric power systems (Saad et al. 1994), and the interpretation of water distribution network state estimates for leak detection and identification (Gabrys and Bargiela 1999). Wen and Lee (1998) show how ANNs can predict a decision-maker's preferences in multiobjective waste load allocation problems, and Bouchart and Goulter (1998) show that ANNs can replicate a decision-maker's "choice" between graphical curves that characterize the risk associated with different reservoir release strategies. Maier and Dandy (2000) provide a complete review of the use of ANNs for predicting and forecasting water resource variables. Modelling issues, including ANN 5 performance criteria, model inputs, network architecture, training, and model validation, are discussed. Details of the water resource applications are also summarized. As evident in the review, the use of ANNs for simulation and forecasting has gained significant popularity since 1992, and particularly since 1997. The majority of applications involve the prediction and/or forecasting of rainfall and streamflow, although a number of applications also relate to water quality. Dawson and Wilby (1998) produce a 15-minute forecast of streamflows with a lead time of six hours and show how ANNs can be used as alternatives to conventional flood forecasting models. Sureerattanan and Phien (1997) use ANNs for daily flow forecasting with rainfall, discharge, and past values of rainfall and discharge as system inputs. Karunanithi et al. (1994) make similar one-day-ahead flow predictions and employ a constructive algorithm that automatically synthesizes a suitable network architecture as part of the training process. In terms of multi-day forecasts, Atiya et al. (1999) find that the standard feed-forward network performs better than a recursive approach where a network forecasts one step ahead and then is applied recursively to forecast k steps ahead. ANN models have been found to forecast monthly reservoir inflows better than statistical models (Raman and Sunilkumar 1995) and better than Auto-Regressive-Moving-Average time-series models at high flows (Jain et al. 1998). Nam et al. (1998) both simulate and forecast monthly inflows, and find that using an error updating method does not improve long-term forecasts as all the available information is already being applied by the network. The use of ANNs in water quality problems has been limited compared to the number of streamflow and rainfall-runoff applications. Maier et al. (1996) use an ANN to produce a 14-day forecast of salinity for the River Murray in South Australia. For the same river, Maier et al. (1998) model the appearance of blue-green algae in the system. Bastarache et al. (1997) estimate acidity and conductivity levels in small streams by using daily flow values and the time of year as inputs. Polynomial 6 networks are also used to aid in the choice of input vectors, and various network architectures are investigated. Raw water quality can impact the treatment processes and costs associated with safe water supply, and therefore the ability to predict water quality changes is important. Zhang and Stanley (1997) use an ANN to forecast raw water colour in the North Saskatchewan River one day ahead. As a result, rather than relying on the traditional reactionary approach, process conditions can be planned and optimized to meet expected changes in raw water quality. The authors find that the dominant network inputs for forecasting colour are turbidity, flow rate, precipitation, and the rates of change of these parameters. They also recommend that up to ten previous days of parameter values be considered when determining potential network inputs. Due to their computational efficiency, ANNs have been used as a replacement for conventional process based models to predict water table depths (Yang et al. 1996). The networks compute data significantly faster than DRAINMOD, a conventional mathematical model that simulates the performance of subsurface drainage systems. The ANNs also require fewer inputs than DRAINMOD to produce comparable results. In another groundwater application, Johnson and Rogers (1995) develop an ANN to replicate mass-extraction and management information that is usually generated by SUTRA, a two-dimensional hybrid finite-element/finite-difference model. The ANN is then used to optimize a pumping strategy, and a total of 250 pumping patterns are identified which meet restoration goals at minimum cost. The rapid computation time allows the completion of a sensitivity analysis which distinguishes between desirable and less desirable results. Rogers and Dowla (1994) present a similar groundwater remediation study where advantage is taken of the ANN'S ability to model flow and transport in a parallel manner. 7 2.2 R i s k -Ba sed Per fo rmance Indicators for Water R e s o u r c e S y s t e m s In recent years, the focus of water quality management has changed dramatically to include concerns regarding long-term sustainability and design for ecological resilience (Beck 1997). As a result, the criteria by which we judge what is "best" are changing. In addition to focusing on system stability (i.e., reliability) in which we keep the environment in some equilibrium state, we have begun to acknowledge that multi-equilibrium states may exist. The concept of ecological resilience was first developed (Holling 1973) in an attempt to represent the degree to which an ecological system is protected from shifting into a less preferred state of equilibrium or non-equilibrium. Fiering and Holling (1974) suggest that a simple measure of ecological resilience is the propensity of any point far from an acceptable state to migrate, within one time interval, to an acceptable state. Hashimoto et al. (1982) develop the first operational definition of resilience for use in water resource applications. Resilience is defined as the probability of the system being in a non-failure state in one period, given that it was in failure the previous period; reliability is defined as the probability of the system residing in a satisfactory state, i.e., the probability of non-failure; and vulnerability is a third indicator that quantifies the magnitude of a failure. These indicators have been applied in various water resource applications, particularly to evaluate reservoir performance (such as Burn et al. 1991; Hashimoto et al. 1982; Moy et al. 1986) and water distribution systems (Xu and Goulter 1999; Zongxue et al. 1998). Water quality applications have been relatively limited (Lence and Ruszczynski 1997; Maier et al. 1999). Rather than maximizing the reliability or resilience of a water supply project explicitly, Hashimoto et al. (1982) treat these indicators as performance measures which are estimated after the selection of an optimal operation strategy. Subsequent water resource studies focus on either reformulation of the resilience surrogate (Burn et al. 1991; Moy et al. 1986) or on application of classical optimization techniques such as deterministic Mixed Integer Linear Programming (MILP) in order to optimize for 8 resilience (Moy et al. 1986). With MILP, however, the length of time period that can be examined is limited and the stochastic nature of inputs cannot be accounted for. Lence and Ruszczynski (1997) integrate resilience into the decision-making process by developing a Stochastic Branch and Bound model to solve the time-dependent stochastic problem of maximizing resilience. This is shown to be comparable to maximizing the probability that the water quality is maintained for a given length of time. More recently, Xu and Goulter (1999) show how the design of a water distribution network can be optimized by taking into account reliability, the estimates of which are obtained using FORM. Maier et al. (1999) derive definitions of reliability, vulnerability, and resilience using FORM, and then apply these performance indicators to a water quality case study of the Willamette River, Oregon. In this case study, tradeoffs are observed between the risk-based indicators at different dissolved oxygen standards and wasteload levels, suggesting that alternative water quality management policies may impact measures of reliability, resilience, and vulnerability in different ways (Maier et al. 1999). Using Genetic Algorithms and a simplified water quality model for the same case study, Vasquez et al. (1999) obtain waste load allocation solutions that provide the optimal tradeoff between treatment cost and water quality reliability. 2.3 F irst -Order Reliability Method (FORM) App l i ca t ions FORM (Madsen et al. 1986; Rackwitz 1976) was originally developed to assess the reliability of structures where limit state design is employed and the performance of a system is expressed in terms of its load (demand) and resistance (capacity). FORM has more recently been applied in water resource engineering, particularly in groundwater applications (Hamed et al. 1995; Sitar et al. 1987; Skaggs and Barry 1997). Melching et al. (1990) use FORM to determine the uncertainty of the peak discharge predictions obtained from a rainfall-runoff model for the Vermillion River watershed, Illinois. A comparison between MFOSM, FORM, and MCS for the same case study is also carried out (Melching 1992). There is good agreement between FORM and MCS for a wide range of storm magnitudes and types; however, MFOSM 9 does not perform as well in cases where nonlinearities are significant. Other examples include the reliability analysis of a coastal dike (Ronold 1990) and the design of a wastewater storage pond (Buchberger and Maidment 1989). There have also been a number of surface water quality applications using FORM. For example, Tung (1990) compares the performance of FORM, Mean-value First-Order Second Moment (MFOSM) analysis, and Monte Carlo Simulation (MCS) for evaluating the probability of violating various dissolved oxygen standards for a hypothetical case study. A similar study is also carried out by Melching and Anmangandla (1992) using the hypothetical case studies of Burges and Lettenmaier (1975) and Tung and Hawthorn (1988). In both papers, the performance of FORM is very similar to that of MCS, however, MFOSM does not perform as well, especially at the extremes of the range of DO standards investigated. As described in Section 2.2, recent water quality applications using FORM include those of Maier et al. (1999) and Vasquez et al. (1999). 10 3 ARTIFICIAL NEURAL NETWORKS (ANNs) ANNs are mathematical mechanisms that consist of simple computational units interlinked by weighted connections. They are commonly used in three main ways (Sarle 1994): i) as models of biological nervous systems and "intelligence"; ii) as real-time adaptive signal processors or controllers implemented in hardware for applications such as robots; and iii) as data analytic tools. For simulation modelling applications, ANNs fall into the third category. The theory behind ANNs is based, in an abstract manner, on the way their biological counterparts function. The concept of "learning" is exploited, albeit on a much smaller and simpler scale. Just as humans apply knowledge gained from past experience to new problems or situations, ANNs take previously solved examples, look for patterns, and then learn these patterns so they can correctly classify new information (Zhang and Stanley 1997). In many instances, ANNs are in fact comparable to statistical or other standard simulation models. All are capable of processing vast amounts of data and of making predictions that are accurate (Sarle 1994). The most obvious statistical interpretation of a multilayer network is that it provides a nonlinear regression that is achieved by optimizing a measure of fit to the training data (Cheng and Titterington 1994). The connection weights are adjusted in much the same way that statistical parameters are calibrated in a statistical model. The main distinction between ANNs and standard simulation models is that the former belong to a class of data-driven approaches while the others are model-driven (Raman and Sunilkumar 1995). Despite their similarities, ANNs provide numerous advantages over standard simulation modelling techniques, in particular the ease with which they may be applied to novel or poorly understood problems (Battiti 1992). In addition, no mathematical algorithm is required, they are fast and flexible due to the simple task performed by each neuron, non-linear relationships can be handled well, and the networks are generally fault-tolerant (Zhang and Stanley 1997). Sarle (1994), 11 however, notes that many engineers apply their networks as black boxes requiring no human intervention - data in, predictions out. As a result, there is a tendency among users to throw a problem at a network blindly in the hope that it will formulate an acceptable solution (Flood and Kartam 1994a). One disadvantage of ANNs, as opposed to statistical models, is the difficulty with which network connection weights can be interpreted to determine the importance and relationships of input variables. The remainder of this chapter describes how ANNs work, discusses network training in detail, and presents some of the issues that affect network performance. 3.1 Neural Network Descr ipt ion A neural network is made up of numerous processing elements called neurons (or perceptrons) that are interconnected and arranged in layers. Neurons are characterized by either a linear or nonlinear transfer function, and each connection is associated with a weighted value. The overall computational mechanism implements a non-linear mapping of Y=K(U) where the mapping function K is determined during the training phase of network development (Bebis and Georgiopoulos 1994). During network training, the connection weights are adjusted so that input patterns U are correctly associated with given output patterns V. The learning rule is a recursive algorithm in which the weights are modified until an acceptable level of error between predicted and actual output is achieved. When connections between neurons are allowed in a single direction, i.e., between neurons in one layer and neurons in the next (but not within the same layer), the network is known as a feed-forward ANN. In this common form, information is passed in one direction through the network where neurons receive a weighted input from each of the neurons in the previous layer. Figure 3-1 shows how the summed input is then processed by the transfer or activation function, with the output being passed to the next layer. Computations are carried out at each layer until the final layer is reached. 12 Figure 3-1 A N N Schemat i c - Three Layer Network 13 3.2 Network Tra in ing During training, input-output examples are repeatedly presented to the ANN. At each iteration, input variables are processed by the neurons and then the calculated output, V, is compared to a known value, V. Based on this comparison, the connection weights are adjusted in an attempt to reduce the error between the two output values. Generally, networks are trained by minimizing the sum-of-squared-differences, Z£( V,-Y/)2, where the summation is over all outputs and over the subset of the training data under consideration. Training continues until a pre-determined stopping criterion has been met. The following sections discuss various network training issues, namely generalization ability, error back-propagation, epoch size, stopping criteria, and model validation. 3.2.1 Generalization and Overtraining The main goal of ANN development is to train a network so that it performs well, not only on the training examples provided, but on the entire realm of possible data that is representative of the problem. The generalization ability is a measure of a network's performance outside the training domain. If a network is overtrained it has learned the intricacies of a particular data set well but has lost its ability to generalize. Figure 3-2 compares a well-trained network to an overtrained network where the predicted curve fits the data points well, but is not representative of the overall trend. 14 Good Generalization Overtraining Figure 3-2 G o o d Genera l izat ion vs. Overtra in ing 15 3.2.2 Error Back-Propagation Error back-propagation (Rumelhart et al. 1986) is the most widely-applied algorithm used in ANN training. It consists of two phases: the forward phase where inputs are fed through the network, and the backward phase where modifications to the connection weights are made based on the overall error measure between computed outputs and actual target values. In the backward phase the output error is reduced by effectively redistributing it backwards through the hidden layers (Jain et al. 1998). The generalized delta rule is used to determine the weight change for each connection which is proportional to the size and direction of the gradient of the error surface. (The error surface is defined by all combinations of U for which K=0, and the gradient is the partial derivative of the error function with respect to the weight in consideration (Maier and Dandy 1998a)). The delta rule is a gradient descent approach which utilizes the chain rule for differentiation, and as such, the neurons must be characterized by differentiable transfer functions. Despite its capabilities, back-propagation suffers from several problems that make the development of an ANN model a task that is neither simple nor straightforward. These include (Hegazy et al. 1994; Jain et al. 1998): i) ill-defined knowledge representation and problem structuring; ii) slow training and sensitivity to initial weights; iii) training trapped in local minima or paralysis; iv) nonguided design of an optimal network configuration for adequate generalization; and v) difficult interpretation of the network weights (black box effect). In addition, the generalized delta rule is slow and tedious, requiring the user to set various algorithmic parameters by trial and error (Sarle 1994). These parameters (discussed in Section 3.3) include the learning rate, momentum coefficient, number of neurons in each layer, their connectivity, and stopping criteria for training. 3.2.3 Epoch Size Epoch size refers to the number of training samples that are presented to the network before each update of the connection weights. (The updates are based on 16 the total error over the epoch.) In batch mode, the entire set of training examples is processed by the network before the weights are updated. In online mode the epoch size is one, and connection weights are updated each time a training example is presented. The main advantage of batch processing is that the error is calculated over the entire training set and therefore the true gradient of the error surface can be obtained. When weights are adjusted based on this overall solution, subsequent network evaluations always result in decreased total error. This is accomplished, however, at the expense of significant computational time if the training set is large. Online processing is faster, however the generally-decreasing error measure tends to fluctuate since weight updates are based on the error associated with a single training example and not the entire training set. A significant advantage of online processing is that the added degree of randomness can often help the network escape local minima. 3.2.4 Stopping Criteria and Network Validation Various stopping criteria are used to determine when network training should be terminated. (Again, if a network is overtrained it loses its ability to generalize.) Common techniques include training for a set number of iterations or training until an acceptable level of error has been reached. These methods generally involve two data sets, one for training and another for network validation. Network output (Y) can be compared to known targets (V) both qualitatively (e.g., visually) and quantitatively using statistical measures such as the root-mean-square error (RMSE), the correlation coefficient (R2), the average absolute error (AAE), and the average absolute percent error (AAPE). As indicated by Karunanithi et al. (1994), the root-mean-square error is a good measure of fit for peak values, whereas the relative error (AAE or AAPE) provides a more balanced perspective of the goodness of fit at moderate values. The measures of error are defined as follows: RMSE = ' -I w A/tr' 17 AAE =-Lyi\/-Y. AAPE= l y ^ l where A/ is the number of samples, v and a represent the mean value and standard deviation of the samples, respectively, and all other variables are as defined above. Evaluating network performance at intervals during the training process is an informative way of assessing training progress. With cross-validation (see Amari et al. 1997), three representative data sets are used: one for training, one for testing, and one for validation. Network connection weights are adjusted based on the error associated with the training set. At predetermined intervals, network performance is also evaluated using the test data set. Since both data sets are representative of the same problem, the error surface for the test examples should decrease at a rate similar to that of the training examples (Flood and Kartam 1994a). At some point, however, the network has fully learned the training examples, and continued training may lead to overtraining. Rising test error is indicative of decreased generalization ability, and at this point the training process is stopped and the network is assumed to be fully trained. Because the test data set is used in the training process, it should not be used for network validation. Network performance should be confirmed using an independent data set, in this case the validation data set. In order for training and validation to be successful, however, it is important that all three data sets be representative of the same problem domain and therefore have similar statistical characteristics. In addition, since ANNs are not generally able to extrapolate, the 18 training patterns should reach the extremes of the problem domain so that the network "learns" those areas of the problem space (Flood and Kartam 1994a). 3.3 Network Arch i tecture and Internal Parameters Network training, and hence overall network performance, is affected by the network architecture chosen and the internal parameters (e.g., learning rate and momentum coefficient) used during the training process. Trial and error experimentation is often employed, and iterative refining of internal parameters, network redesign, and problem reformulation are frequently required (Hegazy et al. 1994). However, defining the goal of the application can help influence many decisions. For example, the primary objective of an application may be to develop a highly accurate model, regardless of its complexity or the time that it takes to train. Alternatively, the goal may be to achieve satisfactory performance with less training, or a simpler model that, once trained, processes data quickly. The following sections summarize the effects of network architecture and internal parameters on network training and performance. Refer to Maier and Dandy (2000) for a complete discussion as well as a recent review of other ANN modelling issues. 3.3.1 Number of Layers Many researchers agree that the quality of a solution given by an ANN depends strongly on the network size used (Bebis and Georgiopoulos 1994). In general, network size affects network complexity and learning time, as well as the generalization capabilities of the network. For practical reasons, choosing a smaller network over a larger one has advantages such as reduced training time, reduced memory requirement, and fewer training examples required. However, this also restricts the number of free parameters in the network and, consequently, the error surface is more complicated and contains more local minima which may hinder the training process (Bebis and Georgiopoulos 1994). 19 ANN models comprise an input layer of nodes, an output layer, and one or more hidden layers in between. There is some disagreement as to whether it is better to use one or two hidden layers. Two hidden layers can provide greater flexibility for modelling complex-shaped error surfaces (Flood and Kartam 1994a), and highly nonlinear functions can sometimes be approximated well with fewer connection weights than would be necessary with a single hidden layer (Sarle 1994). Maier and Dandy (1998b), however, note that one hidden layer has been found to be adequate in most practical ANN applications. As such, it is generally recommended to start with one hidden layer and add more as required. 3.3.2 Number of Nodes The number of nodes in the input and output layers can generally be determined by the dimensionality of the problem (Bebis and Georgiopoulos 1994), i.e., the number of input variables supplied to the network and the number of outputs required are the number of nodes in the input and output layers, respectively. On the other hand, there is no direct or precise way of determining the appropriate number of nodes to include in each hidden layer (Flood and Kartam 1994a). As a result, the number of hidden nodes is often determined by experimentation and trial and error (Hegazy et al. 1994; Raman and Sunilkumar 1995) where a range of different configurations is considered, and the configuration with the best performance is accepted (Flood and Kartam 1994a). 3.3.3 Transfer or Activation Functions Each neuron or node consists of a transfer function that maps its summed input to an output value. The output is then weighted and passed to the next layer. The transfer functions may be linear or non-linear, but must be differentiable for use by the generalized delta rule in error back-propagation. Common transfer functions include: linear, threshold, sigmoidal, radial basis, and hyperbolic tangent functions (see Figure 3-3). Since these transfers map into a bounded range of-1 to 1 or 0 to 1, the original input vectors are often normalized in the training process. This helps 20 to reduce the effects of patterns that represent a region that is relatively narrow in some dimensions while elongated in others (Flood and Kartam 1994a). The choice of transfer functions can affect the speed of convergence and training time of a network since the size of the steps taken in weight space is proportional to the derivative of the transfer function. If the gain of the transfer function (i.e., the slope of the almost linearly varying portion) is large, the derivative of the transfer function, and hence the weight change increments, is larger and training speed increases (Maier and Dandy 1998a). On the other hand, training can slow to a near halt at high or low output values where the derivative of the transfer function approaches zero, and accordingly, the weight change increments approach zero (Hegazy et al. 1994). 3.3.4 Learning Rate and Momentum Coefficient The learning rate is an important internal parameter used in the back-propagation algorithm. It is directly proportional to the weight changes taken to move about the error surface in search of the global minimum. If the weight change is too small, training is very slow and the network is likely to get stuck in a local minimum. A small weight change, however, can be very useful in avoiding oscillations and for getting deeper into valleys or similar regions of minimum error. On the other hand, larger step sizes may allow the network to jump around the error surface and escape local minima to find better solutions. The learning rates used in various water resource ANN applications are given by Maier and Dandy (2000). The momentum coefficient also determines the size and direction of weight changes taken in error space. It can effectively increase the step size across shallow portions of the error surface so that the training process can be accelerated. The momentum coefficient lies between values of 0 and 1. 21 Linear My Threshold V Sigmoidal V Radial Basis Hyperbolic Tangent Figure 3-3 A N N Trans fer Func t i ons 22 3.3.5 Initial Connection Weights Training is sensitive to a network's initial position on the error surface. If initial network positioning is unfavourable the network may get stuck in a sub-optimal local minimum. As a result, randomly selecting the initial weights of the network connections for each training run may start the network at a better location from which an improved result may be found. 23 4 THE FIRST-ORDER RELIABILITY METHOD (FORM) and RISK-BASED PERFORMANCE INDICATORS The following sections define the indicators used to evaluate system performance, summarize FORM, and describe its application in determining water quality performance. The information presented is based on the work of Maier et al. (1999) and Xu and Goulter (1999). 4.1 R i s k -Ba sed Per fo rmance Indicators 4.1.1 Reliability Reliability is a measure of the probability that a system resides in a satisfactory state, S. For a system defined by random variables X, Hashimoto et al. (1982) define the reliability of a system, a, at time t as: a=Pr{XreS} (1) Reliability is also the conjugate of the probability of failure, pt, and can be defined as: a=^-pf (2) 3.1.1 Resilience In water resource engineering, resilience has generally been used to measure how quickly a system recovers from failure, once a failure has occurred. Hashimoto et al. (1982) give two equivalent definitions of resilience, y. One is a function of the expected value, E[ ], of the length of time a system remains unsatisfactory after a failure, 7) (see Equation 3). 24 1 7 = E[Tf] (3) The other definition of resilience is based on the probability that the system will recover from failure in a single time step (Equation 4). 4.2 F irs t -Order Reliability Method (FORM) Theory FORM (Madsen et al. 1986) was originally developed to assess the reliability of structures (Rackwitz 1976), but has recently been used in water resources applications (Maier et al. 1999; Melching and Anmangandla 1992; Sitar et al. 1987; Skaggs and Barry 1997; Tung 1990). It is based on the principle that the performance of an engineering system can be expressed in terms of its load (demand) and resistance (capacity). For example, in a structural case, load and resistance may correspond to the force on a column and its bearing capacity, respectively. Similarly, in a water supply problem, a distribution network's water demand and supply capacity may correspond to the load and resistance parameters. In FORM, each probabilistic event is described by a unique performance function, G(X), which defines events in terms of the basic random variables affecting system performance. If X = (X1; X2, Xj)T is the vector of random variables that influences a system's load (L) and resistance (R), then the performance function, G(X), is commonly written as: where the failure (limit state) surface G(X) = 0 separates all combinations of X that lie in the failure domain (F) from those in the satisfactory or survival domain (S). As 7 = Pr{XM G S Xt e F} = Pr{X, e F and XM e S} Pr{X, G F} (4) G(X) = R-L (5) 25 illustrated in Figure 4-1 for the two-dimensional case, failure occurs when load surpasses resistance, i.e., when G < 0, whereas survival is characterized by a load meeting adequate resisting capacity, i.e., when G > 0. Consequently, the probability of failure, pf, is given as (Sitar et al. 1987): p, = Pr{XeF} = Pr{G(X)<0} = \fx(x)dx (6) G(X) < 0 where fx(x) is the joint probability density function (PDF) of X. Approximate solutions for the integral in Equation 6, and hence the probability of failure, can be obtained using a variety of techniques including Monte Carlo Simulation (MCS) and FORM. Detailed comparisons of FORM and MCS can be found in Maier et al. (1999) and Skaggs and Barry (1997). For reliability computations using FORM, it is convenient to transform the basic random variables X into the standard normal space Z = T(X), so that the elements of Z are statistically independent and give a standard normal distribution. The exact nature of this transformation depends on the properties of random variables X. When the variables X are normal and mutually independent, the transformation is given by (7) where xy and cry are the mean and standard deviation of random variable xj, respectively. For non-normal independent variables X, the equivalent normal distribution can be derived by matching the original cumulative distribution and probability density functions to the approximate or equivalent normal random variable distributions at the failure point (Yen et al. 1986). For non-normal and dependent variables, the variables X can be transformed to standardized normal variables Z using the Rosenblatt transformation (Madsen et al. 1986). 26 Mean point of xi and X2 l*2 Figure 4-1 Fai lure Sur face in Two-D imens iona l R a n d o m Var iab le S p a c e The transformation also maps the limit-state surface defined by G(X) = 0 into standard normal space defined as G(2) = g(r1(2)) = 0, where superscript -1 denotes the inverse of the transformation. An approximation of the failure probability, pt, in this standard normal space is then obtained by replacing the failure surface with a hyperplane at the design point using First-Order Taylor Series expansion. Figure 4-2 illustrates this for the two-dimensional case. The design point, Z*=(zi*,z2*), yields the highest risk of failure, and is the point on the failure surface closest to the mean point for zi and z2, or the point defined by the n variable means. The distance from the mean point to the design point is known as the "reliability index", j8, and can be interpreted as a safety margin that indicates how far the system is from failure when in its mean state. It is given as The reliability index can then be used to obtain the system probability of failure since where 0() is the standard normal cumulative distribution function (CDF). Determination of the design point, and hence /3, is a constrained nonlinear minimization problem and can be formulated as: P = (zTzf=[(zS)2+(z2*)2j (8) Pf=*(-j8) (9) Min j3 = Min Z T Z subject to G(2)=0 (10) 28 Mean point of zi and Z2 z2 Figure 4-2 T rans fo rmed Fai lure Sur face in S tandard Normal S p a c e 29 Note that Equation 9 is only exact if (i) the elements of X are uncorrelated normal variables with a mean of zero and a standard deviation of one, and (ii) the failure surface is a hyperplane. Since these conditions are rarely met in realistic applications, the transformation to standard normal space (as discussed previously) is necessary. In addition, due to the First-Order Taylor Series expansion used, FORM provides only an approximate estimate of the failure probability unless the performance function is itself linear. The degree of nonlinearity in the performance function, and hence the accuracy of FORM, is problem dependent. The sensitivity of the probability of failure estimates obtained with FORM to the uncertainties associated with the random variables can be determined by computing the direction cosines of the variables in standard normal space. For example, the direction cosine of the first random variable is defined as: cos 01 =Zi/j8 (11) where 01 is the angle between the design point vector and the axis of random variable Xi. The sum of the squares of the direction cosines for all random variables is equal to one, and the greater the square of the direction cosine the greater the importance of that variable near the design point. 4.3 F O R M B a s e d Est imates of Reliability and Res i l i ence 4.3.1 Reliability By combining Equations 2 and 9, reliability can be estimated as: a=1-Pf=1-O(-/3) = 0>(j3) (12) 30 As discussed in Section 4.2, the above relationship is only exact if the failure surface is a hyperplane, otherwise it is an approximation. 4.3.2 Resilience In water resource applications, Equation 3 has generally been used to obtain estimates of resilience. This is undertaken by examining a time-series (real or synthetic) of the system performance variables, examining failure events and if a failure occurs, counting the number of consecutive time steps that the system remains in failure. If the conditional probability definition is considered (Equation 4), estimates of resilience can be obtained using FORM. The concept of multiple failure modes is exploited as illustrated below. In many applications it is necessary to consider more than one failure mode. For example, in structural or geotechnical engineering, a beam may fail in bending or in shear and a retaining wall may fail by overturning or by sliding. If there are two failure modes, the probability of failure is given by Pf= Pfi + Pn- Pm = Pr{Gi < 0} + Pr{G2 < 0} - Pr{Gi < 0 and G 2 < 0} (13) where pn and pa are the probabilities of failure due to failure modes 1 and 2, respectively, pm is the joint probability of failure for failure modes 1 and 2 and G\ = Gi(X) and G 2 = G2(X) are the performance functions for failure modes 1 and 2, respectively. The failure probabilities for the individual failure modes (pn and p^ ) can be obtained using Equation 9. The joint probability of failure, pm, is given (Madsen etal. 1986) as: Pfi2 = *(-Pv-P2;Pn) = * ( - h ) * ( - P 2 ) + P'iv(-Pv-P2;y)dy (14) 31 where 0( , ;p) is the cumulative distribution function (CDF) for a bivariate normal vector with zero mean values, unit variances, and correlation coefficient p, and y/( , ;p) is the corresponding probability density function (PDF). The integral in Equation 14 is generally obtained numerically. The correlation coefficient needed to evaluate this integral, P12, is calculated using (Madsen et al. 1986): Z*T Z * 1 P12 = , 1 , T\ = — z : T Z: (15) |z;| & /32 1 2 where Zi* and Z2* are the design points in standard normal space for failure modes 1 and 2, respectively. If we define the performance functions as Gi = Rt-Lt (16) G 2 = L M-f? f + 1 (17) the corresponding individual and joint failure probabilities are given by p f l = Pr{X F E F} = 0(-j31) (18) Pa = Pr{X f + 1 e S} = 0(-/32) (19) p f l 2 = Pr{Xt e F and e S} = 0(-j8i, -/32, p12) (20) The conventional definition of the performance function (Equation 5) is used in Equation 16, however, the order of L and R is reversed in Equation 17, so that the probability of failure, as defined in Equation 6, is actually the probability that the 32 system will return to a non-failure state (see Equation 19). Combining Equations 4, 18, and 20, resilience is given by 7 = (21) 4.3.3 Reliability and Resilience Estimates for Turbidity In water quality systems, loads are generally given by the ambient water quality or environmental conditions, while resistance is usually associated with a particular water quality standard or guideline. System performance can be described using Equation 16. In the case of turbidity and suspended sediment concentrations, the performance function becomes: where TS is a fixed turbidity standard at a critical location, and Ta(X) is the ambient turbidity level, a function of random variables X, and is generally estimated using a water quality response model. Resilience calculations involve the evaluation of a second failure mode as described by Equation 17. For turbidity resilience problems, the second failure function defines system recovery. FORM incorporates the uncertainties inherent in water quality modelling by using random variables. Information about the mean, standard deviation, and distribution of each variable is required, as well as the correlation structure between the variables. In many applications, this information cannot be obtained easily in its entirety, thus expensive data collection programs are required or assumptions must be made based on experience or studies conducted elsewhere. Consequently, it is desirable to keep the number of random variables to a minimum while ensuring that Gi(X) = R-L = TS-Ta(X) (22) G2(X) = L-R = Ta(X)-TS (23) 33 all significant sources of uncertainty are included. Restricting the number of random variables also helps to reduce the computational time required. 4.4 F O R M and Monte Car lo S imulat ion (MCS) MCS involves determining a problem's probability of failure by performing multiple evaluations of the scenario using possible combinations of input parameters. Numerous realizations are carried out, each one with a set of random variables generated based on their statistical properties and on the relationships between them. The system probability of failure is calculated by counting the total number of failures that occur in all the iterations that are carried out. Provided that the number of iterations is sufficiently large, MCS can generally calculate a failure probability close to the exact value (Melching 1992). The major disadvantage of MCS is its high computational cost since the number of realizations required to estimate an accurate probability depends on the unknown failure probability itself. In contrast, FORM only needs a small number of iterations for convergence. Consequently, it is more computationally efficient than MCS, particularly at low failure probabilities. The relative advantage of FORM, however, decreases with a larger number of random variables (Adebar et al. 1994). The performance function becomes more complicated and additional computation is required to compute the gradient of the error surface at each FORM iteration. The main disadvantage of FORM is that the accuracy of its estimates is problem specific and depends highly on the shape of the failure surface involved. Despite this, the first-order approximation given by FORM generally gives good results in standard normal space since the probability density function decays exponentially with distance from the origin. Consequently, most of the probability content in the failure region is in the vicinity of the design point where the first-order expansion is a good approximation of the failure surface (Sitar et al. 1987). 34 Apart from its computational efficiency, a major advantage of FORM is that it can provide a measure of the sensitivity of the failure probability to the input parameters Xwith little additional computational burden. Similar information cannot be directly obtained using MCS. An important characteristic of FORM to note is that it only caters to random variables, not random processes. This can pose a problem when stochastic processes are involved or when time-series information is required to calculate system load or resistance. 35 5 CAPILANO RESERVOIR CASE STUDY An ANN model, based on the Capilano Reservoir under major and long-term drawdown conditions, is developed to simulate and predict turbidity levels at the dam outlet as a function of streamflow, rainfall, rainfall intensity, and meteorological variables. The turbidity model is then linked to a FORM-based algorithm to estimate the system's reliability and resilience in terms of meeting turbidity standards. Under normal operating conditions the Capilano Reservoir undergoes seasonal drawdowns due to variations in water supply and demand. The case study presented, however, represents a hypothetical scenario since the drawdown is maintained for an indefinite period and at very low water levels. The main focus of this research is not to comment on expected turbidity values in such a scenario, but to demonstrate the validity of using ANNs for water quality modelling, and to demonstrate how reservoir performance can be described with risk-based performance indicators. This chapter provides background information on the Capilano Reservoir, presents turbidity issues related to water quality, summarizes the turbidity modelling that has previously been completed, and discusses scenarios of interest in this research. 5.1 B a c k g r o u n d The Capilano Reservoir and Cleveland Dam are located in North Vancouver, British Columbia. The reservoir is one of three reservoirs that provide water to the Greater Vancouver Regional District (GVRD). The Capilano Reservoir is used solely for water supply purposes, supplying approximately 40% of the total demand for the population of 1.8 million, as well as a small flow to a downstream federal fish hatchery. The region is mountainous, receiving high amounts of annual precipitation, and the entire watershed is closed to public access. The Capilano Reservoir is a long and narrow water body (approximately 4.2 km long by 0.5 km wide), with a depth of up to 75 m at the Cleveland Dam face. It is oriented 36 north-west to south-east, however flow near the dam is predominantly to the south. The Upper Capilano River is the major inflow of the reservoir. It enters the reservoir at its shallow north end while minor inflows enter along its length and join the water body at steep side banks. Cleveland Dam is situated at the southern end of the reservoir, and the fish-spawning Lower Capilano River flows downstream of this point. The Capilano Fish Hatchery is supplied with water directly from the reservoir and is located on the left bank of the Lower Capilano River, just downstream of the dam. The reservoir's full supply level, at a water-surface elevation of 146 m, is usually maintained throughout the late fall, winter, and spring seasons. An annual drawdown occurs over the summer period, the amount of which depends predominantly on the weather and the related water demand. The average summer drawdown lowers water levels in the reservoir by 8 to 14 metres which translate into surface elevations ranging from 132 to 138 m. In terms of water quality, the key locations within the reservoir are at the mid-level and low-level dam outlets, located at elevations 115 and 75 m, respectively. These locations are important since drinking water and fisheries supplies are drawn from the mid-level works, and excess discharges for flood control are often made to the Lower Capilano River through the low-level outlets. 5.2 Turbidity Elevated turbidity levels have a negative impact on water quality and can affect its various uses. In surface water systems, turbidity levels are affected by the type of sediments present, rainfall events, rainfall intensity, flow levels, erosion and deposition potential, and wind and wave action (Haan et al. 1994). In addition to these mechanisms, reservoir systems are particularly sensitive to such activities as landslides, deforestation, and major drawdowns where large amounts of sediment can be added to the water body. 37 5.2.1 Turbidity as a Water Quality Indicator Turbidity is a measure of the capacity of a fluid to scatter or absorb light due to elevated levels of particulate (Peavy 1985). It is commonly measured in nephelometric turbidity units (NTU) by determining the amount of light passing through a sample or the amount of light reflected by a sample at a certain angle relative to the light source. Although turbidity levels can indicate the amount of particulate present, they cannot be related directly to weight concentrations of suspended solids because of the effects of size and shape on a solid's ability to scatter light. Colour and fraction of organic matter can also contribute significantly to turbidity. Despite these factors, turbidity is often used as a diagnostic property to infer water quality. For example, water potability is determined by the GVRD using turbidity rather than total suspended solids (TSS). Reasons for this include a greater ease in testing and monitoring, a higher sensitivity at low sediment concentrations (where a significant portion of particles may be too small to be detected by the TSS test), and consistency for comparison with water quality guidelines which also use the turbidity measure. The maximum desirable turbidity level for drinking water is 1 NTU. However, the maximum acceptable limit, measured at the point of consumption and for aesthetic objectives, is set at 5 NTU (BC Environment 1998; Canada 1997). Turbidity levels in drinking water should be below 1 NTU to ensure adequate disinfection, as elevated levels are associated with increased adsorptive sites and particulates that can shield organisms from the full effects of chlorination (Peavy 1985). High turbidity levels are also associated with increased biological nutrient availablility, the potential growth of harmful bacteria, the formation of trihalomethane, and elevated concentrations of heavy metals (Canada 1995). 38 Apart from being a drinking water issue, turbidity is also a fisheries concern due to its relation to suspended solids concentrations. High suspended solids concentrations can affect fisheries growth and survival by clogging or damaging fish gills, by causing abrasion, and by modifying natural fish movements and migration (Wilber 1983). Elevated turbidity levels can also lead to the smothering of eggs and changes in water chemistry to which the fish are not easily adaptable. 5.2.2 Elevated Turbidity Levels within the Capilano Reservoir High intensity and prolonged rainfall events have substantial erosion potential. As a result, even at near-full water levels, slightly elevated turbidities can occur within the Capilano Reservoir following large storms in the watershed. These turbidity levels, however, tend to subside quickly due to sediment deposition, and highly turbid waters do not generally reach the dam outlets or affect water supply. The high inflows associated with large storms also help to increase the dilution and flushing capacity of the system, thereby reducing the effects of added sediments on reservoir turbidity levels. On the other hand, major drawdown scenarios expose large areas of sediment beaches around the reservoir perimeter which pose a risk to overall water quality. In the case of the Capilano Reservoir, soil characteristics within the reservoir vary significantly with location (e.g., in inflow stream deltas, in near-shore areas, at the lake centre, and at the dam face), ranging from cobbles and sands to fine silts and clays (Thurber Engineering 1993). Lenses and layers containing high amounts of organic matter are also common. During drawdown periods, these sediments are subject to erosion and resuspension in the water column and elevated turbidity levels may result if the sediments contain significant quantities of fine material. Water quality in the reservoir has also been impacted by landslide activity. For example, in October 1995 a landslide occurred in an area of high instability in the 39 north-west portion of the reservoir, moving a large amount of debris directly into the water body. The turbidity generated by the landslide reached peak levels of eight to ten times greater than the acceptable level of 5 NTU (Canadian Water Quality Guidelines), and forced the closure of the reservoir as a water supply source for the following five months (Cavill 1997). During this period, water supply for the whole district was provided by the two other reservoirs which service the region. 5.2.3 Historical Reservoir Drawdowns Major reservoir drawdowns have occurred during the February-April periods of 1991, 1992, 1996, and 1999. During these months, the reservoir water levels were lowered significantly below the annual summer low of approximately 135 m, to levels of 124.5 m in 1992 and 122 m in 1996 and 1999. The maximum turbidities measured at the mid-level dam outlet were 18, 14, and 22 NTU during the three drawdowns, respectively (Hatfield Consultants 1999). The 1996 drawdown was undertaken to enable remedial works to be completed within the reservoir basin. Heavy machinery was used to remove large amounts of sediment that had been deposited by the October 1995 landslide, and the area was stabilized to reduce further erosion activity. In 1999, the drawdown enabled in-reservoir drilling to be completed along the east abutment of the dam, and also provided an opportunity to collect relevant data related to the drawdown and its effects on water quality. An extended drawdown is currently scheduled to begin in the fall of 2001 and to be maintained through the spring so that seepage repairs can be made along the east abutment of Cleveland Dam. 5.3 Ex i s t ing Turbid i ty Mode l for Cap i l ano Reservo i r In order to gain a better understanding of turbidity levels that can be expected in the scheduled 2001 drawdown, turbidity modelling exercises have been carried out. Preliminary modelling was performed by Hay and Co. (Hatfield Consultants 1997) and detailed modelling and analysis was completed by Klohn-Crippen Consultants 40 Ltd. (Hatfield Consultants 1999; Klohn-Crippen Consultants 1999). The turbidity model developed by Klohn-Crippen Consultants Ltd. is a linked sediment load and transport model which combines the Universal Soil Loss Equation (USLE) (Haan et al. 1994) with the CE-QUAL-W2 (Cole and Buchak 1995) water quality and hydrodynamic model to route sediment loads through the reservoir system. It is a dynamic model which uses time-series data of inputs and predicts turbidity levels at locations of interest within the reservoir over time. The following two sections provide a summary of this reservoir turbidity model (refer to Klohn-Crippen Consultants 1999). 5.3.1 Estimation of Sediment Loads There are numerous mechanisms that introduce sediment into a reservoir during drawdown. To account for these, sediment loads are derived in three ways: • Sediment from the Upper Capilano River is derived as a function of inflow rate to allow for sediment transport capacity and for erosion and downcutting of the delta; • Rainfall-induced loads from exposed side and end beaches are computed for individual rainfall events using the Universal Soil Loss Equation (see below); and • A constant sediment source function is included to account for wave induced erosion, small tributary delta erosion, groundwater discharge erosion, and other miscellaneous mechanisms. The Universal Soil Loss Equation (USLE) is an empirically-based multiplicative relationship that determines the average annual soil loss per unit area of exposed land, A, as a function of rainfall and runoff (R), soil erodibility (K), slope length (L), slope steepness, (S) cover type (C), and conservation practices (P) (Haan et al. 1994). The USLE equation is expressed as: 41 A= RKLSCP (24) In the Capilano case, the K, L, S, C, and P factors remain constant throughout the drawdown period, and therefore the amount of rain-induced sediment produced varies depending on the R factor and the area of exposed beach. The R factor characterizes individual storm events and is a function of an event's total rainfall, duration, and maximum 30-minute intensity. A high R-factor indicates that a given storm has a large potential, in terms of energy, for soil erosion by way of soil particle detachment and transport. 5.3.2 Surface Water Quality and Hydrodynamic Modelling In any modelling exercise, appropriate water quality and hydrodynamic components must be chosen that can replicate the physical, biological, and chemical characteristics of the waterbody of interest. CE-QUAL-W2 (Cole and Buchak 1995) is a well-known two-dimensional longitudinal-vertical model that is maintained by the US Army Corps of Engineers. It is specifically designed for long and narrow waterbodies like the Capilano Reservoir which exhibit predominantly longitudinal and vertical water quality and hydrodynamic gradients. Its hydrodynamic and water quality components are also coupled as a single modelling unit which enables it to handle large variations in surface water levels and makes it well-suited to modelling drawdown situations. CE-QUAL-W2 simulates the transport of a wide range of water quality constituents based on hydrodynamics, initial temperature and concentration conditions, influent constituent concentrations (from point and diffuse sources), influent temperatures, constituent kinetics, and inflow and outflow boundary conditions. CE-QUAL-W2 is a dynamic model and runs with time-series input to provide the values of water quality parameters, over time, at different locations in the system. Both temperature and suspended sediment concentration affect model hydrodynamics because of density 42 effects and, as a result, CE-QUAL-W2 can be sensitive to influent temperatures (affecting density flows), and to meteorological parameters (particularly wind stress because of its effects on cooling and circulation). For the sediment model developed for the Capilano Reservoir, daily sediment loads are derived using the USLE and historical rainfall data, and are then routed through the reservoir system using CE-QUAL-W2. The hydrodynamic portion of the modelling is based on the time-series of daily inflows and reservoir levels. Due to a lack of available meteorological data, monthly average values of air temperature, dew point temperature, inflow temperature, wind, and cloudiness are used. CE-QUAL-W2, like most water quality simulation programs, models suspended sediment concentrations, and therefore a conversion to turbidity units is employed to complete the modelling process. Water quality and hydrometric data from the 1992 and 1996 drawdown periods are used to calibrate the Klohn-Crippen turbidity model. It is subsequently fine-tuned with data available from the 1999 drawdown (see Klohn-Crippen Consultants 1999). Results from this turbidity model are used to predict expected average and maximum turbidity levels at the dam outlets for a reservoir drawdown to 122 m under average, wet, and dry year scenarios. 5.4 C a s e Study Scena r i o s In developing ANN turbidity models and for evaluating the Capilano Reservoir performance, a number of scenarios are investigated. These scenarios consist of the following operating periods: • Annual • Complete Wet Season (October to April) • Monthly for the Wet Season (seven months, October to April) These scenarios are used to investigate how reservoir performance varies throughout the year. The Annual period scenario serves as a basis for comparison 43 and the Wet Season period scenario generally experiences elevated turbidities due to increased rainfall. (The Wet Season is also, coincidentally, the same period over which the drawdown in 2000 is scheduled.) By looking at system reliability and resilience in each month of the Wet Season it is possible to determine the months in which the risk of elevated turbidities is greatest. Comparisons of the monthly periods can also provide information useful for operational decisions related to water quality and the scheduling of drawdowns. 44 6 ARTIFICIAL NEURAL NETWORK DEVELOPMENT and ESTIMATES OF PERFORMANCE INDICATORS FOR CAPILANO RESERVOIR The first half of this chapter describes the development of ANN models for simulating and forecasting turbidity levels at the mid-level outlet of the Capilano Reservoir. Two ANNs are developed using Neuframe, an ANN development software program. One ANN is for the Annual period and the second is for the 7-month Wet Season period from October to April. Due to a lack of existing turbidity data for major drawdown scenarios, a synthetic data set is initially generated upon which the ANNs are trained. The second half of the chapter illustrates how these networks are integrated with a FORM approach to estimate reservoir performance in terms of meeting turbidity standards. FORM is implemented using a general reliability software package, RELiability ANalysis (RELAN), and estimates of system reliability and resilience are obtained for different scenarios of interest. 6.1 Turbidity Data The available outflow turbidity record for the Capilano Reservoir is limited as continuous monitoring at the mid-level dam outlet began recently in 1993. In terms of major drawdown events, the record is limited to seven months of data derived from the drawdowns that occurred in 1992, 1996, and 1999. Since the aim of the ANN modelling in this research is to simulate and forecast turbidity levels under similar drawdown conditions, it is necessary to synthesize representative turbidity data for network training, testing, and validation. The CE-QUAL-W2 turbidity model developed by Klohn-Crippen Consultants Ltd. is used to generate a synthetic turbidity record for a drawdown scenario where the reservoir is maintained indefinitely at a water level of 125 m. This elevation is 21 m below full supply level, 10 m above the mid-level outlet works, and corresponds to an exposed sediment beach area of approximately 70 ha. The reservoir capacity at 45 125 m is reduced to a volume of 36.8 million m3, nearly half of the 70 million m3 volume at full supply. The CE-QUAL-W2 model was run for the entire period of record for which concurrent records of each input parameter exist, albeit with minor gaps. Concurrent data for hourly rainfall, maximum 30-minute rainfall intensity, and Upper Capilano River inflow are available for the periods March 1964 to October 1972, January 1974 to December 1979, and December 1990 to November 1998. Monthly average values of wind speed, wind direction, air and dew point temperature, cloudiness, and influent stream temperatures are also used. The turbidity record synthesized by CE-QUAL-W2 is based on approximately 22.5 years of historical data. It is representative of the given drawdown scenario and is used in the subsequent ANN model development. 6.2 Artif icial Neural Network (ANN) Deve lopment The ANN modelling exercise is aimed at simulating and forecasting turbidity levels at the reservoir's mid-level outlet when the reservoir is subject to very low water surface elevations. However, because the ANN is trained and validated using synthesized data, the aim is also to investigate how ANNs can be applied to replicate the results of the more complex CE-QUAL-W2 model. Simplifying complex simulation models while maintaining satisfactory performance can present a number of advantages. For example, the CE-QUAL-W2 turbidity model requires the time-consuming calculation of parameters such as the rainfall-runoff factor which is used to derive sediment loads. It also requires initial conditions and complete time-series data as inputs to complete the hydrodynamic simulation and provide turbidity predictions over time. Once an ANN is trained, however, it can quickly process single input vectors and provide single turbidity predictions. These inputs may contain information from previous days, but are not required in a continuous time-series format. Compared to complex water quality simulation models, ANNs can also be adapted for forecasting applications relatively easily. Forecasting ability is advantageous from an operational point of view where, rather than relying on a reactionary approach, advance warning of changes in raw water quality can allow 46 treatment processes to be modified and can improve water supply planning (Zhang and Stanley 1997). The ANN models are developed using Neuframe (Version 3.0.0 Neural Computer Sciences, Ltd., 1997), a commercially available Windows-based neural network development program. The following sections discuss the input parameters that are used in the two ANN turbidity models (Annual and Wet Season), and the reasons for their choices. Using these inputs, various parameter and network architecture combinations are investigated in an attempt to determine optimum ANN performance. The training method is also discussed along with network performance in terms of turbidity forecasting. 6.2.1 Network Inputs The problem of identifying appropriate inputs for ANN time-series modelling can be divided into two tasks: determining which individual parameters and data are relevant, and determining the times at which each of these inputs are important. Inputs from previous periods, e.g. at time t.n (n days before t0), are often included since the output time-series may depend not only on its own previous values, but also on past values of the other input variables involved (Maier and Dandy 1997; Zhang and Stanley 1997). For example, water quality parameters such as flow and temperature often exhibit persistence and are dependent not only on today's events, but also on events from previous days. 6.2.1.1 Relevant Variables A number of different input parameters may be useful for modelling and forecasting turbidity levels at the Capilano Reservoir outlet. In particular, it should be recalled that the elevated turbidity levels associated with large drawdowns often result from rainfall induced erosion of exposed sediment beds. In addition, high inflows introduce more turbidity to the system than low inflows because of a greater 47 sediment carrying capacity. Potential ANN inputs are identified based on these factors and based on the form and type of available data. The following input variables are considered: • Time of year or date; • Meteorological variables: air temperature (°C), dew point temperature (°C), wind speed (km/hr) and direction (rad), and cloudiness (decimal percent); • Reservoir inflow: Upper Capilano River flow (m3/s); • Precipitation: amount of rainfall (mm) in a storm event, where an event is defined as at least 3 mm of precipitation preceded and followed by a minimum 3-hour dry period; • Storm duration: rainfall event duration (hr); • Rainfall intensity: average and maximum 30-minute intensity (mm/hr). In ANN modelling, the time of year can be used as an input to take into account the seasonal effects that characterize many parameters, as well as other factors that are not explicitly included in the input. Atiya et al. (1999) apply a monthly date factor in numerical terms, while Zhang and Stanley (1997) address the issue of "seasonality" by dividing the available data into four sets and training four separate networks, each one representing a different time period. In the Capilano Reservoir case study, the method presented by Bastarache et al. (1997) is applied where daily dates are represented as continuous positions on a circle. The date factors take the form of sine and cosine functions, e.g. s\n(2n(j/k) and cos(2%(j/k)), where j and k are the Julian date and the number of days in the year, respectively. The inclusion of meteorological input variables can also be important due to their effects on seasonal water quality relationships. Air temperature, dew point temperature, inflow temperature, wind speed, wind direction, and cloudiness all 48 impact the transport of water quality constituents and hence their concentrations in different parts of a water body. Cloudiness, C, is a measure of the number of hours that sunlight does not reach the ground during the day. It is defined as: where hg is the number of hours of sunlight reaching the ground, and hp is the total number of potential sunlight hours between dawn until dusk. As mentioned previously, inflow, rainfall, and rainfall intensity all affect the amount of sediment and turbidity introduced into a reservoir system. These inputs are consistent with the variables determined by Zhang and Stanley (1997) as dominant in the cause and effect relationship in colour forecasting. Turbidity levels are directly affected by the amount of colour present in a sample (Peavy 1985), therefore it is likely that the same variables are significant in the Capilano Reservoir case. 6.2.1.2 Parameter Values from Previous Days It is necessary to determine which input parameter values from previous days should be included in the ANN model. In the Capilano Reservoir case only inflow, rainfall, rainfall duration, and rainfall intensity data are available in a daily format, while meteorological information is available as monthly averages. The main task is to determine the maximum lag (nmax days) for each input time-series beyond which time {tnmax) there is no significant effect on the output time-series (Maier and Dandy 1997). This can be accomplished using two approaches. The first applies knowledge of reservoir response in the physical system, and the second makes use of a computer algorithm that automatically determines dominant network inputs. Determining ANN Inputs Using System Knowledge By examination of the inflow, rainfall, and synthesized turbidity records, it is apparent that turbidity response at the dam outlet is delayed following high flow or rainfall events. This is due to the detention characteristics of the reservoir which delay the arrival of sediments at the dam face once they have been introduced into the 49 system, predominantly by rainfall at the upstream end. Knowledge of underlying physical processes should be used wherever possible in developing ANNs (Maier and Dandy 1997) and examination of the flow and rain characteristics can provide useful insight into the range of previous flow values and rain data that may be important input in the modelling exercise. At a water surface elevation of 125 m (as assumed in the case study), the reservoir holding capacity is approximately 36.8 million m3, compared to about 70 million m3 at the 146 m full supply level. The average and maximum daily inflow are 20 m3/s and 295 m3/s, respectively. (The flow record for the concurrent data periods as described in Section 6.1 are plotted in Figure 6-1 and found to include two higher flows, 363 and 472 m3/s. These are considered extreme events for the purposes of this exercise and are thus treated as outliers). The average and maximum inflows, combined with the available reservoir volume, translate into average and minimum retention times of 21.3 and 1.4 days, respectively. Inspection of the turbidity and flow time-series also indicates that negligible turbidity is generated with inflows below the average daily level of 20 m3/s. In addition, the average and maximum rainfall event durations are 13 and 41 hours, respectively, or 0.5 and 1.7 days. Based on the expected retention times and the storm durations given above, a range of previous days' input values is proposed that should be sufficient for predicting turbidity response at the dam face. The minimum time lag, t.nmin, to consider is at nmin = (1.4 + 0.5) = 2 days (based on the minimum retention time plus the average rainfall duration), while the maximum lag, t.nmax, is likely at nmax = (21.3 + 1.7) = 23 days (based on the average retention plus the maximum rainfall duration). It can also be argued that nmin should be 1.4 days based on minimum retention time and a very short but high intensity rainfall event. This scenario, however, is unlikely since a short duration-high intensity storm event does not generally coincide with extended high inflows that characterize minimum retention. For purposes of further investigation, it is decided to retain a conservative range of input flow, rainfall, duration, and intensity data for the time period from foto t25. 50 3001 4001 5001 6001 Days (from the concurrent data periods) 8001 Figure 6-1 Histor ical Daily Average Inflow R e c o r d for Cap i l ano Reservo i r s h o w i n g two extreme va lues (363 and 472 m 3/s) 51 Determining ANN Inputs Using a Fuzzy-Neural Algorithm Although a t0to grange of previous days' values is determined as potential input to the ANN model, it is important to include only those inputs useful in reproducing the turbidity relationship. If there are too many input parameters with similar values, they may interact with each other and produce noise (Zhang and Stanley 1997). Because of this and since network size directly affects training and processing time, an effort is made to identify unnecessary parameters and to reduce the number of input variables without affecting the network's ability to perform well. Neuframe (Neural Computer Sciences Ltd., 1997), the software program used to develop the ANN turbidity models, provides a fuzzy-neural option that develops ANNs with the use of fuzzy-logic rules. The major advantage of the fuzzy-neural algorithm is that it automatically determines which inputs help characterize a given relationship. It produces a self-constructed network that includes only the useful inputs and excludes those that do not improve the network's generalization ability. Other approaches, such as the use of polynomial networks (Bastarache et al. 1997) and the use of ANN sensitivity analysis (Maier and Dandy 1997), can be also be used to determine dominant network inputs. The procedure followed in pruning the potential inputs into a reduced but adequate input set is two-fold. First, a separate network is trained for each of the four time-series inputs (flow, rainfall, duration, and intensity) using the fuzzy-neural algorithm. Each network contains 26 inputs representing the variable values at time fo through L25, as well as the date and meteorological data associated with the target turbidity. Results from these fuzzy-neural trainings indicate which t.n values are useful in reproducing the turbidity relationship. The second part of the input pruning combines the significant t.n variables from each of the four single-variable networks as input in an overall network which contains the four time-series variables. Using the fuzzy-neural algorithm again, the t.n inputs are further reduced by eliminating redundant information and a final set of input data are produced. Table 6-1 indicates the t.n inputs that are dominant for the Annual and Wet Season scenarios. 52 Tab le 6-1 A N N Dominant Input Resu l t s f rom Fuzzy-Neura l A lgor i thm SCENARIO VARIABLE DETAILS Annual Date Meteorological Data Flow (m3/s), f.„ Rainfall (mm), t„ Duration (hr), t„ 30-min max intensity (mm/hr), f.„ Sine and Cosine functions Air temperature, Wind, Cloud n= 1-8, 12, 15, 17, 22, 25 n = 3-5, 16 None n = 3-5, 15, 16, 20, 25 Wet Season Date Meteorological Data Flow (m3/s), f.„ Rainfall (mm), t„ Duration (hr), t.„ 30-min max intensity (mm/hr), tn Sine and Cosine functions Air temperature, Wind, Cloud n = 1-4, 6, 8, 9, 14, 22, 25 n = 3-4 None r> = 3-5, 7, 13 Note that no rainfall duration data are included. This is likely due to a high correlation between rainfall duration and rainfall intensity data. 6.2.1.3 Neural Network Inputs Used in Model Development For both the Annual and Wet Season scenarios, three input variable sets are used to evaluate the performance of potential ANN turbidity models, by investigating the optimum number and best input parameters to include. While a larger network may provide additional information that improves network performance, it also increases training and processing time. The inclusion of excess parameters increases the likelihood of noise which results in poorer performance. The input sets used in the ANN development are described below and include the fuzzy-neural sets listed in Table 6-1, along with variations of these sets that are derived from the overall time range (to to t.25) for which turbidity is likely to be sensitive. 53 Annual Scenario: Set 1 consists of the significant variables identified by the fuzzy-neural algorithm; Set 2 includes flow values from t0 to t.25 to determine the impact of including all possible flow values, as well as rainfall and intensity values from t0 to t10; Set 3 includes a complete set of flow, rainfall, and intensity values for Mo t.25 to investigate if network performance can be improved with the inclusion of all available information. See Table 6-2 for a summary of the Annual scenario dominant inputs. Wet Season Scenario: Set 1 consists of the significant variables identified by the fuzzy-neural algorithm; Set 2 includes additional flow and rainfall inputs so the flow range is more complete and so the tn values of rainfall and intensity data are consistent with each other; Set 3 further extends the ranges of the t„ rainfall and intensity values so they are similar to the potential inputs derived for the Annual scenario. See Table 6-3 for a summary of the Wet Season dominant inputs. Tab le 6-2 A N N Input Sets - A n n u a l INPUT SET VARIABLE DETAILS Input Set 1 Date Sine and Cosine functions (32 variables) Meteorological Data (°C, km/hr, %) Air temperature, Wind, Cloud Flow (m3/s), t.„ /7 = 1-8, 12, 15, 17, 22, 25 Rainfall (mm) / Intensity (mm/hr), t.„ n = 3-5, 15, 16, 20, 25 Input Set 2 Date Sine and Cosine functions (54 variables) Meteorological Data (°C, km/hr, %) Air temperature, Wind, Cloud Flow (m3/s), tn n = 0-25 Rainfall (mm) / Intensity (mm/hr), t.„ /»= 0-10 Input Set 3 Date Sine and Cosine functions (84 variables) Meteorological Data (°C, km/hr, %) Air temperature, Wind, Cloud Flow (m3/s), t„ n = 0-25 Rainfall (mm) / Intensity (mm/hr), t.n n = 0-25 54 Tab le 6-3 A N N Input Sets - Wet S e a s o n INPUT SET VARIABLE DETAILS Input Set 1 Date Sine and Cosine functions (20 variables) Meteorological Data (°C, km/hr) Air temperature, Wind Flow (m3/s), t.n /7 = 1-4, 6, 8, 9, 14, 22, 25 Rainfall (mm), t.n n = 3-4 Intensity (mm/hr), t.n n = 3-5, 7, 13 Input Set 2 Date Sine and Cosine functions (27 variables) Meteorological Data (°C, km/hr) Air temperature, Wind Flow (m3/s), tn n= 1-9, 14, 22, 25 Rainfall (mm) / Intensity (mm/hr), t„ n = 3-5, 7, 13 Input Set 3 Date Sine and Cosine functions (32 variables) Meteorological Data (°C, km/hr) Air temperature, Wind Flow (m3/s), t.n n= 1-9, 14, 22, 25 Rainfall (mm) / Intensity (mm/hr), t.n n = 2-5, 7, 13, 16 6.2.2 Network Architecture Network architecture can have a large impact on modelling performance, however the developer has many degrees of freedom in defining such configurations (Hegazy et al. 1994). Since architecture development consists of determining the number of layers in the ANN, the number of neurons in each layer, the type of network connections, and the type of transfer function, the potential number of combinations is immense. However, the number of nodes in the input and output layers can often be determined by the dimensionality of the problem (Bebis and Georgiopoulos 1994). In the Capilano Reservoir case, for example, the number of nodes in the first layer must equal the number of input variables; and similarly, the number of nodes in the final layer equals one since the aim is to predict a single output value representing the turbidity level at the mid-level dam outlet. For the Capilano Reservoir case model development begins with a single hidden layer (feed-forward and fully connected), and a subsequent layer is added only if performance is deemed unsatisfactory. In this work, a range of different 55 configurations is considered, and that with the best performance is accepted. In developing the turbidity ANN, three architectural configurations are evaluated for each of the input data sets. Bailey and Thompson (1990) suggest that an initial number of hidden nodes should be 75% of the input layer size, therefore the various architectures in Table 6-4 and Table 6-5 were chosen for evaluation. The configurations indicate the number of nodes in the first (input), second (hidden), and final (output) layers, respectively. Tab le 6-4 Network Arch i tectures - Annua l INPUT SET ARCHITECTURE A ARCHITECTURE B ARCH ITECTURE C Input Set 1 32-35-1 32-25-1 32-15-1 Input Set 2 84-80-1 84-60-1 84-40-1 Input Set 3 * T l i 54-50-1 54-35-1 54-25-1 * The architecture configurations indicate the number of nodes in the first, second, and final layers, respectively. Tab le 6-5 Network Arch i tectures - Wet S e a s o n INPUT SET ARCHITECTURE A ARCHITECTURE B ARCH ITECTURE C Input Set 1 20-20-1 20-15-1 20-10-1 Input Set 2 27-25-1 27-20-1 27-15-1 Input Set 3 * T U - i _ : i j 32-25-1 25-20-1 25-15-1 * The architecture configurations indicate the number of nodes in the first, second, and final layers, respectively. The nodes are fully connected in the forward direction, thus creating a feed-forward network. The hyperbolic tangent function is chosen for the Capilano Reservoir turbidity application as this type of processor has been found to perform well with other time-series models, and it is generally associated with quick training since learning speed is proportional to its derivative. 56 6.2.3 Network Training Training is the process by which network weight connections are adjusted until a minimum is reached in the error between predicted and desired output. It is often viewed as synonymous with the term calibration which is commonly associated with statistical and mathematical models. In training, the network is presented with a set of patterns, each comprising an example of the problem to be solved (inputs) and its corresponding solution (target output). The resultant outputs are compared with the target outputs, and the connection weights are then adjusted by small amounts so that the error is reduced (Flood and Kartam 1994b). The ANN turbidity models for the Capilano Reservoir are trained using the Neuframe default momentum coefficient of 0.8 as well as the on-line data processing option. The momentum term increases the weight changes that are made along shallow portions of the error surface, while on-line data processing, as opposed to batch processing, adds a degree of randomness that can help the network escape local minima in the error surface. Four learning rates (0.05, 0.1, 0.15, and 0.2) are used with each architecture, and three training runs are performed for each combination with a set of randomly selected initial connection weights. The aim of randomly selecting the initial connection weights is to potentially start the training process at a more favourable location on the error surface from which an improved solution can be found. The development of the turbidity ANNs is extensive as it involves 12 training runs (4 learning rates x 3 weight randomizations) for each of the architectures shown in Table 6-4 and Table 6-5. This results in a total of 108 training runs for each of the scenarios. While the time involved in completing the training is extensive, it allows for a complete evaluation of possible network performance. As described by Sarle (1994), multilayer ANNs are general-purpose, flexible, non-linear models that, given enough hidden neurons and enough data, can approximate virtually any function to any degree of accuracy. This, however, can lead to 57 overtraining in many applications. If an ANN overtrains, it loses its ability to generalize and perform well on data that were not used during the training process. It is necessary to employ a stopping criterion upon which network training is terminated, and at which time the final connection weight vector is said to represent the network's knowledge of the problem. Commonly applied stopping criteria include using a set number of iterations or training cycles until an acceptable or minimum level of error is reached. The use of test data sets and the cross-validation method is also popular (Amari et al. 1997; Flood and Kartam 1994a); however, Maier and Dandy (1998c) note that if cross-validation is applied, three separate data sets should be used: a training set, a test set, and a validation set. Since the test set is used in the training process (to determine the end of training), a third independent set must be available for validation purposes. Cross-validation is used as the stopping criteria in the turbidity model development for the Capilano Reservoir. The synthesized turbidity record for the Annual and Wet Season periods, along with their associated input parameter values (see Table 6-2 and Table 6-3), are divided into data sets for training, testing, and validation as shown in Table 6-6 and Table 6-7. It is important to ensure that all three sets have similar statistical characteristics and are therefore representative of the same problem domain. If not, training performance and generalization ability can be affected. In addition, training patterns should reach the edges of the range of expected values in all dimensions (Flood and Kartam 1994a) since ANNs cannot usually extrapolate beyond the data with which they have been trained. Tab le 6-6 Data Set Character i s t i c s for Turbidity (NTU) - A n n u a l Number of Samples Mean Std Dev Min Value Max Value Training Set 4110 9.93 3.55 4.01 32.20 Test Set 1350 10.03 3.56 5.16 31.33 Validation Set 1220 9.65 3.45 5.40 26.39 58 Tab le 6-7 Data Set Character i s t i c s for Turbidity (NTU) - Wet S e a s o n Number of Samples Mean Std Dev Min Value Max Value Training Set 2075 11.54 3.46 4.34 31.33 Test Set 700 11.17 3.35 5.01 26.39 Validation Set 720 10.60 3.25 5.41 20.68 6.2.4 Turbidity Forecasting ANNs for water resource parameter forecasting are developed in a similar manner to ANNs that are used for parameter estimation or simulation. The difference is that in forecasting, available data are used to predict water parameter values at some future point in time, t+n, whereas in simulation, available data are used to estimate the parameter's value on the current day, to. Training for the ANN forecasting problem can be accomplished using the same data that are available for the simulation exercise. Network training sets for forecasting are arranged so that the same input parameters are now associated with a future target value, i.e., a known value at a forecast length of +n days. In the forecasting exercises, the ability of ANNs to forecast turbidity at the Capilano Reservoir outlet is evaluated. Six forecasting networks are trained, using data from the Annual scenario, for forecast lengths ranging from +1 to +6 days. Short-term forecasts such as these are useful for decision-making related to reservoir water quality, treatment operations, and potential restrictions that could be placed on water intake for drinking supply or fisheries purposes. The forecast lengths are also on the order of days due to the nature of the ANN input data as daily time-series. In general, forecasting ability decreases as the forecast length increases due to the lack of information from recent periods. The Capilano Reservoir system is dynamic in that it has a large flushing capacity that can dilute and move turbid waters through the system relatively quickly. As a result, long-term forecasts in terms of weeks and 59 months may be successful in replicating seasonal variations in turbidity, but would not necessarily perform well in predicting turbidity peaks that result from specific inflow or rainfall events. The turbidity forecasting networks use the same architecture configuration as the standard simulation network that produces the most favourable results. The momentum coefficient, as in the simulation case, is held fconstant at the default value of 0.8, and the learning rate is varied from 0.1 to 0.2. Various random selections of initial connection weights are allowed during training, and cross-validation is also used as the stopping criterion. 6.3 Per fo rmance Indicator Es t imates The reliability and resilience of the Capilano Reservoir are estimated by linking the ANN turbidity models to a general-purpose reliability software program. Reservoir performance is estimated for the scenarios of interest using the FORM approach developed by Maier et al. (1999), and comparisons are made with MCS. This section describes the software used, the ANN link that is incorporated, relevant inputs, and the method by which reliability and resilience are estimated. 6.3.1 RELiability ANalysis (RELAN) Software RELAN (Foschi and Folz 1990) is a reliability software program that can be used to estimate the probability of failure (and hence reliability) of a given system. RELAN generates a set of random variables which characterize the problem, and then an associated performance function (see Equation 5) is evaluated to determine system performance. RELAN incorporates a number of techniques by which the performance can evaluated; these include FORM (see Chapter 4), Second-Order Reliability Method (SORM), MCS, and each of these three approaches in conjunction with Importance Sampling (IS), and Adaptive Sampling (AS). 60 The Annual and Wet Season turbidity models are linked to RELAN as subroutines in the reliability program (see Appendix I for the subroutine computer code). Using the connection weights obtained in the training process, the ANN subroutine calculates an expected turbidity level based on the set of random variables generated by RELAN. The predicted ANN turbidity, TNN, is compared to a user-defined critical level or standard, Ts, to determine whether the system resides in a satisfactory or unsatisfactory state. The failure function for the Capilano Reservoir turbidity case is defined as: G = TS- TNN (26) 6.3.2 RELAN Inputs - Random Variable Characteristics and Correlations As discussed in Chapter 5, the scenarios of interest with respect to turbidity levels during reservoir drawdown are the Annual and Wet Season scenarios, as well as monthly during the Wet Season. Random variable distributions are determined using frequency analysis and by comparison with standard distributions. The distribution parameters required by RELAN to evaluate system performance are shown in Table 6-8 and the parameter values for each random variable and scenario are presented in Table 6-9 and Table 6-10. Note that model error is included as a random variable in the RELAN evaluations. This model error represents the error associated with the ANN turbidity model (as determined during network validation), and is added to calculated turbidity values before evaluation of the failure function. It is based on linear regression analysis and is assumed to be normally distributed with a mean of zero and standard deviation equal to the root-mean-square-error (RMSE) between the ANN predictions and the synthetic turbidity data upon which the network is trained (Hines and Montgomery 1990). 61 Tab le 6-8 Distr ibution Descr ip tors for R E L A N Distribution Type Descriptor 1 Descriptor 2 Uniform Minimum value Maximum value Normal Mean Standard Deviation Log-Normal Mean Standard Deviation Gamma A = Mean / B = (Mean / (Standard Deviation) 2 Standard Deviation.)2 Tab le 6-9 R a n d o m Var iab le Character i s t i c s - A n n u a l and Wet S e a s o n ANNUAL WET SEASON Random Variable D i s t r i b u t i ° n Descriptor Descriptor Descriptor Descriptor Type 1 2 1 2 Sine Date Factor Uniform -1 1 -1 1 Cosine Date Factor Uniform -1 1 -0.5 1 Tair (deg C) Uniform 1 19 1 10.5 Wind (m/s) Log-normal 1.56 3.6 1 2.5 Cloud Factor Uniform 3 8 5.75 8 Flow (m"Vs) Log-normal 20.42 25.52 23.78 30.51 Rain (mm) Gamma 0.0371 0.1908 0.0381 0.2815 Intensity (mm/hr) Gamma 0.1744 0.2739 0.2204 0.4359 Error Term Normal 0 1.56 0 1.52 Tab le 6-10 R a n d o m Var iab le Character i s t i c s - Wet S e a s o n Months OCTOBER NOVEMBER Random Variable D i s t r i b u t i o n Descriptor Descriptor Descriptor Descriptor Type 1 2 1 2 Sine Date Factor Uniform -1 -0.85 -0.88 -0.48 Cosine Date Factor Uniform 0 0.63 0.57 0.93 Tair (deg C) Uniform 4.6 14.3 2.4 9.6 Wind (m/s) Uniform 1.28 1.59 1.17 1.83 Cloud Factor Uniform 4.8 7.6 6.0 7.0 Flow (mJ/s) Log-normal 23.2 32.18 28.75 36.57 Rain (mm) Gamma 0.03024 0.23994 0.03971 0.33993 Intensity (mm/hr) Gamma 0.17297 0.37597 0.20307 0.47981 Error Term Normal 0 1.52 0 1.52 DECEMBER JANUARY Random Variable D i s t r i b u t i o n Descriptor Descriptor Descriptor Descriptor Type 1 2 1 2 Sine Date Factor Normal* -0.23 0.16 0.3 0.15 Cosine Date Factor Normal* 0.97 0.03 0.92 0.06 Tair (deg C) Uniform 2.3 4.6 2.4 3.9 Wind (m/s) Uniform 1.59 1.66 1.63 1.83 Cloud Factor Uniform 7.0 7.6 7.0 7.6 Flow (nrVs) Log-normal 27.51 38.57 24.5 32.18 Rain (mm) Gamma 0.03943 0.41498 0.03412 0.30352 Intensity (mm/hr) Gamma 0.24233 0.58171 0.24953 0.45654 Error Term Normal 0 1.52 0 1.52 FEBRUARY MARCH APRIL Random Variable D i s t r i b u t i o n Descriptor Descriptor Descriptor Descriptor Descriptor Descriptor Type 1 2 1 2 1 2 Sine Date Factor Uniform 0.55 -0.48 0.88 1 0.875 1 Cosine Date Factor Uniform 0.44 0.93 -0.05 0.47 -0.58 -0.05 Tair (deg C) Uniform 2.3 6.2 3.9 8.5 6.2 11 Wind (m/s) Uniform 1.66 2.04 1.63 2.4 1.51 2.04 Cloud Factor Uniform 6.8 7 6 7.6 4 6.8 Flow (mJ/s) Log-normal 20.27 22.65 20.94 23.56 21.35 18.79 Rain (mm) Gamma 0.04304 0.29132 0.04913 0.26028 0.06017 0.25709 Intensity (mm/hr) Gamma 0.26813 0.49808 0.26102 0.38625 0.24229 0.34245 Error Term Normal 0 1.52 0 1.63 0 1.52 * maximum +1 Random variable generation is not only based on the characteristics shown in Table 6-9 and Table 6-10, but also on the correlation structure between the input variables. In the Capilano Reservoir application, numerous variables must be generated as inputs for the turbidity model since a high number of inputs are involved. As a result, 63 many correlations must be provided in order to represent the existing inter-relationships. For example, if there are 46 random variable inputs, all of which are related to one another, 1035 correlation coefficients are needed. Although the size and shape of the correlation matrix is problem dependent, a computational requirement of RELAN is that it be positive-definite. 6.3.3 Risk-Based Performance Indicator Calculations using RELAN and FORM As discussed in Chapter 4, reliability, a, is the probability of being in a successful state (see Equation 1). Resilience can be defined as the probability of recovering from failure in a single time-step, i.e., the probability of success in Day 2 given failure in Day 1 (see Equation 4). For the Capilano Reservoir case study, success is defined as the situation where turbidity levels at the mid-level dam outlet are below a critical level and thus within an acceptable range, while failure is defined as a violation of the water quality guideline resulting from turbidity levels which exceed the standard. Both reliability and resilience estimates can be obtained using RELAN which provides estimates of the probability of failure in Day 1, pfl, the probability of success in Day 2, ps2, and the total system probability of failure in Day 1 and success in Day 2, pn2, as defined by Equation 27. A12 = Pr{X, e F grX2 e S} (27) Given Equation 27, and by applying Equation 21, resilience can be estimated as = <b(-pv -02;p,2) _ Pf1 +Ps2 -p t 1 2 7 " W ~ Pn ( 2 8 ) 64 As mentioned previously, the probabilities of failure associated with reliability and resilience can be estimated using a number of different methods. RELAN capabilities include FORM and SORM, and MCS which can provide exact calculations if sufficient samples are made. With IS, the FORM solution is used to focus on a sub-region of the error space which is then sampled by a reduced MCS for a given number of simulations. This approach improves estimates of failure probabilities where the error surface is highly non-linear and FORM does not work as well. In the Capilano Reservoir case study, the system reliability and resilience are evaluated by FORM and IS-FORM for the Annual and Wet Season scenarios. Comparisons are also made with MCS results. FORM is applied using a beta (B) error tolerance of 0.001, so that the calculated B values for two consecutive iterations must be within 0.001 of each other. If this tolerance does not allow for convergence of the solution, a B tolerance of 0.01 is used. For these cases, IS requires 5,000 simulations, and for those scenarios also evaluated using MCS, a total of 10,000 simulations are performed. Considerations with MCS Resilience estimates using MCS involve a large computational burden when the system is evaluated for high reliability levels. This is due to the manner in which resilience is calculated by RELAN. In MCS mode, RELAN calculates resilience by counting the number of recoveries there are following a water quality violation, and only evaluates Day 2 turbidity if there is a violation in Day 1. As a result, determining resilience at high reliabilities requires a large number of simulations to produce sufficient failures in Day 1 which are a prerequisite of the resilience evaluation. For example, if a system has a high reliability of 98% (2% unreliable), approximately 500,000 simulations are required in order to generate 10,000 failures in Day 1 from which resilience calculations can be completed. 65 The reliability and resilience measures for the Capilano Reservoir are evaluated for turbidity standards ranging from 4 NTU to 20 NTU, and for the different scenarios listed in Section 5.4. The set of random variables generated by RELAN for each iteration includes inputs for both the Day 1 and Day 2 turbidity calculations since they are dependent on each other. Using these inputs, the turbidity model predicts turbidity levels for two-day sequences. Each turbidity prediction is then compared to a given turbidity standard to determine failure or success. These combinations of failure and/or successes then determine the estimate of system resilience. Because the ANN models use input values at tn days, some of the inputs required for Day 2 turbidity predictions are in fact the same inputs used in Day 1 evaluations. This is the case, for example, where the flow two days ago relative to Day 2 is the same flow as the flow for one day ago relative to Day 1. In these situations, the redundant variables can be shared by Day 1 and Day 2 calculations, thus reducing the total number of variables that need to be generated by RELAN. By reducing the number of random variables, the dimensionality of the correlation matrix is also reduced and the matrix is more likely to be positive-definite. 66 7 RESULTS AND DISCUSSION This chapter presents results of the turbidity modelling and estimation of reservoir performance for the Capilano Reservoir case study. The application of ANNs for both turbidity simulation and short-term forecasting is discussed and validation results are provided. The performance indicators of reliability and resilience are then estimated by linking the ANN models to RELAN. The scenarios of interest (see Section 5.4) are investigated for a range of turbidity standards. Note that the results presented in this section are specific to the Capilano Reservoir case study in which the reservoir water surface is drawndown to 125 m and maintained at this level indefinitely. 7.1 Artif icial Neura l Network Turbidity S imulat ion and Forecas t ing 7.1.1 Turbidity Simulation For the turbidity simulation exercise, three different architectures are evaluated for each of the potential input sets (see Table 6-4 and Table 6-5). While improved training is usually achieved with learning rates of 0.15 or 0.20 (as opposed to 0.10 and 0.05 that are also tested), overall performance is generally insensitive to the size of the input set and the type of architecture chosen. The input size and architecture type do, however, affect the amount of time taken to complete training. For each training run the connection weights are initialized with random values so that an improved solution may be achieved by starting the training process at a more favourable position on the error surface. Because of the large number of training runs performed (108 runs for each scenario), only results from the selected Annual and Wet Season turbidity models are presented. Results from network training, testing, and validation for the Annual and Wet Season models are provided in Table 7-1 and Table 7-2, respectively. The selected Annual turbidity model is trained with a learning rate of 0.20 and uses the 67 32-25-1 configuration (32 input, 25 hidden nodes, 1 output), while the best performing Wet Season ANN has a 32-15-1 structure and is trained with a learning rate of 0.15. The input parameters for both scenarios are shown in Table 7-3. The turbidity models are chosen based on their performance with the test data set. The test turbidities are compared to the target values derived from the synthesized record, and the networks associated with minimum test error are selected. Following network selection, the turbidity models are validated using the validation set of data. Tab le 7-1 A N N Per fo rmance Resu l t s - A n n u a l ERROR TRAINING TESTING VALIDATION 0.849 0.784 0.799 RMSE 1.40 1.70 1.56 AAPE (%) 10.5 11.0 11.2 AAE (NTU) 0.98 1.10 1.12 Tab le 7-2 A N N Per fo rmance Resu l t s - Wet S e a s o n ERROR TRAINING TESTING VALIDATION R' 0.814 0.792 0.783 RMSE 1.50 1.51 1.52 AAPE(%) 9.7 10.7 9.8 AAE (NTU) 1.11 1.14 1.09 68 Tab le 7-3 Se lec ted Turbidity Mode l s - Annua l and Wet S e a s o n SCENARIO INPUT SET ARCHI-TECTURE LEARNING RATE VARIABLE DETAILS / LAGS Annual Input Set 1 32-25-1 (32 Inputs) 0.20 Wet Season Input Set 3 32-15-1 (32 Inputs) 0.15 Date Meteorological Data Flow Rain/Intensity Date Meteorological Data Flow Rain/Intensity Sine and Cosine Air temperature, Wind, Cloud 1-8, 12, 15, 17, 22, 25 3-5, 15, 16, 20, 25 Sine and Cosine Air temperature, Wind 1- 9, 14, 22, 25 2- 5, 7, 13, 16 The error measures used to evaluate network performance (R2, RMSE, AAPE, AAE) often indicate ambiguous results. For instance, the R2 value may increase (indicating improvement), but so may the RMSE, AAE, and AAPE measures which indicates poorer performance. The opposite is also observed where improved RMSE, AAPE, and AAE are achieved but the R2 measure drops below acceptable levels. In general, unless a large contradiction arises between the four measures, the R2 value and the AAE are used to determine best overall network performance. The R2 parameter is widely applied in the comparison of data sets, and the AAE measure is easily translated into meaningful terms as it indicates the expected error margin in units equivalent to those used in the modelling exercise. The R2 value for the selected Annual scenario turbidity network is 0.799, and the AAE is 1.12. This implies that any turbidity prediction is, on average, within 1.12 NTU of its true value. For the Wet Season neural network, the R2 value is slightly less, however turbidity predictions are marginally more accurate, and can be made within 1.09 NTU. These error levels are well within acceptable ranges as performance of the ANNs to within 1.12 and 1.09 NTU compares favourably with the 69 errors associated with previous modelling work and the field sampling. For the 1999 drawdown of the Capilano Reservoir, the differences between continuous turbidity monitoring and grab samples, and between samples measured in the field and measured in the laboratory, are on the order of 4 and 9 NTU, respectively (Klohn-Crippen Consultants 1999). Figure 7-1 through Figure 7-4 present the Annual and Wet Season validation results in time-series and scatter-plot formats. The time-series figures show how the ANN simulation results compare with the synthetic turbidity targets of the validation data sets. Recall that the ANN turbidity models are developed using synthesized turbidity data, generated by CE-QUAL-W2, and are representative of a major drawdown scenario for the Capilano Reservoir (see Section 6.1). To ensure successful ANN performance, different periods of information are joined together so that the data sets used for training, testing, and validation have similar statistical characteristics (see Section 3.2.4). As a result, Figure 7-1 and Figure 7-3 are not true time-series plots as the ANN simulation results and synthetic turbidity data are not plotted against actual dates, but against periods of daily sequences. Note also that the sharp breaks in the Wet Season time-series plot (Figure 7-3) are due to discontinuities within the validation data set since it comprises a number of October to April time periods. The scatter-plot figures (Figure 7-2 and Figure 7-4) present the same validation results and show how ANN estimates compare with the synthetic targets for various turbidity levels. Results from both the Annual and Wet Season scenarios indicate good performance in terms of simulating the seasonal turbidity variations, as well as most turbidities in the 5-15 NTU range. Above 15 NTU, however, the ANN models tend to under-estimate turbidity levels. This is likely due to the limited number of high turbidities, compared to average levels, that are contained in the training data. A bias in the training data toward average values does not present enough high turbidity samples to the network during the training process for it to fully learn the more extreme conditions. 70 71 r I 1 5 10 15 20 25 30 Synthetic Target Turbidity (NTU) Figure 7-2 A N N Val idat ion for the A n n u a l S cena r i o (2) S imulated and Synthet ic Turbid i t ies 72 73 10 15 20 Synthetic Target Turbidity (NTU) 25 30 Figure 7-4 A N N Val idat ion for the Wet S e a s o n (2) S imulated and Synthet ic Turbid i t ies 74 7.1.2 Turbidity Forecasts ANN performance is evaluated for short-term turbidity forecasting. Forecast lengths of +1 to +6 days are evaluated for the Annual scenario, with network architectures set at the 32-25-1 configuration that proved successful in the simulation application (see Table 7-3). The same input variables associated with this configuration are also used in the forecasting exercise. Learning rates between 0.1 and 0.2 are examined, the momentum coefficient is held constant at 0.8, initial connection weights are randomized for each training run, and cross-validation is used as the stopping criterion. The Annual and Wet Season models developed for turbidity simulation are in fact +1day forecasting tools that can be used to forecast turbidity one day in the future, i.e., at time at f+i. This is because the simulation networks use a minimum input parameter lag value of U to simulate the turbidity at time to. (The minimum time lag involved is U for the flow variable). The simulation networks essentially predict turbidity levels one day ahead, equivalent to using information from t0 to forecast turbidity at time f+i. Although the ANN inputs include various meteorological parameters, these values are monthly averages and therefore do not change from day to day unless the forecast is for the first day of the following month. Values of the cosine and sine date functions also change insignificantly from one day to another. As in the turbidity simulation cases, the error measures used to select the best forecasting ANN are often ambiguous. Improved test performance is possible in terms of R2 values, however this is achieved at the expense of poorer measures of RMSE, AAPE, and AAE. A balance between these measures is used to select trained models for predicting turbidity levels at the desired forecast lengths. Validation results from the turbidity forecasting are shown in Table 7-4. 75 Tab le 7-4 A N N Per fo rmance - Turbidity Forecas t ing +1 Day +2 Day +3 Day +4 Day +5 Day +6 Day Forecast Forecast Forecast Forecast Forecast Forecast R 0.799 0.773 0.734 0.709 0.718 0.711 RMSE 1.56 1.63 1.77 1.84 1.82 1.84 AAPE (%) 11.2 12.0 12.9 13.0 12.8 13.2 AAE (NTU) 1.12 1.20 1.27 1.27 1.27 1.28 Figure 7-5 also illustrates the performance of the ANN models for the different forecast lengths. As expected, there is deterioration in model performance as forecast length increases. However, depending on the desired accuracy, this deterioration could be considered acceptable and predictions may remain within reasonable bounds. For example, although the R2 value drops from 0.80 for +1 day forecasts to 0.71 at a forecast length of +6 days, the average error (AAE) for predictions +6 days is 1.3 NTU, an increase of only 0.2 NTU from the best performance observed. If RMSE is considered, however, the increased error between the +1 day forecasts and +6 day forecasts may not be considered acceptable. Again, this depends on the acceptable level of error for a given situation or application. The relatively small decrease in network performance with increased forecast length indicates that the exclusion of recent information, in the form of time-series input from recent days, does not have a significant impact on a network's forecasting ability, i.e., the ANN can provide acceptable short-term turbidity forecasts without using recent flow, rainfall, or rainfall intensity data as inputs. This also suggests that the date factors and meteorological variables are the more dominant model inputs. This is somewhat counter-intuitive since high turbidities during drawdown are generally associated with flow, rainfall, and intensity factors. However, since seasonal turbidity is correlated with date and meteorological variables, these data alone can be used to successfully forecast and predict turbidity levels. 76 2.0 1.8 1.6 1.4 HI 0.8 0.6 0.4 0.2 0.0 - R M S E (NTU) -AAE(NTU) -Rsquare 2 3 4 5 Forecast Length (Days) Figure 7-5 A N N Turbidity Fo reca s t Pe r fo rmance For O n e to S ix Day Fo reca s t s 77 7.2 Eva luat ion of Reservo i r Per fo rmance Indicators Reservoir reliability and resilience for the Annual and Wet Seasons are evaluated using the ANN models trained for these respective periods. The Wet Season turbidity model is also used for evaluation of the Monthly Wet Season scenarios of interest. The following section describes the random variables that are generated in order to evaluate the performance, and results are presented and then discussed for each of the scenarios. 7.2.1 Random Variable Generation and Assignment Resilience estimates involve evaluating turbidity levels for two consecutive days in order to determine the probability of system recovery following a failure. As mentioned in Section 6.3, some of the inputs used for Day 2 turbidity predictions are in fact the same inputs used in Day 1 evaluations. In these situations, some variables are used in both the Day 1 and Day 2 calculations and the total number of variables that need to be generated by RELAN is reduced. Note that meteorological inputs are monthly average values and as such do not change from Day 1 to Day 2 except between months. The change in the date factors is also negligible. Although the Annual turbidity model selected requires an input set of 32 values (see Table 7-3), 18 of these parameters can be shared between Day1 and Day2, so that only 46 random variables ((32*2)-18) have to be generated for the two-day resilience calculation. Similarly, for the Wet Season network, 19 parameters are shared and a total of 45 random variables must be generated as ANN input. In both cases, two additional variables are also generated to take into account the model error associated with the ANN turbidity predictions (see Section 6.3). Random variables for the Annual and Wet Season turbidity networks are shown in Table 7-5 and Table 7-6, respectively. All random variables are generated based on their distribution types, the associated distribution parameters, and a complete set of correlation coefficients. For the Annual scenario this requires 1035 correlation coefficients, 78 while only 990 are required for the Wet Season since there are more shared random variables (see Appendix II for the correlation matrices used). Tab le 7-5 R a n d o m Var iab les for R E L A N - A n n u a l DAY 1 DAY 2 DAYS 1/ 2 Turbidity Time relative Random Time relative Random Shared Inputs to Day 1,f„ Variable to Day 1,f„ Variable Random n = Number n = Number Variables Cosine Date N/A 1 N/A 1 X Sine Date N/A 2 N/A 2 X Air Temp N/A 3 N/A 3 X Wind N/A 4 N/A 4 X Cloud N/A 5 N/A 5 X Flow -1 6 0 33 Flow -2 7 -1 6 X Flow -3 8 -2 7 x Flow -4 g -3 8 X Flow -5 10 -4 9 X Flow -6 11 -5 10 X Flow -7 12 -6 11 x Flow -8 13 -7 12 X Flow -12 14 -11 34 Flow -15 15 -14 35 Flow -17 16 -16 36 Flow -22 17 -21 37 Flow -25 18 -24 38 Rainfall -3 19 -2 39 Rainfall -4 20 -3 19 X Rainfall -5 21 -4 20 X Rainfall -15 22 -14 40 Rainfall -16 23 -15 22 x Rainfall -20 24 -19 41 Rainfall -25 25 -24 42 Intensity -3 26 -2 43 Intensity -4 27 -3 26 X Intensity -5 28 -4 27 X Intensity -15 29 -14 44 Intensity -16 30 -15 29 X Intensity -20 31 -19 45 Intensity -25 32 -24 46 error N/A 47 N/A 48 79 Tab le 7-6 R a n d o m Var iab les for R E L A N - Wet S e a s o n D A Y 1 D A Y 2 D A Y S 1/ 2 Turbidity Time relative Random Time relative Random Shared Inputs to Day 1,^  Variable to Day 1,4, Variable Random n = Number n = Number Variables Sine Date N/A 1 N/A 1 X Cosine Date N/A 2 N/A 2 X Air Temp N/A 3 N/A 3 X Wind N/A 4 N/A 4 X Cloud N/A 5 N/A 5 X Rain -2 6 -1 33 Intensity -2 7 -1 34 Rain -3 8 -2 6 X Intensity -3 9 -2 7 X Rain -4 10 -3 8 X Intensity -4 11 -3 9 X Rain -5 12 -4 10 X Intensity -5 13 -4 11 X Rain -7 14 -6 35 Intensity -7 15 -6 36 Rain -13 16 -12 37 Intensity -13 17 -12 38 Rain -16 18 -15 39 Intensity 16 19 -15 40 Flow -1 20 0 41 Flow -2 21 -1 20 X Flow -3 22 -2 21 X Flow -4 23 -3 22 X Flow -5 24 -4 23 X Flow -6 25 -5 24 X Flow -7 26 -6 25 X Flow -8 27 -7 26 X Flow -9 28 -8 27 X Flow -14 29 -13 42 Flow -17 30 -16 43 Flow -22 31 -21 44 Flow -25 32 -24 45 error N/A 46 N/A 47 7.2.2 Annual Reliability and Resilience The reliability of the Capilano Reservoir in the Annual scenario is shown in Figure 7-6 for a range of turbidity standards. For this case, the FORM estimates do not compare well with the MCS estimates. This indicates that the turbidity problem is non-linear in nature, and that its failure surface cannot be adequately approximated by a linear function, as is assumed with FORM. In this case, the reliability estimated by FORM is approximately 10-20% less than the MCS predictions for the mid-range of standards evaluated. Such a scenario is illustrated in Figure 7-7 where the true area representing probability of success (reliability) is greater than the approximated area given when the failure surface is assumed to be linear. Although FORM does not perform well in comparison to MCS, the FORM estimates are significantly improved with the use of IS. As shown in Figure 7-6, the IS-FORM estimates are very similar to those generated using MCS, indicating that reliability can be estimated satisfactorily by this approach. Since IS-FORM samples the failure region as opposed to the entire failure space as in MCS, fewer simulations are required to achieve accurate results and significant time savings may be achieved. A slight drawback is that at very high and low reliabilities, FORM does not always converge to a solution from which IS can be initiated. For the Capilano Reservoir case, however, the IS-FORM curve suggests accurate reliability estimates can be obtained for the 0.05 to 0.95 reliability range. Depending on the desired information, extrapolation beyond these values may also be considered acceptable. Similar comparisons may be drawn from the MCS, FORM, and IS-FORM resilience results as shown in Figure 7-8. All three resilience-turbidity standard curves indicate a similar trend; however, in this case, the IS-FORM estimates do not compare well with the MCS estimates. In addition, resilience calculations do not converge at turbidity levels over 16 NTU. With FORM, if Day 1 turbidity evaluations do not converge (as is the case at high reliabilities), the Day 2 evaluations are not carried out and resilience cannot be determined. 81 0.0 -I , , , , . , , , 1 1 0 2 4 6 8 10 12 14 16 18 20 Turbidity Standard (NTU) Figure 7-6 Reliability for the Annua l S cena r i o us ing MCS , F O R M , and I S -FORM 82 Figure 7-7 Fai lure Sur face and Under -Es t imat ion of Reliability by the F O R M Approx imat ion 83 1.0 0.9 Figure 7-8 Res i l ience for the Annua l S cena r i o us ing MCS , F O R M , and IS -FORM 84 IS-FORM estimates of resilience are not as accurate as the reliability estimates because of the method by which they are calculated. Resilience is a conditional probability defined as the probability of recovery given failure in the previous time step. In the calculation of resilience, small errors associated with original estimates of failure or recovery probabilities are compounded and may have a significant impact on resilience estimates. This is demonstrated in Table 7-7 and Table 7-8 for resilience calculations at turbidity standards of 8 and 14 NTU, respectively. As shown in Table 7-7, a 1% error in probability estimates can result in resilience estimates ranging from 0.088 to 0.159, or ±4% from the actual calculated IS-FORM value of 0.120. The resilience value of 0.159, calculated assuming a +1.0% error, is equivalent to the Monte Carlo estimate. At higher turbidity standards, the effect of inaccuracies in the reliability estimates is more significant as the failure probabilities are much smaller. For example, at the 14 NTU standard (see Table 7-8), small errors in reliability estimates of 0.5% can lead to resilience estimates ranging from 0.287 to 0.549. The 0.549 resilience value is 12% higher than the actual estimate of 0.425 derived by IS-FORM, and 10% higher that the MCS estimate of 0.492. Tab le 7-7 Ef fect o f Probabi l i ty A c c u r a c y on Res i l i ence Es t imates - 8 N T U Probabilities Estimated by RELAN Calculated from RELAN Estimates Results P(Fi) [1] P(S2) [2] P(Fi inter S2) [3] P(Fi union S2) [4] = [1]+[2]-[3] Reliability = 1-[1] Resilience = [4]/[1] Actual .7304 .2655 .9085 .0874 .2696 .120 + 1.0% .7404 .2755 .8985 .1174 .2596 .159 -1.0% .7204 .2555 .9124 .0635 .2796 .088 85 Tab le 7-8 Ef fect of Probabi l i ty A c c u r a c y on Res i l ience Es t imates - 14 N T U Probabilities Estimated by RELAN Calculated from RELAN Estimates Results P(Fi) [1] P(S 2) [2] P(F! inter S 2 ) [3] P(FT union S 2) [4] = [1]+[2]-[3] Reliability = 1-[1] Resilience = [4]/[1] Actual .0987 .8940 .9508 .0419 .9013 .424 + 0.5% .1037 .8990 .9458 .0569 .8963 .549 - 0.5% .0937 .8890 .9558 .0269 .9063 .287 7.2.3 Wet Season Reliability and Resilience As with the Annual scenario, reliabilities obtained using IS-FORM for the Wet Season scenario compare well with solutions obtained by MCS. Figure 7-9 illustrates this and also compares these results with those from the Annual scenario. As expected, due to increased rains and inflow, the Capilano Reservoir is less reliable over the winter season than over the entire year. However, the difference is relatively small, averaging approximately 7% within the middle range of turbidity standards and significantly less toward the extremes. Again, FORM is unable to converge within the acceptable error tolerance at high reliabilities. In the Wet Season scenario this means that.system reliability could not be determined by IS-FORM for turbidity standards above 14 NTU. System resilience for the Wet Season scenario is shown in Figure 7-10 where both MCS and IS-FORM results are plotted. Annual scenario results are shown for comparison. As in the Annual scenario, the IS-FORM estimates for the Wet Season are less than the MCS estimates by about 8% in the mid-turbidity standard range. Overall IS-FORM results also indicate, as with reliability, that the Capilano Reservoir system is less resilient in the Wet Season than on average over the entire year. Again, this is to be expected due to the higher amounts of sediment-generating rainfall, and greater rainfall intensity and flows that are associated with this period. 86 Turbidity Standard (NTU) Figure 7-9 Reliability for the Annua l and Wet S e a s o n Scena r i o s us ing M C S and IS -FORM 87 0.9 0.8 4 6 8 10 12 14 16 18 20 Turbidity Standard (NTU) Figure 7-10 Res i l ience for the Annua l and Wet S e a s o n S cena r i o s us ing M C S and I S -FORM 88 Interestingly, Figure 7-10 also shows that IS-FORM resilience estimates for the 12 and 14 NTU standards are essentially equal for both the Annual and Wet Season scenarios. Similar results are observed for the MCS estimates at 12 and 16 NTU. Combining the reliability and resilience results provides insight as to the reservoir's performance at different turbidity standards in the two scenarios. It appears that although the reservoir is less reliable in the Wet Season, its resilience at these standards is comparable to that of the Annual scenario. In effect, the system may be more likely to fail in the winter than at other times, but, should it fail, it is just as likely to recover by the following day. The high flows associated with the Wet Season may help dilute the turbid waters that cause the water quality violation, and to move them quickly through the system before the next time period. 7.2.4 Reliability and Resilience for the Wet Season Months Figure 7-11 indicates how system reliability varies from month to month during the Wet Season. The shape of each tradeoff curve is similar, however the late fall months of October, November, and December are particularly poor in terms of reliability. In these months, minimum turbidity levels for the drawdown scenario are about 8 NTU compared with the more acceptable levels of 4-6 NTU during the later months from January to April. For the 10 NTU turbidity standard above which an alternate drinking-water supply must be provided, monthly reliability improves from approximately 0.15 in December to 0.40 in January. This considerable increase of 25% can be useful for making operational and managerial decisions that relate to reservoir drawdowns and the water quality impacts that may result from low water levels. Resilience values for the Wet Season Months are presented in Figure 7-12. Although slightly less obvious than in the reliability case, resilience for the October to December period is also poorer than the subsequent January to April months. The 89 resilience values are incomplete at higher reliabilities as the FORM calculations are not able to converge at these levels. Figure 7-11 Reliability for the Wet S e a s o n Months us ing I S -FORM 90 1.0 Turbidity Standard (NTU) Figure 7-12 Res i l ience for the Wet S e a s o n Months us ing I S -FORM 91 7.2.5 Random Variable Sensitivities and Dominant ANN Inputs An advantage of using IS-FORM to quantify reservoir performance is that a sensitivity analysis can be conducted which indicates the dominant random variables in terms of estimating reservoir reliability and resilience. As the random variables are associated with specific ANN input parameters, the sensitivity analysis also provides an indication of the important turbidity simulation and forecasting inputs. Although the evaluation of reservoir performance at different turbidity standards is a unique problem characterized by a unique failure surface, the sensitivities of the random variables were found to be similar for all the standards tested. Figure 7-13 and Figure 7-14 show the sensitivity of the reliability estimates to each of the random variables in the Annual and Wet Season scenarios at the 10 NTU turbidity standard. Note that the date factors and meteorological inputs are variables 1-5 for both scenarios; however, the flow, rainfall, and intensity inputs are assigned different numbers in each case (refer to Table 7-5 and Table 7-6 for random variable numbers and descriptions). Results from the sensitivity analysis confirm indications from the ANN forecasting exercises. The Annual ANN turbidity model is sensitive to the date factors and meteorological information provided. All previous values of flow (variables 6-18) are also important parameters, while rain and intensity (variables 19-32) play a less significant role. Variables 33-46 are the additional inputs required for Day2 turbidity evaluations. They also indicate that previous flow values are more important than the rainfall and intensity values. Similar information can be gained from the sensitivities of the Wet Season ANN. In this case, however, it appears that the date factors are most dominant, while all other factors, including the meteorological information, are of equal significance in estimating turbidity levels. In addition, the ANN for the Wet Season is considerably less sensitive to flow inputs than that for the Annual scenario. Both of these results are consistent with the findings that the exclusion of recent lagged data, as is 92 conducted in the forecasting exercise, does not have a significant impact on the ANN'S ability to predict turbidity levels at the dam outlet. The date factors (and meteorological information in the Annual case) appear to provide sufficient information to replicate the important seasonal variations in turbidity. Any peaks on top of the seasonal trend are then accounted for by the flow, rainfall, and intensity information. 93 0.80 0.60 r Ih 111. JLLu 1 ^ 05 — c o i n r ^ c n — c o i n t ^ c 5 T - l c o ^ ! ^ g ) _ _ _ C N I C M C M C M C N C O H C O C O C O C O ' * ' ^ - ' * l l f l l l L Random Variable Number Figure 7-13 Sensit ivity of Reliability Es t imates to R a n d o m Var iab les for the Annua l S cenar i o and 10 N T U Turbidity S tandard 94 0.80 0.60 040 8 0.20 c •f* 0.00 "55 c W -0.20 -0.40 f i ' m s oi r co m f- o — c o r n e a l — co m a> — c o m — — — — — C N C M C N J C N C M C O C O C O C O C O T t f ^ r -0.60 -0.80 Random Variable Number Figure 7-14 Sensitivity of Reliability Estimates to Random Variables for the Wet Season Scenario and 10 NTU Turbidity Standard 95 7.2.6 Application of Importance Sampling (IS) to Reduce Computational Burden Significant time savings can be achieved by evaluating reservoir performance using IS-FORM in comparison with MCS. This is particularly true for resilience calculations where high system reliability requires thousands of initial MCSs to produce a sufficient number of failure scenarios that can then be evaluated for recovery. Typical reliability and resilience calculations are performed using IS-FORM with 5,000 samples, or with 10,000 MCSs. A reduced sample size with IS-FORM can accurately determine failure probabilities because the standard FORM solution is first used to focus on the failure region of the problem space. With a Pentium II computer, 200 MHz processor, and 128 MB of RAM, typical reliability calculations for a single turbidity standard take approximately 6.8 minutes using IS-FORM. In comparison, it takes 7.5 minutes to complete 10,000 MCSs. Although the time savings are not significant for reliability estimates, IS-FORM is a fast alternative to MCS for resilience calculations. Using IS-FORM, a single resilience calculation with 5,000 samples takes 7.0 minutes; however the amount of time required using MCS depends on the reliability of the system at the given turbidity standard. For example, at a standard of 8 NTU, 12,500 simulations are required to generate 10,000 failures in Day 1 from which system resilience is then evaluated. The computational time is 9.4 minutes. At higher reliabilities, however, more simulations are needed to generate the prerequisite 10,000 failures in Day 1. For a standard of 16 NTU, producing sufficient failure and resilience scenarios involves 1.1x106 MCSs at a computational cost of 13.8 hours. Table 7-9 indicates other computational times associated with Wet Season resilience calculations for different turbidity standards. 96 Tab le 7-9 Computat iona l T ime for Res i l ience Es t imates - Wet S e a s o n Number of METHOD STANDARD RELIABILITY Simulations Required to Generate 10,000 Day 1 Failures TIME REQUIREMENT IS-FORM using 5,000 samples Any standard for which FORM converges Depends on the standard N/A 7.0 minutes 8 NTU .197 12,500 9.4 minutes MCS using 10,000 samples 12 NTU 14 NTU .713 .891 34,800 91,700 26.1 minutes 1.1 hours 16 NTU .991 1,100,000 13.8 hours 97 8 CONCLUSIONS and FUTURE WORK Two ANNs are developed to simulate Annual and Wet Season (October to April) turbidity levels at the mid-level outlet of Capilano Reservoir under major drawdown conditions. Results show that the seasonal variation in turbidity levels are well reproduced, as are turbidities in the 5-15 NTU range. However, the ANNs tend to under-estimate turbidity levels greater than 15 NTU, likely due to limited available examples, and thus insufficient training at this high end. The average absolute error (AAE) associated with the ANN models is 1.1 NTU, well within acceptable error measures that are associated with field and laboratory sampling techniques. The ANN models are successful in replicating simulated turbidities achieved using the two-dimensional CE-QUAL-W2 model, and they can be used as an alternative modelling technique. Advantages of using ANNs over traditional response models include faster computational time, use of fewer input variables, and no requirement for input data in full time-series format. ANNs are also trained to provide short-term turbidity forecasts for the Annual scenario. One- and two-day forecasts are associated with a negligible increase in error, and forecast lengths of up to six days are possible depending on the level of accuracy that is desired. The ability to forecast turbidity levels is particularly useful for providing management with accurate information upon which treatment operations and water supply decisions can be made. When linked to a reliability analysis program, the ANN turbidity models enable reservoir performance to be evaluated in terms of meeting turbidity standards at the water supply intake and dam outlet. Inputs to the turbidity models are generated as random variables, and reservoir reliability and resilience are estimated using FORM. FORM, in conjunction with IS, is shown to produce similar reliability estimates as MCS. For estimating resilience, however, IS-FORM produces results that are slightly lower than those achieved by MCS. This is due to the manner in which 98 resilience is calculated, where small inaccuracies in initial failure probability estimates are compounded and can have a large impact on estimated resilience values. Reliability and resilience at different turbidity standards are estimated for the Annual period, Wet Season, and Monthly periods in the Wet Season. Results from the scenarios of interest indicate that during a major drawdown higher turbidity levels are expected over the October to December period. Although the reservoir has a lower reliability in the Wet Season than over the Annual period, the reservoir exhibits comparable resilience at certain turbidity standards. This implies that although turbidity levels may surpass a standard more often in the Wet Season, the reservoir is just as likely to recover in a single day as compared to the Annual scenario. Areas of future work include the development of new indicators to evaluate the performance of water quality systems, and the application of such measures as tools in operational decision-making. In particular, alternative measures of resilience may be developed that provide improved descriptions of reservoir system performance. The ANN and FORM approach developed in this work may also be applied to other reservoirs and to a wide range of future water quality applications. As the approach is not limited to turbidity applications, ANNs can be developed for the simulation and forecasting of other water quality parameters such as temperature, DO, and alkalinity, as well as water resource variables such as rainfall-runoff and streamflow. The use of ANNs for real-time or predictive water quality management would also be beneficial to management as treatment operations and water-use restrictions can be modified more efficiently based on expected changes in raw water quality. Future work may also investigate how modifications to the IS-FORM approach might improve estimates of system performance for highly non-linear problems. 99 REFERENCES Adebar, P., Foschi, R., and Yao, F. (1994). "Predicting strength variability of concrete offshore structures." Journal of Structural Engineering, 120(7), 2108-2122. Amari, S.-l., Murata, N., Muller, K.-R., Finke, M., and Yang, H. H. (1997). "Asymptotic statistical theory of overtraining and cross-validation." IEEE Transactions on Neural Networks, 8(5), 985-995. Atiya, A. F., El-Shoura, S. M., Shaheen, S. I., and El-Sherif, M. (1999). "A comparison between neural-network forecasting techniques - Case study: River flow forecasting." IEEE Transactions on neural networks, 10(2), 402-409. Bailey, D., and Thompson, D. (1990). "Developing neural-network applications." Al Expert, September, 34-41. Bastarache, D., El-Jabi, N., Turkkan, N., and Clair, T. A. (1997). "Predicting conductivity and acidity for small streams using neural networks." Canadian Journal of Civil Engineering, 24(6), 1030-1039. Battiti, R. (1992). "First- and second-order methods for learning: between steepest descent and newton's method." Neural Computation, 4, 141-166. BC Environment, W. Q. S. (1998). "British Columbia water quality guidelines (criteria): 1998 edition.", Ministry of Environment, Lands and Parks, Victoria. Bebis, G., and Georgiopoulos, M. (1994). "Feed-forward neural networks." IEEE Potentials(5), 27-31. Beck, M. B. (1997). "Applying system analysis.... a new agenda." Water Science and Technology, 36(5), 1-17. Bouchart, F. J.-C, and Goulter, I. C. (1998). "Is rational decision making appropriate for management of irrigation reservoirs?" Journal of Water Resources Planning and Management, 124(6), 301-309. 100 Buchberger, S. G., and Maidment, D. R. (1989). "Design of wastewater storage ponds at land treatment sites. II: Equilibrium storage performance functions." Journal of Environmental Engineering, 115(4), 704-723. Burges, S. J., and Lettenmaier, D. P. (1975). "Probabilistic methods in stream quality management." Water Resources Bulletin, 11(1), 115-130. Burn, D. H., Venema, H. D., and Simonovic, S. P. (1991). "Risk-based performance criteria for real-time reservoir operation." Canadian Journal of Civil Engineering, 18(1), 36-42. Canada. (1995). "Canadian Water Quality Guidelines." , Canadian Council of Ministers of the Environment, Ottawa. Canada. (1997). "Guidelines for Canadian Drinking Water Quality, 6th edition." , Health and Welfare Canada, Ottawa. Cavill, B. (1997). "Capilano landslide remedial works." Land and Water, 41(4), 16-19. Cheng, B., and Titterington, D. M. (1994). "Neural networks: A review from a statistical perspective." Statistical Science, 9(1), 2-54. Cole, T. M., and Buchak, E. M. (1995). "CE-QUAL-W2: A Two-Dimensional, Laterally Averaged, Hydrodynamic and Water Quality Model, Version 2.0, User Manual." EL-95-1, US Army Corps of Engineers, Washington, DC. Dawson, C. W., and Wilby, R. (1998). "Flow forecast in two flood-prone UK catchments using real hydrometric data." Hydrological Sciences Journal, 43(1), 47-66. Fahlman, S. E., and Lebiere, C. (1990). "The cascaded-correlation learning architecture." CMU-CS-90-100, Carnegie Mellon University, Pittsburgh. Fiering, M. B., and Holling, C. S. (1974). "Management... ecosystems." Agro-Ecosystems, 1, 301 -321. Flood, I., and Kartam, N. (1994a). "Neural networks in civil engineering. I: Principles and understanding." Journal of Computing in Civil Engineering, 8(2), 131-148. Flood, I., and Kartam, N. (1994b). "Neural networks in civil engineering. II: Systems and application." Journal of Computing in Civil Engineering, 8(2), 149-162. 101 Foschi, R. O., and Folz, B. (1990). "RELAN: RELiability ANalysis." , University of British Columbia, Vancouver, BC. Gabrys, B., and Bargiela, A. (1999). "Neural networks based decision support in presence of uncertainties." Journal of Water Resources Planning and Management, 125(5), 272-280. Garrett, J. H. (1994). "Editorial : Where and why artificial neural networks are applicable in civil engineering." Journal of Computing in Civil Engineering, 8(2), 129-130. Haan, C. T., Barfield, B. J., and Hayes, J. C. (1994). Design Hydrology and Sedimentology for Small Catchments, Academic Press, San Diego. Hamed, M. M., Conte, J. P., and Bedient, P. B. (1995). "Probabilistic screening tool for ground-water contamination assessment." Journal of Environmental Engineering, 121(11), 767-775. Hashimoto, T., Stedinger, J. R., and Loucks, D. P. (1982). "Reliability, resiliency, and vulnerability criteria for water resource system performance evaluation." Water Resources Research, 18(1), 14-20. Hatfield Consultants, L. (1997). "Appendix 1: Capilano Reservoir Sediment/Turbidity Analysis, Interim Environmental Report, Cleveland Dam - East Abutment Seepage Control", Report to Klohn-Crippen Consultants Ltd., Vancouver. Hatfield Consultants, L. (1999). "Appendix A1: Capilano Reservoir Sediment/Turbidity Analysis, Interim Environmental Report, Cleveland Dam -East Abutment Seepage Control - Detailed Design." , Report to Klohn-Crippen Consultants Ltd., Vancouver. Hegazy, T., Fazio, P., and Moselhi, O. (1994). "Developing practical neural network applications using back-propagation." Microcomputers in Civil Engineering, 9, 145-159. Hines, W. W., and Montgomery, D. C. (1990). "Ch. 14: Simple Linear Regression and Correlation." Probability and Statistics in Engineering and Management Sciences, John Wiley & Sons, New York. Holling, C. S. (1973). "Resilience... ecological systems." Annual Review of Ecological Systems, 4, 1-23. 102 Jain, S. K., Das, A., and Srivastava, D. K. (1998). "Application of ANN for reservoir inflow prediction and operation." Journal of Water Resources Planning and Management, 125(5), 263-271. Johnson, V. M., and Rogers, L. L. (1995). "Location analysis in ground-water remediation using neural networks." Ground Water, 33(5), 749-758. Karunanithi, N., Grenney, W. J., Whitley, D., and Bovee, K. (1994). "Neural networks for river flow prediction." Journal of Computing in Civil Engineering, 8(2), 201-219. Klohn-Crippen Consultants, L. (1999). "Reservoir Turbidity Analysis, Cleveland Dam - East Abutment Seepage Control - Detailed Design." , Report to the Greater Vancouver Water District, Richmond. Lence, B. J., and Ruszczynski, A. (1997). "Managing water quality under uncertainty: application of a new stochastic branch and bound method." Risk, Reliability, Uncertainty and Robustness of Water Resources Systems, Cambridge University Press, Cambridge, UK, in press. Madsen, H. O., Krenk, S., and Link, N. C. (1986). Methods of Structural Safety, Prentice-Hall, Englewoor Cliffs, N.J. Maier, H. R., and Dandy, G. C. (1996). "The use of artificial neural networks for the prediction of water quality parameters." Water Resources Research, 32(4), 1013-1022. Maier, H. R., and Dandy, G. C. (1997). "Determining inputs for neural network models of multivariate time series." Microcomputers in Civil Engineering, 12, 353-368. Maier, H. R., and Dandy, G. C. (1998a). "The effect of internal parameters and geometry on the performance of back-propagation neural networks: an empirical study." Environmental Modelling and Software, 13, 193-209. Maier, H. R., and Dandy, G. C. (1998b). "Neural network based modelling of environmental variables: A systematic approach." Mathematical and Computer Modelling, accepted 1998. 103 Maier, H. R., and Dandy, G. C. (1998c). "Understanding the behavious and optimising the performance of back-propagation neural networks: an empirical study." Environmental Modelling and Software, 13, 179-191. Maier, H. R., and Dandy, G. C. (2000). "Neural networks for the prediction and forecasting of water resources variables: A review of modelling issues and applications." Environmental Modelling & Software, 15, 101-124. Maier, H. R., Dandy, G. C, and Burch, M. D. (1998). "Use of artificial neural networks for modelling cyanobacteria Anabaena spp. in the River Murray, South Australia." Environmental Modelling and Software, 105, 257-272. Maier, H. R., Lence, B. J., Tolson, B. A., and Foschi, R. O. (1999). "First-order reliability method for estimating reliability, vulnerability and resilience." Water Resources Research, submitted August 1999. McCulloch, W. S., and Pitts, W. (1943). "A logical calculus of the ideas imminent in nervous activity." Bulletin of Mathematical Biophysics, 5, 115-133. Melching, C. S. (1992). "An improved first-order reliability approach for assessing uncertainties in hydrologic modeling." Journal of Hydrology, 132, 157-177. Melching, C. S., and Anmangandla, S. (1992). "Improved first-order uncertainty method for water-quality modeling." Journal of Environmental Engineering, 118(5), 791-805. Melching, C. S., Yen, B. C, and Wenzel, H. G. J. (1990). "A reliability estimation in modeling watershed runoff with uncertainties." Water Resources Research, 26(10), 2275-2286. Moy, W.-S., Cohon, J. L, and ReVelle, C. S. (1986). "A programing model for analysis of the reliability, resilience, and vulnerability of a water supply reservoir." Water Resources Research, 22(4), 489-498. Nam, L. H., Phien, H. N., and Gupta, A. D. (1998). "Filtering and forecasting of monthly streamflows by backpropagation neural networks with an error updating method." Water Resources Journal, 198, 29-39. Peavy, H. S. (1985). Environmental Engineering, McGraw-Hill, USA. Rackwitz, R. (1976). "Practical probabilistic approaches to design." , Comite European du Beton, Paris, France. 104 Raman, H., and Chandramouli, V. (1996). "Deriving a general operating policay for reservoirs using neural network." Journal of water resources planning and managemnet, 122(5), 342-347. Raman, H., and Sunilkumar, N. (1995). "Multivariate modelling of water resources time series using artificial neural networks." Hydrological Sciences Journal, 40(2), 145-163. Rogers, L. L, and Dowla, F. U. (1994). "Optimization of groundwater remediation using artificial neural networks with parallel solute transport modeling." Water Resources Research, 30(2), 457-481. Ronold, K. O. (1990). "Reliability analysis of a coastal dike." Coastal Engineering, 14, 43-56. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). "Learning internal representations by error propagation." Parallel Distributed Processing, D. E. Rumelhart and J. L. McClelland, eds., MIT Press, Cambridge, 318-364. Saad, M., Turgeon, A., Bigras, P., and Duquette, R. (1994). "Learning disaggregation technique for the operation of long-term hydroelectric power systems." Water Resources Research, 30(11), 3195-3202. Sarle, W. S. "Neural networks and statistical models." Nineteenth Annual SAS Users Group International Conference, 1-13. Sitar, N., Cawfield, J. D., and Der Kiureghian, A. (1987). "First-order reliability approach to stochastic analysis of subsurface flow and contaminant transport." Water Resources Research, 23(5), 794-804. Skaggs, T. H., and Barry, D. A. (1997). "The first-order reliability method of predicting cumulative mass flux in heterogeneous porous formations." Water Resources Research, 33(6), 1485-1494. Sureerattanan, S., and Phien, H. N. (1997). "Back-propagation networks for daily streamflow forecasting." Water Resources Journal, ST/ESCAP/SER.C/195, 1-7. Thurber Engineering, L. (1993). "Assessment of Turbidity-Generating Sediment Sources and Transport in the Capilano, Seymour, and Coquitlam Watersheds.", Report to the Greater Vancouver Water District, Vancouver. 105 Tung, B. C. (1990). "Evaluating the probability of violating dissolved oxygen standard." Ecological Modelling, 51, 193-204. Tung, Y.-K., and Hawthorn, W. E. (1988). "Assessment of probability distribution of dissolved oxygen deficit." Journal of Environmental Engineering, 114(6), 1421-1435. Vasquez, J. A., Maier, H., R., Lence, B. J., and Tolson, B. A. (1999). "Achieving water qulity system reliability using genetic algorithms." ASCE, accepted 1999. Vicens, G. J., Rodriguez-lturbe, I., and Schaake, J. C. J. (1975). "A bayesian framework for the use of regional information in hydrology." Water Resources Research, 11 (3), 405-414. Wen, C.-G., and Lee, C.-S. (1998). "A neural network approach to multiobjective optimization for water quality management in a river basin." Water Resources Research, 427-435. Wilber, C. G. (1983). Turbidity in the Aquatic Environment, C.C. Thomas, Springfield, Illinois. Xu, C, and Goulter, I. C. (1999). "Reliability-based optimal design of water distribution networks." Journal of Water Resources Planning and Management, 125(6), 352-362. Yang, C. C, Prasher, S. O., and Lacroix, R. (1996). "Application of artificial neural networks to land drainange engineering." Transactions of the ASAE, 38(2), 525-533. Yen, B. C, Cheng, S. T., and Melching, C. S. (1986). "First-order reliability analysis." Stochastic and risk analysis in hydraulic engineering, B. C. Yen, ed., Water Resources Publications, Littleton, Col, 1-36. Zhang, Q., and Stanley, S. J. (1997). "Forecasting raw-water quality parameters for the North Saskatchewan river by neural network modeling." Water Research, 31(9), 2340-2350. Zongxue, X., Jinno, K., Kawamura, A., Takesaki, S., and Ito, K. (1998). "Performance risk analysis for Fukuoka water supply system." Water Resources Management, 12(1), 13-30. 106 Appendix I ANN Turbidity Subroutines for RELAN FORTRAN code for the Annual and Wet Season turbidity ANNs that is incorporated in RELAN to calculate turbidity levels based on the random variables generated by RELAN. 107 Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * * C * USER SUBROUTINES FOR CAPILANO RESERVOIR TURBIDITY MODELLING * C * * C * ANNUAL SCENARIO * C * * Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE DETERM (IMODE) IMPLICIT REAL*8 (A - H, 0 - Z) DIMENSION NNOD(4), BIAS(3,99), WEIGHT(3,99,99) CHARACTER*12 f i l e n a m e , o u t f i l e COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT,MODE f i l e n a m e = 'T2_a4.txt' o u t f i l e = 'ANNout.txt' OPEN (5, FILE = filename) IF(IMODE.GT.1) GOTO 2000 C Set up the Network A r c h i t e c t u r e and Weights READ (5,100) NLAY READ (5,110) (NNOD(I), 1=1,NLAY) READ (5,125) WTS C For each l a y e r s t a r t i n g at Layer 2 DO 2 00 L=2,NLAY READ (5,127) LAYER C Read i n the b i a s f o r each node i n t h a t l a y e r DO 200 N=l,NNOD(L) READ (5,130) BIAS(L-1,N) C Get the c o n n e c t i o n weights from nodes i n p r e v i o u s l a y e r DO 200 W=l,NNOD(L-l) IF (N.GE.10) GO TO 90 READ (5,140) WEIGHT(L-1,N,W) GO TO 95 90 READ (5,145) WEIGHT(L-1,N,W) 95 WRITE(*,*) 2 00 CONTINUE C Input the C r i t i c a l T u r b i d i t y l e v e l WRITE (*,150) READ (*,*) TCRIT C Input the type of c a l c u l a t i o n ( l f o r FORM, 2 f o r MCS) WRITE (*,160) READ (*,*) MODE 100 FORMAT (/, 16X, 12) 110 FORMAT (12X,I2) 120 FORMAT (' LAYERS= ', 12, '; NODES= ',12,2X,12,2X,12) 125 FORMAT (Al) 127 FORMAT (5X, 12) 1 128 FORMAT (' Layer ', 12,' weights and b i a s e s ' ) 108 130 FORMAT (15X, F20.17) 135 FORMAT (' B i a s f o r node 12,' i s ' , F20.17) 140 FORMAT (24X, F20.17) 145 FORMAT (25X, F20.17) 150 FORMAT (' What i s the c r i t i c a l t u r b i d i t y l e v e l ( i n NTU) f o r which +you would l i k e t o e v a l u a t e r e l i a b i l i t y ? The l e v e l s h o u l d be +within the range 0-30 (NTU). 1,/) 160 FORMAT (' What type of e v a l u a t i o n would you l i k e t o do? E n t e r 1 +for FORM, o r 2 f o r Monte C a r l o s i m u l a t i o n . ' ) • CLOSE(UNIT=5) 2000 CONTINUE RETURN end ! end of the DETERM s u b r o u t i n e C ******************************************************* Q *********************************************************************** SUBROUTINE GFUN (X, N, IMODE, GXP) IMPLICIT REAL*8 (A - H, O - Z) DIMENSION NNOD(4), BIAS(3,99), WEIGHT(3,99,99), X(N),V(N) COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT, MODE K=2*NNOD(1) IF(IMODE.EQ.l) THEN ! For "Day 1" C a l c u l a t i o n s DO 500 1=1,NNOD(1) V(I)=X(I) 50 0 CONTINUE CALL NEURALNET(V,TTRUE) ! Add the random e r r o r term t o take i n t o account model e r r o r between ! Neuframe and Qual-W2. The e r r o r i s n o r m a l l y d i s t r i b u t e d w i t h a ! mean of zero and s t d dev equal t o the MSE. I t i s u n c o r r e l a t e d . TTRUE=TTRUE+X(4 7) IF(MODE.EQ.l) THEN GXP= TCRIT - TTRUE ! [1] ELSEIF(MODE.EQ.2) THEN GXP= TTRUE - TCRIT ! [2] ENDIF' c Use [1] f o r FORM e v a l u a t i o n ( p r o v i d e s p r o b a b i l i t y of f a i l u r e Day 1) c Use [2] f o r MCS where c i f GXP < 0 the t u r b i d i t y i s ACCEPTABLE c i f GXP > 0 the t u r b i d i t y i s UNACCEPTABLE c The GXP must be POSITIVE ( i . e . t u r b i d i t y f a i l u r e f o r day 1) f o r c mode/day 2 t o be e v a l u a t e d . I f not, i t jumps d i r e c t l y t o the next c i t e r a t i o n . The " p r o b a b i l i t y of f a i l u r e " t h a t i s output (based on c -ve GXP v a l u e s ) , i s i n f a c t the p r o b a b i l i t y of t u r b i d i t y l e v e l s c b e i n g a c c e p t a b l e on Day 1 OR Day 2. 1 0 9 i f ( G X P . G T . O ) THEN ! Counts the number of t u r b i d i t y f a i l u r e s MM=MM+1 END IF J1=J1+1 ! T o t a l e v a l u a t i o n s f o r mode 1 E L S E I F ( I M O D E . E Q . 2 ) THEN ! GIVEN THAT THERE WAS A TURBIDITY FAILURE ! IN DAY 1, EVALUATE DAY 2 DO 600 I= l ,NNOD(l ) V( I )=X(I+(NNOD( l ) ) ) 60 0 CONTINUE C In some c a s e s , the i n p u t v a r i a b l e s are the same as v a l u e s from the C day b e f o r e . The r e m a i n i n g i n p u t a r e g e n e r a t e d by R e l a n . V ( l ) =X(1) V(2) =X(2) V(3) =X(3) V(4) =X(4) V(5) =X(5) V(6) =X(33) V(7) =X(6) V(8) =X(7) V(9) =X(8) V(10 )=X(9) V ( l l )=X(10) V(12 )=X(11) V(13 )=X(12) V(14 )=X(34) V(15 )=X(35) V(16 ) =X(36) V(17 )=X(37) V(18 )=X(38) V(19 =X(39) V(20 =X(19) V(21 =X(20) V(22 =X(40) V(23 =X(22) V(24 =X(41) V(25 =X(42) V(26 =X(43) V(27 =X(26) V(28 =X(27) V(29 =X(44) V(30 =X(29) V(31 =X(45) V(32 =X(46) CALL NEURALNET(V,TTRUE) Add the random e r r o r term t o take i n t o account model e r r o r between Neuframe and Qual -W2. The e r r o r i s n o r m a l l y d i s t r i b u t e d w i t h a mean of z e r o and s t d dev e q u a l t o the MSE. I t i s u n c o r r e l a t e d . 110 TTRUE=TTRUE+X(48) [2] Counts the number of RECOVERIES, i . e . a c c e p t a b l e t u r b i d i t i e s i n Day 2 c a l c s . J2=J2+1 ! T o t a l e v a l u a t i o n s f o r mode 2 ENDIF ! END OF RELIABILITY AND RESILIENCE CALCS. w r i t e (*,*) w r i t e (*,710) MM, J l , LL, J2 700 FORMAT (' T h i s i s FAILURE MODE: ',12,/, + ' The CALCULATED T u r b i d i t y i s : ', F7.3,/, + ' The CRTITICAL T u r b i d i t y L e v e l i s : ', F7.3,/, + ' and the GXP i s c a l c u l a t e d t o be: ' , F8.3,/, + ' Note: a p o s i t i v e GXP i n d i c a t e s u n a c c e p t a b l e t u r b i d i t y . ' , / ) 710 FORMAT (' T h i s i s f a i l u r e ',110,' of',110,' t e s t s f o r FailMode 1' + ,/,' T h i s i s r e c o v e r y ',110,' of',110,' t e s t s f o r FailMode 2',/) RETURN END C ************************************************ rj * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE NEURALNET(XIN,XTRUE) IMPLICIT REAL*8 (A - H, O - Z) DIMENSION NNOD(4), XSCALE(99), XTRAN(4,99), BIAS(3,99), + WEIGHT(3,99,99), XIN(99) REAL*8 MAXV(99), MINV(99) COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT, MODE OPEN (6, FILE = 'FNNquery3.txt') C The max and min v a l u e s are c u r r e n t l y r e a d i n from an i n p u t f i l e . READ (6,210) (MAXV(I), 1=1,NNOD(1)),(MINV(I), 1=1,NNOD(1)) 210 FORMAT (2F7.3, 30F7.2,/, 2F7.3, 30F7.2,/ ) C S c a l i n g and T r a n s f e r f o r the f i r s t l a y e r DO 251 1=1,NNOD(1) XSCALE(I)= ((XIN(I)-MINV(I))*0.8/(MAXV(I)-MINV(I)))+0.1 XTRAN(1,I)= DTANH(XSCALE(I)) 2 51 CONTINUE C C a l c u l a t i o n s f o r the remainding l a y e r s DO 400 L=2,NLAY DO 400 NN=1,NNOD(L) YTOT =0.0 DO 60 I=l,NNOD(L-l) YIN = XTRAN(L-1,I)*WEIGHT(L-1,NN,I) YTOT= YTOT + YIN 60 CONTINUE XTRAN(L,NN) = DTANH(YTOT + BIAS(L-1,NN)) 4 00 CONTINUE GXP= TTRUE - TCRIT if(GXP.LT.O) THEN LL=LL+1 END IF 111 C Unscale (based on max/min v a l u e s used i n t r a i n i n g ) f o r the True Turb. XMAX = 31.33 XMIN = 0.0 0 XTRUE = (XTRAN(NLAY,NNOD(NLAY) )- 0.1)*(XMAX-XMIN)/0.8+XMIN CLOSE (UNIT = 6) CLOSE (UNIT =11) RETURN END C ************************************************ 112 Q ******************************************** c * * C * USER SUBROUTINES FOR CAPILANO RESERVOIR TURBIDITY MODELLING * C * * C * WET SEASON * C * * C *********************************************************************** SUBROUTINE DETERM (IMODE) IMPLICIT REAL*8 (A - H, O - Z) DIMENSION NNOD(4), BIAS(3,99), WEIGHT(3,99,99) CHARACTER*12 fi l e n a m e , o u t f i l e COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT,MODE fil e n a m e = 'WetArch.txt' o u t f i l e = 'ANNout.txt' OPEN (5, FILE = filename) IF(IMODE.GT.l) GOTO 2000 C Set up the Network A r c h i t e c t u r e and Weights READ (5,100) NLAY READ (5,110) (NNOD ( I ) , 1 = 1,NLAY) C WRITE (9,120) NLAY, (NNOD ( I ) , 1 = 1,NLAY) READ (5,125) WTS C For each l a y e r s t a r t i n g at Layer 2 DO 200 L=2,NLAY READ (5,127) LAYER C Read i n the b i a s f o r each node i n t h a t l a y e r DO 200 N=l,NNOD(L) READ (5,130) BIAS(L-1,N) C Get the c o n n e c t i o n weights from nodes i n p r e v i o u s l a y e r DO 200 W=l,NNOD(L-l) IF (N.GE.10) GO TO 90 READ (5,140) WEIGHT(L-1,N,W) GO TO 95 90 READ (5,145) WEIGHT(L-l,N,W) 95 WRITE (*,*) 2 00 CONTINUE c Input the C r i t i c a l T u r b i d i t y l e v e l w r i t e (*,150) read (*,*) TCRIT c Input type of e v a l u a t i o n (FORM o r MCS) WRITE (*,160) READ (*,*) MODE 100 FORMAT (/, 16X, 12) 110 FORMAT (12X,I2) 120 FORMAT (' LAYERS= ', 12, '; NODES= ',12,2X,12,2X,12) 125 FORMAT (Al) 127 FORMAT (5X, 12) 113 128 FORMAT (' Layer ', 12,' weights and b i a s e s ' ) 130 FORMAT (15X, F20.17) 135 FORMAT (' B i a s f o r node ', 12,' i s ' , F20.17) 140 FORMAT (24X, F20.17) 145 FORMAT (25X, F20.17) 150 FORMAT (' What i s the c r i t i c a l t u r b i d i t y l e v e l ( i n NTU) f o r which +you would l i k e t o e v a l u a t e r e l i a b i l i t y ? The l e v e l s h o u l d be +within the range 0-30 (NTU).',/) 160 FORMAT (' What type o f e v a l u a t i o n would you l i k e t o do? E n t e r 1 +for FORM, o r 2 f o r Monte C a r l o s i m u l a t i o n . ' ) CLOSE(UNIT=5) 2 00 0 CONTINUE RETURN end ! end of the DETERM s u b r o u t i n e Q ************************************************************** c ************************************************* SUBROUTINE GFUN (X, N, IMODE, GXP) IMPLICIT REAL*8 (A - H, O - Z) DIMENSION NNOD(4), BIAS(3,99), WEIGHT(3,99,99), X(N),V(N) COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT, MODE K=2*NNOD(1) IF(IMODE.EQ.l) THEN ! For "Day 1" C a l c u l a t i o n s DO 500 1=1,NNOD(1) V(I)=X(I) 500 CONTINUE CALL NEURALNET(V,TTRUE) ! Add the random e r r o r term t o take i n t o account model e r r o r between ! Neuframe and Qual-W2. The e r r o r i s n o r m a l l y d i s t r i b u t e d w i t h a ! mean of zero and s t d dev equal t o the MSE. I t i s u n c o r r e l a t e d . TTRUE=TTRUE+X(46) IF(MODE.EQ.l) THEN GXP= TCRIT - TTRUE ELSEIF(MODE.EQ.2) THEN GXP= TTRUE - TCRIT END IF c FOR RESILIENCE (2 f a i l u r e modes) c Use [1] f o r FORM e v a l u a t i o n ( p r o v i d e s p r o b a b i l i t y of f a i l u r e Day 1) c Use [2] f o r MCS where c i f GXP < 0 the t u r b i d i t y i s ACCEPTABLE c i f GXP > 0 the t u r b i d i t y i s UNACCEPTABLE c The GXP must be POSITIVE ( i . e . t u r b i d i t y f a i l u r e f o r day 1) f o r c mode/day 2 t o be e v a l u a t e d . I f not, i t jumps d i r e c t l y t o the next c b e i n g a c c e p t a b l e on Day 1 i t e r a t i o n . The " p r o b a b i l i t y of f a i l u r e " ! [1] ! [2] 114 t h a t i s output (based on -ve GXP v a l u e s ) , i s i n f a c t the p r o b a b i l i t y of t u r b i d i t y l e v e l s b e i n g a c c e p t a b l e on Day 1 OR Day 2. if(GXP.GT.O) THEN MM=MM+1 ! Counts the number of t u r b i d i t y f a i l u r e s END IF J1=J1+1 ! T o t a l e v a l u a t i o n s f o r mode 1 ELSEIF(IMODE.EQ.2) THEN ! GIVEN THAT THERE WAS A TURBIDITY FAILURE ! IN DAY 1, EVALUATE DAY 2 600 DO 600 1=1,NNOD(1) V(I)=X(I+(NNOD(l))) CONTINUE S e t t i n g these v a r i a b l e s as c o n s t a n t s based on v a l u e s the day b e f o r e For the annual s c e n a r i o , see c o r r e l a t i o n m a t r i x V ( l ) =X(1) V(2) =X(2) V(3) =X(3) V(4) =X(4) V(5)=X(5) V(6)=X(33) V(7)=X(34) V(8)=X(6) V(9) =X(7) V(10)=X(8) V ( l l ) = X (9) V(12)=X(10) V(13)=X(11) V(14)=X(35) V(15)=X(36) V(16)=X(37) V(17)=X(38) V(18) =X(39) V(19)=X(40) V(20) =X(41) V(21) =X(20) V(22) =X (21) V(23)=X(22) V(24)=X(23) V(25)=X(24) V(26) =X(25) V(27) =X(26) V(28) =X(27) V(29) =X (42) V(30) =X(43) V(31)=X(44) V(32) =X(45) CALL NEURALNET(V,TTRUE) ! Add the random e r r o r term t o take i n t o account model e r r o r between ! Neuframe and Qual-W2. The e r r o r i s n o r m a l l y d i s t r i b u t e d w i t h a 115 mean of zero and std dev equal to the MSE. It i s uncorrelated. TTRUE=TTRUE+X(47) GXP= TTRUE - TCRIT ! [2] c Use [2] for FORM (provides prob. of acceptable t u r b i d i t y i n Day 2) c Use [2] for MCS: c If GXP < 0 the t u r b i d i t y i s ACCEPTABLE c and there i s RECOVERY !!! c If GXP > 0 the t u r b i d i t y i s UNACCEPTABLE if(GXP.LT.O) THEN ! Counts the number of RECOVERIES, i . e . LL=LL+l ! acceptable t u r b i d i t i e s i n Day 2 calcs. END IF J2=J2+1 ! Total evaluations for mode 2 ENDIF ! END OF RELIABILITY AND RESILIENCE CALCS. write (*,*) write (*,710) MM, J l , LL, J2 700 FORMAT (' This i s FAILURE MODE: ',12,/, + ' The CALCULATED Turbidity i s : ', F7.3,/, + ' The CRTITICAL Turbidity Level i s : ', F7.3,/, + ' and the GXP i s calculated to be: ' , F8.3,/, + ' Note: a p o s i t i v e GXP indicates unacceptable t u r b i d i t y . 1 , / ) 710 FORMAT (' This i s f a i l u r e ',110,' o f , 110,' tests for FailMode 1' +,/,' This i s recovery ',110,' of',110,' tests for FailMode 2',/) RETURN END Q ******************************************** rj * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE NEURALNET(XIN,XTRUE) IMPLICIT REAL*8 (A - H, 0 - Z) DIMENSION NNOD(4), XSCALE(99), XTRAN(4,99), BIAS(3,99), + WEIGHT(3,99,99), XIN(99) REAL*8 MAXV(99), MINV(99) COMMON/DCY/BIAS,WEIGHT,NNOD,NLAY,TCRIT, MODE OPEN (6, FILE = 'WetScale.txt') C The max and min values are currently read i n from an input f i l e . READ (6,210) (MAXV(I), 1=1,NNOD(1)),(MINV(I), I=l,NNOD(l)) 210 FORMAT (2F7.3, 30F7.2,/, 2F7.3, 30F7.2,/ ) C Scaling and Transfer for the f i r s t layer DO 251 1=1,NNOD(1) XSCALE(I)= ((XIN(I)-MINV(I))*0.8/(MAXV(I)-MINV(I)))+0.1 XTRAN(1,I)= DTANH(XSCALE (I)) 251 CONTINUE C Calculations for the remainding layers 116 DO 4 00 L=2,NLAY DO 400 NN=1,NNOD(L) YTOT = 0 . 0 DO 60 1=1,NNOD(L-l ) YIN = X T R A N ( L - l , I ) * W E I G H T ( L - 1 , N N , I ) YTOT= YTOT + YIN 6 0 CONTINUE XTRAN(L,NN) = DTANH(YTOT + B I A S ( L - 1 , N N ) ) 4 00 CONTINUE C U n s c a l e (based on max/min v a l u e s used i n t r a i n i n g ) f o r the T r u e T u r b . XMAX = 31.33 XMIN = 0.00 XTRUE = (XTRAN(NLAY,NNOD(NLAY)) -0 .1 )* (XMAX-XMIN) /0 .8+XMIN CLOSE (UNIT = 6) CLOSE (UNIT =11) RETURN END C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 117 Appendix II Correlation Matrices for RELAN Calculations The correlation coefficients that describe the relationships between the random variable inputs used by the Annual and Wet Season ANNs. 118 CORRELATION MATRIX ANNUAL SCENARIO Random 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Variable cos sin tair wind cloud Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q12 Q15 Q17 Q22 1 cos 1.000 -0.053 -0.886 0.296 0.928 0.110 0.108 0.105 0.103 0.100 0.098 0.093 0.092 0.083 0.073 0.068 0.052 2 sin 1.000 -0.348 0.733 0.163 0.118 0.117 0.118 0.117 0.116 0.116 0.117 0.116 0.122 0.129 0.130 0.138 3 tair 1.000 -0.5B5 -0.928 -0.189 -0.186 -0.185 -0.182 -0.177 -0.175 -0.173 -0.172 -0.165 -0.158 -0.150 -0.133 4 wind 1.000 0.515 0.114 0.113 0.114 0.113 0.110 0.109 0.109 0.107 0.113 0.113 0.114 0.111 5 cloud 1.000 0.129 0.127 0.128 0.127 0.123 0.122 0.118 0.116 0.110 0.101 0.091 0.074 6 Q1 1.000 0.653 0.426 0.330 0.268 0.238 0.224 0.189 0.129 0.119 0.128 0.141 7 Q2 1.000 0.653 0.424 0.327 0.268 0.238 0.228 0.131 0.112 0.132 0.155 8 Q3 1.000 0.652 0.423 0.327 0.269 0.239 0.149 0.126 0.119 0.156 9 Q4 1.000 0.656 0.426 0.331 0.267 0.155 0.125 0.112 0.149 10 Q5 1.000 0.657 0.428 0.328 0.186 0.124 0.125 0.142 11 Q6 1.000 0.656 0.426 0.222 0.143 0.124 0.125 12 Q7 1.000 0.658 0.230 0.152 0.124 0.129 13 Q8 1.000 0.264 0.185 0.145 0.114 14 Q12 1.000 0.328 0.235 0.123 15 Q15 1.000 0.426 0.186 16 Q17 1.000 0.232 17 Q22 1.000 18 Q25 19 R3 20 R4 21 R5 22 R15 23 R16 24 R20 25 R25 26 13 27 14 28 15 29 115 30 116 31 I20 32 I25 33 Q0 34 Q11 35 Q14 36 Q16 37 Q21 38 Q24 39 R2 40 R14 41 R19 42 R24 43 12 44 114 45 119 46 I24 119 CORRELATION MATRIX ANNUAL SCENARIO (CONTINUED) Random 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Variable Q25 R3 R4 R5 R15 R16 R20 R25 13 14 15 115 116 I20 I25 Q0 Q11 1 C O S 0.037 0.230 0.231 0.230 0.223 0.224 0.220 0.210 0.171 0.170 0.171 0.167 0.168 0.165 0.160 0.113 0.086 2 sin 0.142 -0.009 -0.004 -0.001 0.038 0.043 0.056 0.078 -0.019 -0.018 -0.017 0.013 0.015 0.027 0.044 0.117 0.119 3 tair -0.119 -0.219 -0.221 -0.221 -0.229 -0.232 -0.228 -0.225 -0.164 -0.163 -0.165 -0.169 -0.171 -0.167 -0.171 -0.190 -0.167 4 wind 0.111 0.076 0.077 0.082 0.104 0.106 0.115 0.128 0.051 0.051 0.055 0.068 0.071 0.075 0.086 0.116 0.113 5 cloud 0.060 0.213 0.214 0.216 0.218 0.220 0.215 0.211 0.161 0.160 0.162 0.159 0.161 0.155 0.157 0.130 0.112 6 Q1 0.107 0.304 0.214 0.148 0.051 0.064 0.060 0.055 0.272 0.188 0.124 0.058 0.058 0.043 0.032 0.654 0.135 7 Q2 0.115 0.485 0.306 0.212 0.040 0.051 0.071 0.057 0.417 0.270 0.188 0.044 0.059 0.053 0.039 0.427 0.152 8 Q3 0.129 0.420 0.483 0.306 0.045 0.040 0.076 0.080 0.375 0.415 0.273 0.025 0.044 0.052 0.054 0.332 0.157 9 Q4 0.136 0.121 0.420 0.486 0.038 0.046 0.067 0.087 0.144 0.382 0.422 0.023 0.025 0.054 0.060 0.271 0.189 10 Q5 0.152 0.082 0.121 0.422 0.046 0.038 0.061 0.085 0.084 0.157 0.388 0.030 0.023 0.054 0.063 0.239 0.224 11 Q6 0.152 0.068 0.082 0.123 0.066 0.048 0.049 0.060 0.058 0.090 0.160 0.042 0.032 0.056 0.044 0.224 0.233 12 Q7 0.150 0.079 0.069 0.083 0.076 0.068 0.037 0.071 0.048 0.062 0.092 0.046 0.044 0.038 0.054 0.189 0.263 13 Q8 0.146 0.080 0.079 0.069 0.105 0.077 0.041 0.070 0.045 0.048 0.063 0.081 0.048 0.019 0.049 0.156 0.329 14 Q12 0.107 0.038 0.039 0.035 0.216 0.152 0.077 0.036 0.013 0.015 0.018 0.197 0.135 0.048 0.037 0.129 0.660 15 Q15 0.122 0.031 0.015 0.018 0.423 0.490 0.121 0.044 0.010 -0.002 0.006 0.383 0.426 0.108 0.028 0.133 0.265 16 Q17 0.147 0.036 0.024 0.032 0.081 0.123 0.209 0.067 0.027 0.017 0.008 0.087 0.157 0.193 0.043 0.147 0.226 17 Q22 0.310 0.060 0.049 0.037 0.030 0.052 0.075 0.209 0.046 0.030 0.025 0.015 0.032 0.077 0.192 0.132 0.124 18 Q25 1.000 0.034 0.030 0.056 0.012 0.027 0.065 0.417 0.028 0.029 0.045 0.003 0.007 0.035 0.372 0.110 0.114 19 R3 1.000 0.224 0.114 0.059 0.062 0.065 0.063 0.747 0.248 0.119 0.050 0.056 0.042 0.048 0.213 0.040 20 R4 1.000 0.226 0.054 0.059 0.081 0.060 0.264 0.744 0.251 0.039 0.050 0.053 0.061 0.149 0.040 21 R5 1.000 0.023 0.055 0.081 0.078 0.128 0.268 0.748 0.005 0.040 0.058 0.063 0.124 0.056 22 R15 1.000 0.238 0.077 0.023 0.033 0.030 0.013 0.751 0.266 0.067 0.002 0.064 0.150 23 R16 1.000 0.078 0.054 0.032 0.032 0.032 0.276 0.752 0.078 0.036 0.070 0.122 24 R20 1.000 0.077 0.062 0.065 0.062 0.061 0.080 0.751 0.067 0.087 0.067 25 R25 1.000 0.044 0.059 0.046 0.011 0.019 0.060 0.754 0.032 0.047 26 13 1.000 0.321 0.138 0.040 0.038 0.051 0.051 0.190 0.017 27 14 1.000 0.329 0.020 0.039 0.043 0.064 0.125 0.020 28 15 1.000 -0.001 0.021 0.042 0.047 0.107 0.034 29 115 1.000 0.338 0.064 -0.005 0.057 0.133 30 116 1.000 0.084 0.012 0.056 0.109 31 I20 1.000 0.062 0.064 0.041 32 I25 1.000 0.017 0.052 33 Q0 1.000 0.132 34 Q11 1.000 35 Q14 36 Q16 37 Q21 38 Q24 39 R2 40 R14 41 R19 42 R24 43 12 44 114 45 119 46 I24 120 CORRELATION MATRIX ANNUAL SCENARIO (CONTINUED) Random 35 36 37 38 39 40 41 42 43 44 45 46 Variable Q14 Q16 Q21 Q24 R2 R14 R19 R24 12 114 119 124 1 cos 0.075 0.071 0.056 0.042 0.231 0.223 0.221 0.215 0.171 0.166 0.166 0.163 2 sin 0.128 0.130 0.135 0.141 -0.012 0.033 0.052 0.073 -0.022 0.009 0.024 0.041 3 tair -0.160 -0.155 -0.136 -0.125 -0.218 -0.225 -0.227 -0.227 -0.162 -0.166 -0.168 -0.172 4 wind 0.113 0.113 0.112 0.113 0.071 0.099 0.111 0.124 0.048 0.066 0.073 0.086 5 cloud 0.103 0.096 0.079 0.065 0.211 0.217 0.215 0.213 0.159 0.159 0.156 0.159 6 Q1 0.113 0.133 0.157 0.113 0.484 0.043 0.071 0.058 0.416 0.044 0.052 0.039 7 Q2 0.126 0.119 0.156 0.130 0.422 0.048 0.075 0.082 0.375 0.026 0.050 0.054 8 Q3 0.125 0.112 0.150 0.138 0.122 0.038 0.068 0.087 0.145 0.023 0.054 0.059 9 Q4 0.126 0.125 0.143 0.154 0.083 0.049 0.062 0.084 0.086 0.031 0.055 0.062 10 Q5 0.145 0.124 0.125 0.154 0.069 0.068 0.050 0.059 0.058 0.041 0.056 0.043 11 Q6 0.152 0.124 0.131 0.150 0.080 0.080 0.038 0.071 0.048 0.047 0.040 0.055 12 Q7 0.183 0.145 0.117 0.145 0.079 0.104 0.042 0.070 0.044 0,079 0.021 0.050 13 Q8 0.223 0.153 0.107 0.124 0.056 0.111 0.035 0.066 0.035 0.088 0.018 0.051 14 Q12 0.426 0.267 0.143 0.121 0.021 0.321 0.106 0.041 0.009 0.284 0.081 0.019 15 Q15 0.659 0.660 0.226 0.140 0.024 0.122 0.150 0.063 0.021 0.157 0.133 0.040 16 Q17 0.329 0.657 0.263 0.177 0.052 0.067 0.313 0.106 0.034 0.061 0.283 0.082 17 Q22 0.153 0.225 0.649 0.410 0.059 0.032 0.065 0.314 0.048 0.011 0.057 0.279 18 Q25 0.124 0.140 0.250 0.647 0.011 0.013 0.049 0.111 0.010 -0.005 0.029 0.145 19 R3 0.014 0.024 0.050 0.031 0.226 0.053 0.079 0.066 0.263 0.038 0.052 0.061 20 R4 0.018 0.031 0.037 0.057 0.115 0.026 0.081 0.077 0.128 0.009 0.058 0.062 21 R5 0.028 0.015 0.045 0.058 0.097 0.056 0.056 0.079 0.091 0.040 0.057 0.065 22 R15 0.488 0.123 0.052 0.026 0.063 0.233 0.078 0.055 0.033 0.272 0.078 0.036 23 R16 0.320 0.424 0.073 0.030 0.056 0.118 0.099 0.063 0.047 0.140 0.096 0.036 24 R20 0.111 0.148 0.115 0.070 0.050 0.093 0.237 0.078 0.044 0.058 0.273 0.076 25 R25 0.029 0.057 0.144 0.489 0.047 0.047 0.087 0.240 0.036 0.025 0.054 0.275 26 13 -0.001 0.020 0.031 0.029 0.251 0.030 0.066 0.061 0.324 0.020 0.044 0.063 27 14 0.005 0.008 0.025 0.045 0.119 0.013 0.063 0.042 0.137 0.000 0.044 0.045 28 15 0.009 -0.002 0.037 0.045 0.095 0.027 0.045 0.047 0.090 0.019 0.050 0.048 29 115 0.425 0.158 0.032 0.006 0.056 0.261 0.080 0.021 0.040 0.333 0.084 0.013 30 116 0.286 0.384 0.041 0.010 0.058 0.123 0.097 0.047 0.053 0.154 0.103 0.024 31 I20 0.090 0.131 0.149 0.041 0.033 0.055 0.265 0.077 0.037 0.039 0.336 0.081 32 I25 0.014 0.037 0.127 0.421 0.032 0.031 0.053 0.265 0.029 0.014 0.037 0.337 33 Q0 0.120 0.129 0.141 0.107 0.304 0.054 0.060 0.055 0.271 0.058 0.042 0.033 34 Q11 0.330 0.233 0.123 0.105 0.040 0.226 0.077 0.037 0.015 0.202 0.048 0.038 35 Q14 1.000 0.427 0.187 0.121 0.031 0.424 0.122 0.046 0.011 0.383 0.109 0.029 36 Q16 1.000 0.233 0.146 0.035 0.081 0.210 0.075 0.029 0.087 0.194 0.046 37 Q21 1.000 0.310 0.060 0.033 0.078 0.207 0.046 0.016 0.081 0.190 38 Q24 1.000 0.034 0.012 0.067 0.417 0.028 0.001 0.037 0.372 39 R2 1.000 0.059 0.064 0.066 0.747 0.049 0.041 0.049 40 R14 1.000 0.077 0.024 0.033 0.751 0.067 0.003 41 R19 1.000 0.077 0.061 0.061 0.751 0.066 42 R24 1.000 0.046 0.012 0.060 0.754 43 12 1.000 0.039 0.050 0.052 44 114 1.000 0.064 -0.005 45 119 1.000 0.062 46 I24 1.000 CORRELATION MATRIX WET SEASON Random 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Variable s i n C O S t a i r w i n d c l o u d R2 12 R3 13 R4 14 R5 15 R7 17 R13 113 1 s i n 1.000 -0.417 -0.099 0.796 -0.006 -0.094 -0.106 -0.088 -0.101 -0.083 -0.099 -0.075 -0.091 -0.061 -0.081 -0.038 -0.060 2 c o s 1.000 -0.793 -0.520 0.690 0.131 0.104 0.131 0.105 0.131 0.107 0.130 0.104 0.133 0.116 0.139 0.123 3 t a i r 1.000 -0.005 -0.800 -0.086 -0.054 -0.092 -0.063 -0.094 -0.069 -0.098 -0.072 -0.113 -0.093 -0.134 -0.110 4 w i n d 1.000 -0.156 -0.084 -0.088 -0.077 -0.083 -0.076 -0.079 -0.067 -0.072 -0.055 -0.062 -0.038 -0.048 5 c l o u d 1.000 0.053 0.042 0.061 0.049 0.064 0.055 0.069 0.057 0.085 0.083 0.115 0.098 6 R2 1.000 0.768 0.189 0.232 0.078 0.108 0.062 0.071 0.032 0.045 0.023 0.029 7 12 1.000 0.241 0.318 0.126 0.152 0.081 0.085 0.025 0.040 0.012 0.017 8 R3 1.000 0.768 0.186 0.231 0.078 0.108 0.053 0.061 -0.004 -0.007 9 13 1.000 0.245 0.320 0.126 0.151 0.064 0.073 -0.010 -0.017 10 R4 1.000 0.772 0.186 0.232 0.063 0.074 0.027 0.032 11 14 1.000 0.245 0.319 0.081 0.086 0.005 0.013 12 R5 1.000 0.771 0.076 0.110 0.040 0.022 13 15 1.000 0.124 0.151 0.035 0.018 14 R7 1.000 0.776 0.061 0.037 15 17 1.000 0.045 0.037 16 R13 1.000 0.775 17 113 1.000 18 R16 19 116 20 Q1 21 Q2 22 Q3 23 Q4 24 Q5 25 Q6 26 Q7 27 Q8 28 Q9 29 Q14 30 Q17 31 Q22 32 Q25 33 R1 34 11 35 R6 36 16 37 R12 38 112 39 R15 40 115 41 Q0 42 Q13 43 Q16 44 Q21 45 Q24 122 CORRELATION MATRIX WET SEASON (CONTINUED) Random Variable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 sin cos t a i r 18 R16 | -0.025 I -0.142 19 116 20 Q1 21 Q2 22 Q3 23 Q4 24 Q5 25 Q6 26 Q7 27 Q8 jOOSO ^0066 ^0064 -0,059 | -0.055 | -0.051 | - Q . M 7 | _Q.04A I -n n « I 28 Q9 29 30 31 Q14 Q17 Q22 32 Q25 33 R1 34 11 35 0.123 0.064 0. 067 0.069 0.074 0. 39 40 41 42 43 44 45 w i n d -0.031 C l O U d 0.121 •Q.n9|-0.047|-0.0501 -0.059 I -0.0681-0.074 I 079 [ 0.082 0.085 0.090 f a -0.043|-0.0191-0.007 | 0.020 I 0. 035 -0.098 -0.109 091 0.091 0.099 0. R2 12 R3 13 R4 14 R5 15 R7 17 R13 113 R16 116 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q14 Q17 Q22 Q25 R1 11 R6 16 R12 112 R15 115 Q0 Q13 Q16 Q21 Q24 -0.040 -0.029 -0.028 -0.022 -0. 0-104 0.044 0.050 0.067 ~~0. -0.082 -0.091 -a096Ua 100 0.095 0.133 0.10 0181 -0.016 | -0.013 -0.0081-0. 098[-0.1091-Q.1221 -0.137 | -0.1431 -Of 023 | 0.048 0.491 0.415 I 0.1 0. 0.027 0.028 0.007 0.01 0.002 0.023 0.012 0.024 0. 0.062 077 0.085 0.093 0.1 0091-0.0031 0.014 0.03lTa 02 0.067 0.052 Ta 0 0 | 0-104 0.107 0.113 I 0.114 I 049 0.064 -0.085 -0.C 0.058 0.472 0.422 I 0. 070 | 0.073 0.049 0.02610 0.120 0.114 0.059 O.C 159] 0.093 0.056 0.049 0. 021 0.046 0.059 0. 0036 | 0.309 0.492 0.414 I 0. 046 | 0.039 0.011 0.010 I 0. 050 I 005 | 0.189 | 0.243 I 0034 | 0.318 0.473 0.423 I 0.1 102 | 0067 0.052 0.070 0. 8 | 0-021 0.212 0.311 I 0. S3 I 0091 0.056 0.049 I 0. "69 I 0413 | 0.100 0.065 I 0.052 I 0. 074 0.049 -0.001 T a 0 4 6 I 0039 | -0.005 0.028 0.062 I 0. 043 0.064 0.015 0.232 0.3 026 0.058 0.032 0.075 0.125 I 0017 | 0.220 | 0.318 0.473 I 0.421 I 0.156 I 071 0.073 0.005 0. 043 0.108 0.154 0026 | 0.140 0.210 O .313T0 0089 | 0.056 | 0.051 0.046 I 0.004 I 0. 0014 | 0.135 | 0.218 0.321 I 0.474 I 0. 491 | 0.414 0.101 0.067Ta 011 0045 0.023 0.063 O.C 015 0,045 0.042 0.072 0.088 I 054 | 0.071 0.018 0.022 0. 0.025 0.088 0.110 f a 421 0.156 0.090 0.058 0. 002 | 0.006 0.084 0.117 fai 139 | 0.214 0.317 0.495 0. 050 0.007 0.009 | a 411 | 0.102 0.068 0.021 I 0 0 069 0.017 0.014 0. 0.075 I 0.080 0.010 0.012 ~0. 1.000 [ 0.780 0.035 0.021 0,007 I Q 1 000 0.044 0.044 0.021 ~0~ 39 | 0.224 | 0.328 0.482 0.426 I 0.1 032 0.059 0.055 0.066 039 0.063 0.059 0.072 010 | 0.050 0.050 0.059 0.043 61 026 0.052 0.065 0.088 0.100 I 0.115 I 0 1 4 a I 0.091 0.011 0.010 0. 040 | 0.049 0.040 0.036] 021 0.037 0.041 0.083 0. 0.104 0.084 0.019 0. 089 0.117 0.149 0. 026 0.023 0.007 018 0 012 0.026 0.052 0. 009 0.010 0.023 0.042 "a 160 | 0.063 0.008 0.010 0. 026 0.022 061 0.089 0.329 0. 108 | 0.049 0.020 0.05BT(X 1.000 | 0.613 0 J 6 7 0.265 0. 1.000 0.614 0.367 0. 043 | 0.079 0.332 0.161 f a 202 0.171 0.158 0.118 055 046 J 0.010 I 0.056 I 0.054~ 264 [ 0.202 | 0.172 0.164 0.120 I 0. 0080 | 0.033 0.054 0.080 I 0.051 I 0. 1 000 | 0.616 | 0.368 0.266 0.204 I 0.175 I 0. 052 0.060 0.098 0.061 I 0. 416 0.424 101 0.159 1 000 | 0.616 0.370 I 0.270 I 0. 166 0.050 0.045 0.095 I 0. 206 | 0.175 0.053 0.038 I 0. 077 0.066 0.093 091 0.081 0.052 J.000 0 617 0.372 0.271 0.207 0.077 0.058 0.081 0.101 0.069 0.057 0.048 _ l£ 0 0 |a616_ _ 0 £ 3 _ _0273_ _O084_ _0_056_ _O060_ W T^2~U^] ^00__0^_O378__0_121 0.057 0.067 0.096 "0I4I 1 000 [0.623 0.169 0.081_ 0050 0.090 ~0~028 1 000 0.176 0.086 0.040 0.061 Q.Q32 | Q. 1.000 |a280 0^ 86 0.065 0.008 "a 1.000 | 0.182 0.083 0.041 ~0 1.000 0.263 0.023 ~OC 0.040 0.012 1.014 015 .054 045 1 000 | 0.033 [ 0.020 | 1 000 | 0.767 I 1.000 123 CORRELATION MATRIX WET SEASON (CONTINUED) Random Variable 35 R6 36 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 37 R12 38 112 39 R15 40 115 41 Q0 s i n c o s t a i r -0.069 -0.086 -0.0421 -0.064 0.133 0.110 0.135 I 0.120 -0.100 -0.079 -0.1301 -0.105 W i n d -0.063 -0.068 I -0.0401 -0.052 C l O U d | 0.073 | 0.069 | 0.103 | 0.094 R2 12 R3 13 R4 14 R5 15 R7 17 R13 113 R16 116 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q14 Q17 Q22 Q25 R1 11 R6 16 R12 112 R15 115 QO Q13 Q16 Q21 Q24 0.055 0.060 -0.005 -0.007 0.065 0.073 -0.011 -0.01 0.063 0.072 0.026 0.031 0.081 0.085 0.005 0.012 0.077 0.108 0.039 0.022 0.124 0.150 0.033 0.018 0.187 0.233 0.030 0.025 0.244 0.320 0.023 0.013 0.186 0.247 0.038 0.045 0.233 0.323 0.029 0.040 0.033 0.025 0.192 0.253 0.026 0.016 0.237 0.327 -0.017 -0.013 0.051 0.064 -0.019 -0.020 0.067 0.082 0.113 0.116 0.014 0.015 0.140 0.136 0.024 0.021 0.212 0.220 0.051 0.037 0.314 0.324 0.063 0.039 0.491 0.478 0.086 0.080 0.413 0.424 0.096 0.088 0.100 0.159 0.114 0.117 0.067 0.089 0.147 0.147 0.055 0.059 0.225 0.235 0.025 0.011 0.069 0.089 0.007 0.001 0.079 0.058 0.041 0.054 0.004 0.004 0.058 0.062 0.019 0.014 0.032 0.043 0.024 0.033 0.023 0.038 0.013 0.020 1.000 0.774 0.059 0.037 1.000 0.043 0.034 1.000 0.775 1.000 42 Q13 -0.029 0.135 -0.138 -0.031 0.11 0.031 0.01 0.021 0.003 0.023 0.012 -0.016 -0.01 0.036 0.035 0.077 0.122 0.203 0.248 0.020 0.008 0.017 0.013 0.024 0.050 0.062 0.091 0.105 0.501 0.073 0.022 0.007 0.018 0.020 0.020 -0.001 0.063 0.076 .000 43 Q16 -0.053 0.124 -0.1 -0.042 0.107 0.037 0.037 0.025 0.027 0.015 •0.01 -0.020 0.022 0.020 0.102 0.145 0.262 0.336 0.043 0.022 0.008 0.009 0.021 0.039 0.043 0.081 0.094 0.487 0.091 0.013 0.009 0.046 0.055 0.021 0.003 0.070 0.081 0.779 44 Q21 -0.068 0.061 -0.043 -0.029 0.038 0.308 0.31 0.21 0.140 0.135 0.112 0.114 0.078 0.01 0.023 0.043 0.044 0.615 0.368 0.265 0.202 0.158 0.116 0.079 0.042 0.078 0.076 0.489 0.468 0.090 1.084 1.011 0.033 .043 .000 -0.025 0.090 0.114 0.000 -0.003 0.005 0.005 0.021 0.008 0.027 0.012 0.041 0.419 0.428 0.214 0.225 0.052 0.052 0.052 0.214 0.625 0.213 0.079 0.021 0.010 0.022 0.010 0.102 0.162 0.326 0.329 0.034 .000 -0.012 0.099 -0.122 0.023 0.11 0.025 0.030 0.020 0.009 -0.002 0.010 0.061 0.065 0.426 0.428 0.060 0.043 0.035 0.054 0.054 0.054 0.080 0.085 0.121 0.378 0.625 0.176 0.080 0.049 0.045 0.008 .009 0.081 .061 0.108 162 0.054 45 46 Q24 124 0.014 0.031 0.163 J.100 0.096 0.041 -0.132 -0.144 -0.172 0.046 0.062 0.086 0.118 0.117 0.159 0.056 0.030 0.039 0.062 0.042 0.054 0.046 0.023 0.059 0.045 0.042 0.062 0.029 0.058 0.043 0.036 0.061 0.055 0.040 0.059 0.050 0.052 0.064 0.051 0.031 0.035 0.019 0.029 0.044 0.040 0.026 0.006 0.082 0.015 0.000 0.279 0.077 0.024 0.145 0.059 0.012 0.061 0.098 0.056 0.062 0.094 0.076 0.065 0.089 0.080 0.036 0.080 0.101 0.036 0.059 0.100 0.076 0.068 0.095 0.275 0.052 0.087 0.063 0.040 0.060 0.061 0.066 0.045 0.048 0.124 0.056 0.211 0.116 0.013 0.024 0.617 0.366 0.081 0.200 0.618 0.337 0.059 0.005 0.033 0.065 0.016 0.038 0.049 0.051 0.029 0.044 0.052 0.046 0.019 0.026 0.190 0.008 0.011 0.372 0.049 0.019 0.049 0.047 0.008 0.003 0.080 | 0.050 0.066 0.086 0.061 0.754 0.181 I 0.081 0.052 .000 0.265 -0.005 1.000 0.062 Appendix III RELAN Results Failure probability results from RELAN (using MCS, FORM, and IS-FORM) that are used to calculate estimates of reliability and resilience. 125 ANNUAL SCENARIO (MCS and FORM) Turbidity MCS MCS Standard Reliability Resilience 2 4.5 0.0311 0.0219 6 0.0893 0.0590 8 0.2625 0.1591 10 0.5282 0.2754 12 0.7568 0.3907 14 0.9019 0.4916 16 0.9687 0.6332 18 0.9924 0.7579 20 Prob. of Prob. of Prob. of Failure Recovery System (Model) (Mode 2) Failure Joint Prob FORM FORM Reliability Resilience 0.9707 0.9431 0.8323 0.6588 0.4535 0.2564 0.1084 0.0302 0.0216 0.0570 0.1681 0.3414 0:5464 0.7433 0.8911 0.9796 0.9669 0.9327 0.9079 0.9013 0.9159 0.9480 0.0127 0.0332 0.0676 0.0923 0.0986 0.0838 0.0515 0.0293 0.0569 0.1677 0.3412 0.5465 0.7436 0.8916 0.9698 0.0131 0.0352 0.0813 0.1402 0.2175 0.3269 0.4750 ANNUAL SCENARIO (MCS and IS-FORM) Turbidity MCS MCS Standard Reliability Resilience 2 4.5 0.0311 0.0219 6 0.0893 0.0590 8 0.2625 0.1591 10 0.5282 0.2754 12 0.7568 0.3907 14 0.9019 0.4916 16 0.9687 0.6332 18 0.9924 0.7579 20 Prob. of Prob. of Prob. of Failure Recovery System (Model) (Mode2) Failure Joint Prob IS-FORM IS-FORM Reliability Resilience 0.9710 0.9114 0.7304 0.4832 0.2453 0.0987 0.0301 0.0067 0.0338 0.0891 0.2655 0.5226 0.7474 0.8940 0.9658 0.9827 0.9528 0.9024 0.9024 0.9180 0.9508 0.9787 0.0221 0.0477 0.0935 0.1034 0.0748 0.0420 0.0172 0.0290 0.0886 0.2696 0.5168 0.7547 0.9013 0.9699 0.9933 0.0228 0.0523 0.1280 0.2139 0.3048 0.4251 0.5713 WET SEASON SCENARIO (MCS and IS-FORM) Turbidity Standard MCS Reliability MCS Resilience Prob. of Failure (Mode 1) Prob. of Recovery (Mode 2) Prob. of System Failure Joint Prob IS-FORM Reliability IS-FORM Resilience 4 0.0083 0.0090 0.9924 0.0078 0.9941 0.0060 0.0076 0.0061 6 0.0537 0.0381 0.9462 0.0583 0.9693 0.0352 0.0538 0.0372 8 0.1966 0.1253 0.8060 0.1996 0.9332 0.0724 0.1940 0.0898 10 0.4486 0.2370 0.5568 0.4416 0.9030 0.0954 0.4432 0.1713 12 0.713075 0.3956 0.2979 0.6980 0.9058 0.0901 0.7021 0.3024 14 0.89139 0.5560 0.1205 0.8840 0.9559 0.0486 0.8795 0.4033 14.5 0.0895 0.9150 0.9672 0.0374 0.9105 0.4172 16 0.969072 0.6444 18 0.991048 0.7502 126 MONTHLY RESULTS (IS-FORM) Turbidity Standard Prob. of Prob. of Prob. of Failure Recovery System (Model) (Mode2) Failure Joint Prob IS-FORM IS-FORM Reliability Resilience OCTOBER 4 6 8 10 12 14 16 18 20 0.9820 0.8868 0.6266 0.3135 0.1066 0.0386 0.0189 0.1131 0.3675 0.6920 0.8974 0.9708 0.9859 0.9301 0.8645 0.8823 0.9398 0.9812 0.0151 0.0697 0.1296 0.1231 0.0642 0.0282 0.0180 0.1132 0.3734 0.6865 0.8934 0.9614 0.0153 0.0787 0.2068 0.3928 0.6025 0.7307 NOVEMBER 4 6 9 10 12 14 16 18 20 0.9886 0.9618 0.7814 0.4344 0.1638 0.0386 0.0125 0.0386 0.2127 0.5598 0.8516 0.9708 0.9897 0.9694 0.8804 0.8427 0.9185 0.9812 0.0114 0.0310 0.1137 0.1515 0.0969 0.0282 0.0114 0.0382 0.2186 0.5656 0.8362 0.9614 0.0116 0.0322 0.1455 0.3487 0.5915 0.7307 DECEMBER 4 6 8.5 10 12 14 16 18 20 0.9652 0.8518 0.5560 0.2407 0.0828 0.0208 0.0336 0.1387 0.4454 0.7716 0.9284 0.9800 0.9740 0.9103 0.8617 0.9020 0.9569 0.9842 0.0247 0.0802 0.1397 0.1103 0.0543 0.0166 0.0348 0.1482 0.4440 0.7593 0.9172 0.9792 0.0256 0.0942 0.2513 0.4583 0.6556 0.7970 JANUARY 4 7 8 10 12 14 16 18 20 0.9570 0.8860 0.6116 0.2840 0.1018 0.0218 0.0418 0.1072 0.3880 0.7248 0.9144 0.9796 0.9691 0.9291 0.8683 0.8908 0.9510 0.9839 0.0297 0.0641 0.1314 0.1180 0.0652 0.0175 0.0430 0.1140 0.3884 0.7160 0.8982 0.9782 0.0310 0.0723 0.2148 0.4155 0.6405 0.8021 FEBRUARY 4 6 8 10 12 14 0.9842 0.8668 0.5630 0.2238 0.0510 0.0152 0.1250 0.4409 0.7760 0.9528 0.9868 0.9197 0.8471 0.8827 0.9642 0.0126 0.0721 0.1568 0.1171 0.0396 0.0158 0.1332 0.4370 0.7762 0.9490 0.0128 0.0832 0.2785 0.5234 0.7767 127 Turbidity Standard Prob. of Failure (Mode 1) Prob. of Recovery (Mode 2) Prob. of System Failure Joint Prob IS-FORM IS-FORM Reliability Resilience FEBRUAF :Y (con't.) 15 18 20 0.0217 0.9802 0.9837 0.0182 0.9783 0.8388 MARCH A H 5 0.9902 0.0101 0.9908 0.0095 0.0098 0.0096 6 0.9598 0.0371 0.9656 0.0313 0.0402 0.0326 8 0.7580 0.2345 0.8598 0.1327 0.2420 0.1750 10 0.3804 0.6152 0.8314 0.1642 0.6196 0.4316 12 0.1028 0.9002 0.9307 0.0723 0.8972 0.7034 14 16 0.0229 0.9846 0.9872 0.0203 0.9771 0.8872 18 20 APRIL 4 0.9950 0.0047 0.9952 0.0045 0.0050 0.0045 6 0.9346 0.0652 0.9467 0.0531 0.0654 0.0568 8 0.6628 0.3331 0.8253 0.1707 0.3372 0.2575 10 0.2639 0.7338 0.8476 0.1501 0.7361 0.5688 12 0.0601 0.9470 0.9575 0.0496 0.9399 0.8245 13 16 0.0247 0.9788 0.9815 0.0220 0.9753 0.8927 18 20 128 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0064040/manifest

Comment

Related Items