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A control system framework for simulating forest landscape management Hafer, Mark A. 1997

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A CONTROL SYSTEM FRAMEWORK FOR SIMULATING FOREST LANDSCAPE MANAGEMENT by Mark A. Hafer B.Eng., Carleton University, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department Of Forest Resources Management) We accept this thesis as conforming to ^jhe required ^tandard  THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1997 © MARK A. HAFER, 1997  In  presenting  this  degree at the  thesis  in  partial  University of  fulfilment  of  of  department  this or  thesis for by  his  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  her  publication of this thesis for  representatives.  Library shall make  It  financial gain shall not  is  granted  by the  understood  of  ftrTlArJ  ,  fo&HjJlCAd •  The University of British Columbia Vancouver, Canada  Date JUAU  DE-6 (2/88)  ^  , m  i  .  it  that  head of copying  my or  be allowed without my written  permission.  Department  an advanced  agree that permission for extensive  scholarly purposes may be  or  for  M c V ^ ^ i / Y ^  ABSTRACT  Traditional forest planning models, based on mathematical programming techniques for constrained optimization, have failed to adequately address the new realities of managing forest landscapes so as to sustain in perpetuity a full range of intrinsic values and human benefits. This thesis proposes a general simulation modelling framework for the analysis and management of forest landscapes. The framework, based on state space techniques for modelling of complex dynamic systems, addresses many of the requirements of modern forest planning analysis. Application of the general framework is demonstrated in the development of a specific simulation modelfordesigning spatially explicit harvest schedules that balance biodiversity andtimberproduction objectives. A state space model is developed to predict the dynamics stand level structure over a landscape represented in raster format. This portion of the model is shown to provide realistic predictions of the temporal development of stand structure, and demonstrates the potential of the state spaceformatforrepresenting the physical attributes and complex dynamics offorestlandscapes. An heuristic control procedure based on simple multi-criteria decision making techniques is developed to determine the spatial and temporal sequence of management actions according to their combined effect on the objectives for the landscape. To demonstrate and evaluate the feasibility of the proposed approach, the completed model is used to simulate the management of a 3850 hectare landscape over 150 years using a 5 yeartimestep. Scenarios are presented to show achieved levels of serai stage, patch size, andtimberoutputs under different weightings of the corresponding management objectives. The model  ii  was able to maintain the system output within the desired limits when harvest volume was the only specified management objective. The desired harvest levels were achieved primarily through clearcut and heavy thinning activities. In the case of managing for serai stage, patch size, and timber objectives the model was an effective tool for exploring the trade-offs required to find a balance between all seven objectives. Serai stage objectives were fully met, although at the expense of violating the upper bound on harvest levels in several simulation periods. Patch size objectives were met in some periods, and were significantly improved in all other periods compared to the scenario involving only the volume objective. This broad mix of landscape outputs was achieved using a much wider range of harvest treatment intensities than in the single objective case. The case study demonstrates that it is feasible to represent a detailed description of stand level dynamics in a spatially explicit forest level model using readily available desktop computing systems. Such a simulation model is intended to have application as a tool for designing alternative management strategies to provide input to an iterative multi-objective planning process.  111  TABLE OF CONTENTS  ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES v LIST OF TABLES vi ACKNOWLEDGMENTS vii INTRODUCTION 1 FOREST PLANNING MODELS: PAST, PRESENT AND FUTURE 6 THE DYNAMICS OF FOREST STRUCTURE: A FRAMEWORK FOR SIMULATION-MODELLING; 14 Introduction 14 A general state space framework 15 A state space model offorestlandscape dynamics 20 Stage Progression Rate ...23 Stage Survival Rate 25 Regeneration Rate 26 Putting it all together 27 A MULTI-OBJECTIVE APPROACH TO SIMULATING FOREST MANAGEMENT DECISIONS ...28 Introduction 28 A multi-criteria management decision model 28 Management Decision Alternatives 30 Management Decision Criteria 31 EXPLORING THE MODEL ....40 Introduction 40 Evaluating the Landscape State submodel 40 Managing For Biodiversity And Timber: A Case Study 48 Results 53 Discussion 63 CONCLUSION 74 LITERATURE CITED 78 APPENDIX 1 88  iv  LIST OF FIGURES  Number  Page  Figure 1: Landscape management system 15 Figure 2: Performance criterion for a single system output variable with minimum and maximum desired levels 38 Figure 3: North aspect, even-aged stocking, no disturbance 43 Figure 4: South aspect, even-aged stocking, no disturbance 44 Figure 5: North aspect, even-aged stocking, light thinning 46 Figure 6: North aspect, even-aged stocking, heavy thinning 47 Figure 7: Initial diameter distributions for each serai stage 51 Figure 8: Initial serai stage distributionforhypothetical landscape 52 Figure 9: Scenario 1 harvest volume time series— Volume objective only 54 Figure 10: Scenario 1 serai stage time series— Volume objective only 55 Figure 11: Scenario 1 patch size time series— Volume objective only 56 Figure 12: Scenario 1 scheduled treatments 57 Figure 13: Scenario 2 harvest volume time series— Volume and serai stage objectives 59 Figure 14: Scenario 2 serai stage time series— Volume and serai stage objectives 60 Figure 15: Scenario 2 patch size time series— Volume and serai stage objectives61 Figure 16: Scenario 2 scheduled treatments 62 Figure 17: Scenario 3 harvest volume time series— Volume, serai stage and patch size objectives 64 Figure 18: Scenario 3 serai stage time series— Volume, serai stage and patch size objectives 65 Figure 19: Scenario 3 patch size time series— Volume, serai stage and patch size objectives  66  Figure 20: Scenario 3 scheduled treatments 67 Figure 21: Harvest volume time series for modified criteria evaluationVolume, serai stage and patch size objectives 70 Figure 22: Serai stage time series for modified criteria evaluation— Volume, serai stage and patch size objectives 71 Figure 23: Patch size time series for modified criteria evaluation— Volume, serai stage and patch size objectives 72 Figure 24: Scheduled treatmentsformodified criteria evaluation 73  v  LIST OF TABLES  Table 1: Serai Stage Distribution Goals Table 2: Patch Size Distribution Goals Table 3: Growth model comparison matrix and parameter values Table 4: Treatment regime Table 5: Weights for MCDM Control test cases  vi  34 35 42 50 50  ACKNOWLEDGMENTS  I wish to thank the members of my committee, Drs A.F. Howard, J.D. Nelson, D.E. Tait, and W. Klenner, for their encouragement and intellectual input. In particular, I thank my supervisor, Dr. Howard,forproviding the financial supportforthis project, andforallowing me the freedom to pursue and develop my own ideas. I also thank Glenn Sutherland, David Daust, Bob MacDonald, Ian Cameron and Jordan Tanz for their invaluable input and assistance. Most of all, I wish to thank my partner in life, Terri Wilmon, without whose moral support, patience and understanding this work would not have been possible.  vu  Chapter  1  INTRODUCTION  Contemporary forest resource managers must plan their activities in a much different context than in the past. Traditionally, long-term forest management planning has been concerned primarily with predicting a maximum rate of wood volume extraction that is expected to be sustainable according to various criteria of economic optimality (Wykoff et al., 1982). Analyses of this sort focus on predicting the expected production of a single output (timber) from the forest. Theforestis generally modelled as a deterministic linear system (McCarthy and Burgman, 1995), neglecting stochastic natural disturbance events. The dynamics of the future growing stock volume in response to prescribed management actions is characterized by a predetermined set of yield tables which express volume as a function of stand age for different combinations of tree species, stocking level, site quality, and prescribed treatment (e.g. Stage et al, 1988; BCMOF, 1992). Present public attitudes towardsforestmanagement reflect a growing concern about the impacts of harvesting activity on biological diversity, water quality, habitat value, community stability, and the like. These concerns are relevant to forests throughout the world, from tropical to temperate zones. The issues are, to greater or lesser degrees, new toforestry,and have made the planning offorestharvesting activities a far more complex problem involving multiple and often conflicting objectives. As competing demands of the landscape escalate it is important that management decisions be derived from a much broader perspective. This requires a shift in both the conceptual and analytical  1  models applied in the planning process so that emphasis is placed on the longterm dynamics and structural condition of the forest created by proposed harvesting actions, and that a wider range of potential outputs derived from those actions are given consideration. As Hansen et al. (1991) have put it, "the challenge now is to design and effectively manage such multi-purpose landscapes". A great deal of research and experimentation has occurred in recent years as natural resource managers attempt to come to terms with the new realities of forest management. Much of this work has concentrated on the problem of sustaining the natural biological diversity of forest landscapes while also providing an adequate future supply of wood volume (see Burton et al., 1992), and a new paradigm of ecosystem-based landscape management appears to be emerging as the dominant conceptual framework. Many different approaches to implementing ecosystem-based management plans have been proposed and documented in the literature (e.g. Mladenoff et al., 1994; Diaz and Apostol, 1992; Booth et al, 1993; Daust, 1994), yet a central theme in all of them is the recognition of forest landscapes as ecologically diverse systems composed of myriad abiotic and biotic structures distributed (and in the case of many wildlife species, mobile) throughout three spatial dimensions (e.g. latitude, longitude, and elevation), and which are interconnected and shaped by processes that operate across a wide range of spatial and temporal scales (Holling, 1986; Holling, 1992). Disturbance processes, both natural and anthropogenic, interact at a range of spatial and temporal scales, and are an intrinsic part of the dynamic behaviour of these complex systems (Franklin and Forman, 1987; Hansen et al., 1991). Predictive models used to develop and evaluate long term harvesting strategies must therefore characterize a wider range of forest structures and processes than simply the production of wood volume as a function of stand age.  2  In British Columbia, theforestland base occupies more than half of the province by area, roughly 51.8 million hectares (Harding, 1994), most of which is managed by the British Columbia Ministry of Forests (BCMOF). British Columbia'sforestscontain 49.7% of the softwood timber volume in Canada (Forestry Canada, 1990), and they represent an incredible diversity of ecosystems that are world famous for their natural beauty. Consequently, and in response to strong local, national and international public pressure, British Columbia has made significant steps in recent years to move toward an ecosystem based approach to management of the provincialforest.The results of this effort over the last 5 years are contained in the recently enacted Forest Practices Code (FPC) (BCMOF, 1995a). The FPC represents an attempt to foster a more balanced and sustainable approach toforestuse by  • managingforeststo meet present needs without compromising the needs of future generations, • providing stewardship offorestsbased on an ethic of respect for the land, • balancing productive, spiritual, ecological and recreational values of forests to meet the economic and cultural needs of peoples and communities, including First Nations, • conserving biological diversity, soil, water, fish, wildlife, scenic diversity and otherforestresources, and • restoring damaged ecologies; ", (preamble, BCMOF, 1995a). As part of the FPC, numerous Guidebooks have been developed to provide information and recommendations to enable resource decision makers in the province to "exercise their professional judgment in developing site specific management strategies and prescriptions to accommodate resource management objectives specified in higher level plans" (pg. iii, BCMOF 1995b).  3  Recommendations for conserving the natural biological diversity of BC forests are addressed in several of the guidebooks, but primarily in the Biodiversity Guidebook (BCMOF, 1995b). The recommendations are both broad and restrictive, but are intended as aflexibleguide to designing rational management plans to achieve the multiple objectives of sustaining biodiversity and timber harvest operations on the managed landscape. In order to successfully apply the FPC guidelines for biodiversity management when developing harvesting plans, analysts need computer based tools to help expedite the process of identifying alternative management scenarios (in the form of spatially explicit schedules of future harvest actions) that achieve some degree of balance between the potentially conflicting objectives of timber production and biodiversity conservation. Such tools need to permit a better representation of the dynamic and spatial complexity of forest landscape structure than existing models do, and should also allow the analyst to examine the trade-offs between competing objectives that will inevitably be required to design sound, responsible, and equitable management plans. This thesis presents an analytical framework for landscape management simulation that has the potential to provide a foundation for developing computer based models mat meet these requirements. The objective of the thesis is to explore and demonstrate the potential of the proposed modelling framework. State-space control system analysis techniques are used to provide a rich description of stand level attributes that can be feasibly integrated into spatially explicit landscape level planning models. Multi-criteria decision making (MCDM) techniques are used to provide an heuristic control algorithm for determining the spatial and temporal sequence of management actions on the landscape. The benefits of the proposed framework include the following: the dynamics of stand and landscape structural attributes can be predicted  4  across a range of spatial scales; stochastic disturbance processes can be readily incorporated into the framework (although this has not been explicitly pursued in the thesis); landscape attributes, such as serai stage, can be defined on the basis of stand structure rather than stand age; a full spectrum of management action intensities can be explored; the spatial resolution of the model is not constrained by the initial definition of harvest units; multiple objectives for the desired state of the managed forest can be explicitly represented. Chapter 2 reviews the literature on past and present analytical techniques used in forest planning models, and discusses some of the features required of planning models in order to address modern forest management issues. The state space framework is presented in general terms in chapter 3, and then developed for the specific application of simulating the long-term dynamics of forest stand structure at both stand and landscape scales. Chapter 4 extends the framework to include a MCDM-based feedback control model for multiple-objective harvest scheduling decisions, and develops specific indicators for measuring the system performance in terms of timber harvest and biodiversity objectives. In chapter 5 the model is applied to a hypothetical landscape (characteristic of interior Douglas-fir forests in the Kamloops Forest Region of the BCMOF). The purpose of the analysis in this chapter is to demonstrate and explore the feasibility of the framework itself, rather than to pursue a practical management design or analysis for the model landscape. Chapter 6 provides a brief summation and concluding remarks.  5  Chapter  2  FOREST PLANNING MODELS: PAST, PRESENT AND FUTURE  Traditional forest management models are based on a view of the forest as a production system to be managed for die maximum sustainable flow of wood fibre. The concept is an old one, with its origins in the work of Faustman in the last century (Faustman, 1968 as cited in Hilborn et al., 1995). From the late 1940s into the early part of the 1970s, the objective fortimberactivity planning was to set a maximum harvest level during the conversion from old-growth to plantation forests, and thereafter to harvest timber at the long run sustained yield (Johnson, 1981; Williams, 1990). Flarvest rate determination was made by manual calculation using simple mathematical formulae (Hann and Brodie, 1981; Williams, 1990). During the 1970s, growing awareness of and demand for non-timber values from the forest, and recognition that the landbase was limited prompted changes in the focus of forest management planning in North America. Newly legislated responsibilities to plan for non-timber resources in addition to the traditionalfibreflow concerns, combined with the advent of computer technology, led to the development of computer-based models for forest management planning. Forest planning has traditionally taken place at three distinct levels (Murray and Church, 1993). At the strategic planning level, forest-wide goals are developed for harvest levels, wildlife habitat and other broad scale concerns over periods of many decades. The tactical planning level is concerned with the georeferenced implementation of activities specified in strategic plans, while operational planning deals primarily with the detailed implementation of  6  tactical plans at spatial scales of individual cutblocks and access roads. More recently, recognition of the impacts of forest management activities across scales of time and space has blurred the distinctions between these levels of forest planning (Tanz, 1991). The most widely used computer-based forest planning models at all levels of the planning process are based on linear programming (LP) (O'Hara et al, 1989), a mathematical programming (MP) technique for the constrained optimization of linear systems. One of the earliest published LP-based forest planning models was Timber RAM (Navon, 1971 as cited in O'Hara et al., 1989), and subsequent refinements of the technique produced other models such as MAXIMILLION (Ware and Clutter 1971), MUSYC (Johnson and Jones, 1979), and FORPLAN (Potter et al., 1979; Stuart and Johnson, 1985). Johnson and Scheurman (1977, as cited in Davis and Johnson, 1987) classify LP techniques for forest activity scheduling problems into model I and model II categories, based on the way in which the decision variables are defined. In general, however, LP methods divide the forest into strata of similar timber types, and then allocate continuous fractions of each strata to a particular harvesting regime so as to maximize the chosen objective function (typically harvested wood volume or net present value) subject to a set of constraints. These techniques are well established and documented elsewhere (e.g. Johnson and Scheurman, 1977; Iverson and Alston, 1986). As the focus of forest management planning has evolved to include a wider range of biological and social factors, it has become necessary for planners to be as concerned about where and how to apply harvesting actions as they had previously been about how much and how fast to harvest (Johnson, 1981). LP techniques have proven to be limited in their ability to solve such realistic forest management problems involving multiple objectives and the spatial  7  allocation of management actions. The aggregation of forested hectares into timber strata that is necessary to "shoehorn" (Brodie and Sessions, 1991) the activity scheduling problem into the LP framework, along with the use of continuous decision variables to represent the hectares of each strata precludes the analysis or control of the geographic location and juxtaposition of activities within the forest (O'Hara et al., 1989; Snyder and ReVelle, 1996). An attempt was made to address this problem in version 2 of FORPLAN by means of the 'coordinated allocation of choices', however this approach results in a problem of unmanageable size and limited choices (O'Hara et al., 1989). Other attempts have been made to address the spatially constrained operations scheduling problem within the LP framework (e.g. Mealey et al., 1982; Thompson et al. 1973 as cited in O'Hara et al., 1989), however these approaches have not succeeded in producing integral solutions. Numerous other problems with the traditional LP formulations of forest management models have been noted in the literature: their linearity and deterministic nature (Bare and Mendoza, 1990; Mendoza and Sprouse, 1989); the use of a single objective function and "the common use of LP to optimize a given system rather than to design an optimal solution" (Bare and Mendoza, 1990); the drastic simplifications of the problem required to obtain a solution, and the large size of problem formulations and solution spaces (Brack and Marshall, 1992; Brack and Marshall, 1996); their inability to represent the dynamic complexity and diversity of forest ecosystems (Mendoza and Sprouse, 1989); their inherent unsuitability for application to systems in which parameter values, decision consequences and management objectives may change as the system behaviour progresses (Casti, 1983); they are extraordinarily complicated, and require a high degree of specialized knowledge to be used to their full potential (Tanz and Howard, 1991). The emphasis on viewing harvest scheduling as an optimization problem has also been criticized. Optimization is a mathematical concept from the analysis  8  of functions. In practical problems such as forest management, its primary value is that it can provide a theoretical baseline against which to compare the performance of various alternative strategies. However, the many simplifications that are required to formulate a problem so that an optimal solution can be obtained mean that the solution is unlikely to be fully realizable. Yet it is all too easyfordecision makers to accept the predicted optimal solution as a realistic target to be achieved. Furthermore, in managing forest landscapes for the many values of importance to stakeholders and the general public, there are likely to be many alternative management policies which can provide similar levels offorestperformance that would be satisfactory to all concerned, but these alternatives are not identified in an optimization analysis (Tanz and Howard, 1991). Alston and Iverson (1987, as cited in Mendoza and Sprouse, 1989) suggest that modelling should be used as a tool for developing a better understanding of the question rather than to provide the answer (see also Korzukhin et al, 1996, and Walters, 1986). Various alternative modelling approaches have been developed more recently in an attempt to overcome some of the limitations noted above. Mendoza and Sprouse (1989) applied fuzzy systems theory in an LP framework to generate satisfactory solution alternatives for multiple objective planning problems, and Bare and Mendoza (1990) demonstrated a variation on the LP theme, called de novo  programming, as a meansforoptimizing multi-objective forest planning  problems in a non-spatial context. Holland et al. (1994) incorporated one aspect of biological diversity, the diversity of forest stand structure, into a standard LP harvest scheduling framework in order to evaluate the tradeoffs between timber production and the maintenance of vegetative structural diversity. Their approach was not truly a multi-objective optimization technique, however, since each LP solution they generated used only a single factor (one of volume harvested, net present value, basal area diversity, species  9  diversity, vertical crown diversity, and the arithmetic average of the latter three factors) in the objective function. Goal programming is another variation of the standard LP method that has been applied to non-spatial multiple objective problems in forest planning (e.g. Field et al., 1980 and Kao and Brodie, 1979, both as cited in Walker, 1985). Thefirstresearchers to attempt spatially explicit harvest scheduling were Kirby et al. (1980, as cited in Brodie and Sessions, 1991), who applied mixed integer programming (MIP) to maintain the spatial integrity of harvest units (see also Bare et al. 1984 as cited in O'Hara et al., 1989). Decision variables are defined as discrete and indivisible forest stands, and the spatial considerations are expressed as constraints on the adjacency of recently harvested units. A great deal of research effort has since gone into efficiently expressing these spatial and temporal constraints on harvest unit adjacency (see Snyder and ReVelle, 1996). It has been found, however, that integer programming (IP) is only suitable for small geographic areas since the technique requires consideration of all possible combinations of integer variables, making the solution of even modestly large problems cumbersome (Arthaud and Rose, 1996; Brodie and Sessions, 1991). Standard IP/MIP techniques are spatially limited by the preservation of the initial stand definitions, and are unable to represent the nonlinear responses of non-timber resources to the spatio-temporal configuration of the scheduled harvest actions (Hof and Joyce, 1992). IP/MIP techniques also suffer from many of the other problems discussed above in association with traditional LP techniques. Hof and Joyce and Hof et al. published a series of papers (Hof and Joyce, 1992; Hof and Joyce, 1993: Hof et al., 1994) exploring alternative IP formulations for the problem of determining patterns of harvest activities that are spatially and temporally optimal with respect to several wildlife habitat criteria. Thefinalpaper in this series made use of a raster representation of the landbase (rather than polygonal), thus  10  removing the problem of the preservation of initial stand definitions cited above. In addition, they incorporated non-linearities into the objective function in a piece-wise linear fashion, and introduced a probabilistic treatment of habitat connectivity thus addressing some of the criticisms of MP techniques discussed above. The authors acknowledge, however, that their work was strictly an exploratory effort that required many simplifying assumptions in order to produce a tractable problem. In contrast to MP based techniques, many heuristic methodsforspatially explicit activity scheduling have been developed more recently. These procedures generally attempt to find integer solutions that provide high, although not necessarily optimal, objective function values for spatially constrained scheduling problems. Brodie and Sessions (1991) classify and review many of these techniques: examples are Monte Carlo integer programming (MOP) (Nelson and Brodie, 1990), the scheduling and network analysis program (SNAP) developed by Sessions and Sessions (1990, as cited in Brodie and Sessions, 1991), and the spatially constrained resource allocation model (SCRAM) of O'Hara et al. (1989). Other heuristic techniques have been applied to forest management planning problems in which the basic scheme is ' to improve upon an initial feasible activity schedule by exchanging unscheduled harvest units for scheduled harvest units. Examples of this approach are Interchange (Murray and Church, 1995); Tabu Search (Murray and Church, 1995), and Simulated Annealing (Lockwood and Moore, 1993; Murray and Church, 1995). Finally, simulation techniques have been applied to forest planning problems, where satisficing schedules of management activities are determined by projecting timber inventories and scheduling unitsforharvest sequentially through time. This sort of approach has been applied in both stratified (Wang et al., 1987; Tanz et al, 1990, both as cited in Tanz, 1991) and spatially explicit contexts (Nelson et al, 1993).  11  The objectivesforforestmanagement planning are much broader in scope than they once were, encompassing many conflicting factors across a wide range of spatial and temporal scales. Even short term, site specific operational plans may have long-term consequences at large spatial scales, and they must be developed and evaluated with this in mind. In order to be useful as decision support systems to resource managers dealing with the complexity of issues and interests involved in contemporary land management, forest planning models need to possess certain features that have not yet been fully realized in the techniques reviewed above. In order to successfully manage forests for the production of both wood fibre and non-timber values like biodiversity, it is likely that a wider range of harvesting and silvicultural techniques will be required (Hann and Bare, 1979; BCMOF, 1995b). Forest planning models must be able to support this by allowing for the explicit representation of a broad spectrum of treatment intensities. They must also provide mechanisms for specifying the desired system performance in terms of multiple design objectives, and for projecting the values of the variables required to measure system performance against those objectives. Resource managers need to consider far more than simply the distribution of timber volume across a given landscape. The planning process now requires that an equivalent emphasis be placed on the spatial and structural configuration of the living forest as it relates to wildlife habitat, scenic beauty, recreational opportunities, and hydrological processes (BCMOF, 1995a). Thus, planning models must be designed to permit a more comprehensive description of the structure and dynamics of both forest and landscape across a range of spatial and temporal scales.  12  Natural disturbance processes are now recognized as a necessary and intrinsic component of ecosystem dynamics (Attiwill, 1994; Li and Apps, 1995; Holling, 1986; Holling, 1992; Sprugel and Bormann, 1981; Denslow, 1985). Management activities are an additional disturbance process that is superimposed on the natural disturbance regime (even though the latter is frequently modified through various planned suppression programs), and the cumulative effects have a substantial impact on the structure and dynamics of wildlife, vegetation and landscapes across a range of spatial and temporal scales (Hansen at al., 1991). Consequently, natural disturbance processes should not be ignored in the design and evaluation of alternative management strategies. Modelling techniques should therefore possess an analytical structure in which the impact of stochastic disturbances on the dynamics offorestand landscape structure may be represented. The preceding discussion is by no means a comprehensive list of the characteristics requiredformodels to successfully address contemporary forest planning problems. However, the issues raised have been the primary motivators underlying the modelling framework developed in the following chapters.  13  Chapter  3  THE DYNAMICS OF FOREST STRUCTURE: A FRAMEWORK FOR SIMULATION MODELLING  Introduction The basic management question relatingtomanipulation of the forest cover is "what treatment should be applied to each stand each year to best meet the objectives for the forest?", where treatment is understood to include inaction as well as action (Hann and Brodie, 1981). Traditional approaches to forest yield forecasting and harvest scheduling are often based on simple growth curves relating the development of stand or forest level variables as a function of age, for a specific set of site conditions and management actions. These are pre-determined representations of future forest development which were adequate at a time when management planning was primarily interested in timber production. However, achieving multiple objectives for a spectrum of forest related attributes with different intrinsic values requires more intensive management of the landscape not contained in protected natural areas. This in turn means that models developed for prediction and assessment of long term harvest strategies must embody the potential paths of future forest development under a much wider range of conditions and circumstances. A simple illustration of the managed forest landscape as a feedback control system is shown in Figure 1. The dynamic behaviour of the intrinsic structures and processes that make up the natural landscape system are embodied in the box labeled 'Landscape State Transition Process'. The landscape state is modified by other processes such as natural and anthropogenic disturbances,  14  and the system produces certain outputs over time. These outputs need not be outputs in the extractive sense of the word; for example, they may also represent the quantity of certain intrinsic structures that remain within the system. The 'Management Control Process' embodies a description of the desired state of the system and its outputs, and a set of permissible management interventions that may be applied to the system. It applies knowledge of the present system state obtained via the feedback loop to determine the management intervention that should be applied at any point in time so as to drive the system toward its desired state.  Inputs—  Landscape State Transition Process  -• Outputs  Management Control Process  Figure 1: Landscape management system.  Figure 1 serves two purposes. First, it emphasizes that management actions are an integral part of the overall system dynamics.  Second, it serves as a  conceptual framework for the development of models of managed forest landscapes. The details of the 'Landscape State Transition Process' box in Figure 1 are developed in the remainder of this chapter. The 'Management Control Process' box is developed in the next chapter.  A general state space framework The model developed in this thesis is loosely based on the concepts of state space systems analysis (Garcia, 1994). There are several advantages to adopting  15  such a framework for landscape dynamics model development. Systems analysis and modelling is well suited to describing the dynamics of complex non-linear systems (Garcia, 1994; Casti, 1983). The techniques are well developed, with advanced methods and procedures that go far beyond those applied here. These methods have found successful application in a variety of fields, including many related to the study and management of natural systems (e.g. global climate models, hydrological models, pollution dispersal, wildlife population analysis, stand growth projection, economic analysis). The systems analysis method also introduces a certain level of mathematical rigor to the process of expressing and analyzing multiple resource problems. It is explicit in terms of which aspects of the system are represented in the model, and which are not. It is useful to start by identifying how space and time are characterized in the following development. Since the model will ultimately be implemented as a computer simulation, it is convenient to adopt a discrete-space, discrete-time formulation. To represent the spatial distribution of landscape attributes, the area of interest is divided up into a regular grid of cells of equal area. This is analogous to the raster data representation used in many geographic information systems (GIS) (Cromley, 1992), and thus is appropriate for digital implementation. It also has the advantage of permitting a more accurate description of continuously distributed landscape attributes (e.g. topography) than does a polygonal representation based on planned harvest units. In principle, the cells could be of any regular shape, but for simplicity I assume they are square , of dimension Ax-. The landscape attributes that we choose to 1  i I also ignore any spatial effects introduced by die curvature of the earth's surface, and assume that the landscape is projected onto a plane cartesian coordinate system, with elevation expressed as an attribute of each cell. The magnitude of these neglected effects will of course depend on the spatial extent of the landscape being modelled. Furthermore, since most raster GIS products facilitate the correction of spatial data sets to account for such factors, this assumption should not be limiting.  16  incorporate in the model are represented within each cell by a set (or vector) of N numbers (a„ i—1..N) that each denote the average value of a given attribute  for the entire cell. It is not possible to discern the spatial distribution of attributes within a cell . This vector of attribute values represents the current 2  state of that portion of the landscape contained within a given cell. The state of the complete landscape system is thus given by the entire collection of such state vectors for every cell in the two dimensional grid. This is like a photograph of the landscape: it is a static representation of the system at a specific instant in time. Other measures of landscape state, at equivalent or aggregated scales, may be derived from this cellular-level description. To express the dynamic behaviour of the forest, a system of equations is developed to predict the level of each attribute a, in a cell at time t+ A/ as a function of the various attribute levels in that cell at time t, and of the inputs to and outputs from the cell that occur during the intervening interval of time A/. For each attribute this can be written in a general form such as (' + 0 =  (0 + g* i (0 + • • • + &M N (0 + r it) a  a  t  [1]  where g is a coefficient expressing the influence of attribute a. on the v  trajectory of attribute a , r (t) represents the net input of attribute i from t  i  external sources (this could include inputs and outputs of natural and anthropogenic origin), and the time step A,? has been set to 1 for notational clarity.  2  Where greater spatial resolution is required, a smaller cell size may be adopted. Multi-scale approaches to spatial dynamics analysis can also be used, in which areas of exceptional interest are examined at a finer level of resolution, but within the context of boundary conditions derived from coarse resolution analysis of a larger encompassing landscape.  17  The system of N equations of the form given by equation 1 describes the state dynamics for a cell, and can be written in matrix notation as a(/ +1) = Ga(0 + r(t) where  a(t) = [a  l  a  2  ••• a,  •••  a  N  ]  is the column vector of N state variables at time £,  gi  G=  gi  g IN  ga  g NN  gNl  is the matrix of coefficients describing the interdependence of the various state variables (often called a state transition matrix); and  is the column vector of net inputs (input - output) to each state variable. In a typical state space formulation of system dynamics, this state transition equation is augmented by an output function to provide a general analytical description of the temporal interdependence of landscape state variables and outputs for that portion of the landscape area represented by a single cell in the spatial grid. The general form of state space model for a cell is thus  18  a(t +1) = Ga(f) + r(t)  [2]  b(t) = Fa(f) where  f  is the column vector of M output variables at time t, and fm  fu  F=  MN  Ml  is the matrix of coefficients describing the dependence of the M output variables on the N state variables. The complete description of the landscape system thus consists of a two-dimensional array (the discrete-space grid of individual landscape cells) of matrix equations of the form of equation 2. This approach to modelling spatial dynamics is very similar to that employed by Fahrig and Merriam, 1985, for modelling animal population survival in patchy environments except that they incorporate between patch as well as within patch processes. One way to visualize the model is as a set of raster map layers overlaid on top of each other. Each horizontal slice through this stack (one raster map layer) represents the spatial distribution of one specific landscape attribute, or state variable. The vertical column of cells resting on top of a specific landscape grid cell corresponds to the state vector of attribute values for that cell. Equation 2 expresses in functional form the temporal interactions between the different attribute map layers.  19  As developed so far, the state space framework models the interaction between landscape attributes within a cell. It does not explicitly express the relationships between attributes in neighbouring cells. This is a simplifying assumption made in the development of the model presented in this thesis, but it is relaxed in a limited sense in a subsequent chapter. It should be noted that equation 2 constitutes a general form for the description of a linear system, and that other forms are possible for systems in which the relationships between state variables are non-linear. Equation 2 itself can in fact be extended to incorporate some non-linear aspects of the system behaviour through appropriate definitions of the elements of G, r and F, as will been seen later in this chapter.  A state space model of forest landscape dynamics The form of model given by equation 2 should be familiar to biologists with a background in population ecology. Leslie matrix models used in simulations of animal population dynamics (Leslie, 1945 as cited in Getz and Haight, 1989) are essentially a special case in which the non-zero elements of G represent the combined effect of growth and mortality rates, and  is used to represent the  number of births. Demographic matrix models of this sort have also been applied to plant populations (e.g. Harcombe, 1987), and stand level forest resource management problems (Buongiorno and Michie, 1980; Michie, 1985; Usher, 1966 as cited in Getz and Haight, 1989). Getz and Haight (1989) have developed a general formulation of this type of model for simulating the demographic processes in stage-structured populations (e.g. fish, forests, and wildlife). For trees, the population life stages are defined as diameter classes (height classes can also be used), so the state variables used to describe a forest stand are the number of trees per unit area in each diameter class. In their version of equation 2 the coefficients in both G and r are permitted to be  20  functions of an arbitrary measure of the current population density, permitting the growth, survival, and reproduction processes for the population to be expressed as density dependent rates. This extension to the linear state model permits the inclusion of non-linear effects while preserving the relatively simple structure of the linear system description. It thus provides one possible formulation of the dynamics of forest stand growth, and is well suited to simulating a full range of selection harvest actions (see for example Haight, 1987 and Haight, 1990). The resulting growth and yield model is of the whole stand - distance independent variety (Hann and Bare, 1979). I have used the general structure proposed by Getz and Haight (1989) as a starting point for expressing the dynamics of the forest cover component of the landscape. The system model could easily be extended to integrate other landscape features such as the dynamics of snags and coarse woody debris, both of which are important factors in developing management strategies that protect biodiversity through the conservation and generation of old-growth forest structure (BCMOF, 1995b; Spies and Franklin, 1988). The formulation is also applicable for simulating wildlife population dynamics, thus it may provide a useful framework for integration of tree and wildlife population dynamics in a single spatially explicit simulation model (Holt et al., 1995). The form of the Getz and Haight model is  3  a(/ +1) = G(b(0)a(0 + r(b(/)) b(t) = Fa(t)  3  P  ]  The symbology used here differs from that in Getz and Haight (1989). I have changed it to maintain consistency with the general state space framework equations presented earlier in this chapter.  21  where a(t) =  (a (t),...,a (t))' l  N  is a vector in which system attribute a (t) equals the number of trees in the i'th t  stage class at time t,  W))=|  (l-fl(b))si(b)  o  /#>i(b)  (l-A(b)H(b)  < 0  )  •  •  .  0 :  •••  -  0  0  i  0  ;  ; I [4]  /5,-,(b)V-,(b) ^(b)  is a matrix representing the density dependent progression (p (b)) and t  survival (s. (b)) rate processes between and within each stage class;  b(0 = ( 6 , ( 0 , - A ( 0 ) ' is a vector of stand density metrics, and r(b(0) = (r (b(0),...,/' (b(0))' 1  JV  is a vector of inputs to each stage class (in practice only r (t) is non-zero, x  representing the number of trees entering the smallest stage class as a result of the natural regeneration process). It is important to note that the structure of equation 4 assumes that the growth of trees in the population is monotonic, and that the stages can be defined so that, for a small enough At each individual tree either remains in the same class, or progresses into the next larger class. This formulation also treats each tree in a given stage class as equivalent, regardless of how long it has remained in that class (Getz and Haight, 1989).  22  Each stage progression rate parameter p (b) is a function that expresses the i  density dependent proportion of individuals in stage class i at time t that move into the next stage class at time it + At), while the stage survival rate parameters s (b) are functions denoting the fraction of individuals in stage t  class i that survive the time interval (t, t + At) . Note that p (b) =0 by N  definition. The generic stand dynamics model expressed by equations 3 and4 is a relatively simple framework into which a great degree of complexity can be built, as will be shown in the rest of this chapter. In order to apply this model, speciesspecific expressions must be developed for the functions p (b), s. (b) and t  . Ideally, these expressions should be developed from the best available stand growth data available for the species and stand types in the area of study. As such an approach is beyond the scope of this thesis, I have chosen to derive the required expressions from the various model equations that make up the inland empire variant of the Stand Prognosis Model (also known as the Forest Vegetation Simulator, or FVS) (Wykoff, Crookston, and Stage, 1982; Wykoff, 1986). 4  Stage Progression Rate The progression rate for each diameter class is given by  p (b) = dob (b)/W I  l  [5]  where dob,(b) is the average periodic outside-bark diameter increment for diameter class i, and Wis the diameter class width (Davis and Johnson, 1987). 4  Prognosis incorporates different models for large and small trees, however I have used strictly the large-tree models to simplify the present development. The procedure would be analogous for implementing the small-tree models where appropriate.  23  I have defined the stand density vector b in this instance as the stand cumulative basal area distribution, such that the i* element is  or, equivalently,  b,  =^(DBHJ\2) a, 2  [6] b^b^+^iDBHJllfa,  for i=l,2,..,N (number of diameter classes) . Other stand density measures are 5  readily determined from the elements of this vector . In equation 6, DBH is 6  ;  the midpoint diameter of the i* diameter class. Because each tree in a given stage class is treated as equivalent regardless of how long it has remained in that class, all trees in stage class / are assumed to have a diameter at breast height equal to DBH . Furthermore, all trees in stage class /' are assigned the same ;  height, HT , calculated from the following equation ;  HT = exp o /(DBH t  C  +C  t  +1)  + 4.5  [7]  7  where C and C, are species specific constants. The reader should refer to 0  Wykoff et al. (1982) for the relevant values of these constants. 5  Each element of this vector is in units of ft . The use of imperial units at this stage of the model development is necessary because die Prognosis model equations are all expressed in these units.  6  This is one place in the model formulation in which the influence of neighbouring landscape cells could be taken into account quite easily, by calculating b over a given spatial domain that may include cells adjacent to the one for which growth is being projected.  7  Tree height (HT, feet) (equation 6, Wykoff, Crookston, and Stage, 1982)  2  24  An expression for dob,{b) can be derived from the equations that make up the diameter increment submodel of die inland empire variant of the Forest Vegetation Simulator (FVS- also known as the Stand Prognosis Model) (Wykoff, Crookston, and Stage, 1982; Wykoff, 1986). These equations are used to predict dds , the squared inside-bark diameter increment per decade for all t  trees in stage class i, and integrate many factors such as topographic position (through slope, elevation and aspect), habitat type, geographic region, and stand density. The detailed model equations are given in Appendix 1. Once ddSj is known, it is converted to the ten-year inside bark diameter increment through the following equation,  DG, = ^(DBHjkf  +dds  t  - (DBH /k)  [8]  8  i  where k is a species specific outside- to inside-bark diameter conversion factor, and DG; is the inside-bark 10 year diameter increment for all trees in stage class i.  The relevant values of k are given in Wykoff et al. (1982). Finally dob (b) is t  determined from dob (b) = (k-DG /10)-At i  i  [9]  and substituted into equation 5 to arrive at the stage progression rate for a given diameter class. Stage Survival Rate The mortality model used in FVS is based on Hamilton's empirical logistic mortality model (Hamilton, 1986). His equation, together with a set of 6  8  Inside bark diameter increment per decade (DG, inches/10 years) (equation 7, Wykoff, Crookston, and Stage, 1982)  25  modifying assumptions (Hamilton, 1990), predicts the annual probability of mortality for a tree of given DBH as a function of stand density. The equation is used here to predict the annual probability of mortality for all trees in a given diameter class, and is P = {1 + exp[-(* - 0.2223\0DBH t  m t  + 0.04605085/4 - 11.2007ADBH, 1/2  + 0.554421/ DBH, - 0.2A6301RELDBH, - 6.07129ADBH, I DBH,)]}'  1  [10] where Pj is the annual probability of mortality for trees in stage class i, k is a species-specific constant, ADBH is the average annual outside-bark diameter t  increment in stage class i from the previous projection period (in inches), BA is the stand basal area at the start of the current projection period, in ft (and is 2  derived from equation 6 as BA = b ) and RELDBH is DBH,/(arithmetic N  t  mean stand DBH). The relevant values of k are given by Hamilton (1990). This expression is used to calculate the survival function s,(b) for each diameter class in equation 4 as follows.  *,(b)=(i-^r  [«]  Regeneration Rate The regeneration vector, r(b), has not been implemented from the regeneration sub-model of FVS (Ferguson et al., 1986; Ferguson and Crookston, 1991), since the latter is too complex for implementation in the present framework. Furthermore, there is no reliable data from which to develop a model of the ingrowth process for interior Douglas-fir stands (the 'location' of the hypothetical landscape used in Chapter 5) in British Columbia (Alfaro and Maclauchan, 1992). Therefore, a simple arbitrary linear function of total stems per unit area has been implemented. The slope and intercept 26  parameters can be chosen from available regeneration rate study data, or can be adjusted in order to improve the agreement between yield predictions of this model and those of FVS for equivalent stands. Putting it all together In order to apply this stand development model in a spatial context, each landscape cell is assumed to represent a stand (or portion thereof) in which all trees are assumed to be of the leading species, distributed across N stage classes,-and which grows in accordance with the stage progression and input terms of equation 3. Equations 5 through 9, and the equations in Appendix 1 determine the stage progression rate for trees in a given diameter class occupying a given landscape cell. They reflect the influence of site on stand growth through the appearance of location and habitat type constants, as well as slope, elevation, and aspect. Since habitat type and tree species vary across a landscape, they must be introduced to the spatial model as additional attributes of each landscape cell. In the implementation used here, the two attributes have been combined into one by defining a "stand type" attribute for each cell as the combination of biogeoclimatic sub-zone and leading tree species. This approach treats the 9  tree species growing on a site as constant through time (which is clearly unrealistic over the long term), however this limitation could be relaxed in future implementations of the model. The topographic variables are also introduced into the spatial model as constant landscape attributes for each cell. These attributes are readily obtained from the digital terrain modelling functions available in most GIS systems, and are likely to be relevant attributes for modelling other aspects of the landscape state (e.g. hydrologic flow patterns and yields; winter range habitat suitability for large ungulates).  27  Chapter  4  A MULTI-OBJECTIVE APPROACH TO SIMULATING FOREST MANAGEMENT DECISIONS.  Introduction The impact of extractive forest management decisions on the landscape can be introduced into the simulation framework developed in the previous chapter by adding a new term to equation 3. a(t +1) = G(b(0)a(0 + r(b(0) - h(/) b(t) = Fa(t)  [ 1 2 1  where h(7) = (/»,(t),...,h (t))' is a harvest control vector in which h (t) is the N  t  number of trees to be removed from the i* stage class at time t. In the context of designing multi-objective harvest management schedules for forested landscapes, it is the time series of h(t) vectors for every cell in the landscape over the planning horizon that we seek to determine by simulating the system behaviour. Equation 12 is used to advance the state of each cell in the forest landscape from one time step to the next for a given spatial distribution of harvest control vectors.  A multi-criteria management decision model The analytical techniques of multiple criteria decision making (MCDM) represent one possible approach to implementing a feedback control 9  Biogeoclimatic sub-zones are somewhat analogous to the habitat types used by the Prognosis model.  28  mechanism for determining the spatial distribution of harvest control vectors in a given time step. MCDM methods are part of a large body of literature expounding mathematical approaches to modelling and assisting various aspects of the decision making process (Hwang and Masud, 1979; Hwang and Yoon, 1981). Howard (1991) has reviewed some of the techniques with specific reference to forestry applications. Some applications of particular relevance to forest landscape management are Smith and Theberge (1987), Kangas (1992) and Kangas and Kuusipalo (1993). All MCDM techniques apply, more or less, the following basic steps (Howard, 1991): 1. State, in natural language, the desired management objectives; 2. Formulate explicit performance criteria in functional form by which the achievement of each of the objectives can be measured for a particular course of action (i.e. a decision alternative); 3. Specify a finite set of decision alternatives, and evaluate each criterion for each alternative; 4. Transform, as required, the criterion values onto commensurate scales or units; 5. Assign weights to each criteria to reflect the desired preference structure for the different management objectives; 6. Apply an MCDM algorithm to calculate a rank for each alternative from the scaled criteria values; 7. Choose the preferred alternative. Simple additive weighting (SAW) is one MCDM technique used for ranking 10  the set of decision alternatives (Churchman and Ackoff, 1954, as cited by Howard, 1991). SAW is appealing in its conceptual simplicity and its suitability to integrating multiple design objectives in the decision making process. Also, it is mathematically similar to some of the approaches used in models for evaluating and ranking suitable wildlife habitat (e.g. Schreier et al, 1993), thus permitting the natural expression of habitat suitability as a management  1 0  SAW is a simplified derivative of the Analytic Hierarchy Process developed by Zahedi (1986), and used by Kangas and Kuusipalo (1993).  29  objective. In general terms, a set of Q criteria (expressed on commensurate scales) for alternative j can be represented as a column vector  and a set of Q weights (which usually are chosen to sum to 1) as a row vector  w = [w,  ••• w ] . g  The rank for thealternative is then calculated by the vector product Uj = wc,  [13]  which is just the weighted sum of the individual criteria. The alternative with the numerically largest rank is selected as the preferred alternative. Howard and Nelson (1993) demonstrated the use of this technique to generate area based harvest schedules via simulation on a simple age-based raster forest. Their approach applies the SAW decision model iteratively within each time step to choose the best remaining eligible cell to schedule for clear-cut harvest in the current period. This iterative decision procedure is adopted here as the mechanism for implementing the feedback management process of Figure 1, by using it to determine the spatial distribution of harvest vectors to be applied to the landscape in the execution of equation 12. The method is general, permitting the expression of a range of landscape management objectives. Management Decision Alternatives A management action, or treatment, is defined here as a vector whose elements represent the proportion of the total stems in each diameter class to be  30  removed . This permits the representation of any intensity of cut from very 11  light spacings to a full regeneration cut. Landscape cells are grouped into "cutblocks", and a cutblock can be considered as a candidate for a particular treatment when the structural state of it's forest cover satisfies specific eligibility criteria. The current structural state of a cutblock can be determined by calculating the structural state variables from the diameter class vectors for each cell, and then averaging these cell statistics for all member cells in the cutblock.  These structural criteria could take many forms, such as the  minimum and maximum values for the arithmetic mean outside diameter (cm at breast height) and the total basal area density (m /ha). Such an approach 2  provides a mechanism for targeting particular management actions at specific points in the structural evolution of a stand. During each simulation time step, some subset of cutblocks will each be eligible for one or more management actions. The set of all potential cutblock/treatment opportunities constitutes the management decision alternatives. Once a cutblock/treatment combination has been added to the management schedule, other actions involving the same cutblock are no longer available as decision alternatives for the duration of the time step. Using this definition for the set of decision alternatives implies that the decision algorithm simultaneously determines both the location and type of the next management action to be applied to the landscape. Management Decision Criteria Concern about the potential for loss of biological diversity in forest landscapes has lead to a great deal of effort in formulating various conservative policies and strategies for managing the mosaic of landscape ecosystems. In British Columbia such efforts have led to the development of the Biodiversity 1 1  In its present form, the model only permits "extractive" actions directed at the forest cover.  31  Guidebook (BCMOF 1995b) component of the Forest Practices Code (BCMOF 1995a). The guidebook suggests acceptable standards and conditions for various elements of forest structure at both stand and landscape scales. The intent of these guidelines is to produce managed landscapes in which the temporal and spatial mosaic of habitat patches is within the range of variation expected for unmanaged landscapes developing under natural disturbance regimes. It is hoped that by designing management to produce a quasi-natural mosaic of ecosystem types in a variety of developmental stages that landscapes can be maintained in a state that is favourable to the habitat needs of the widest possible range of species, while still providing an acceptable level of extractive resource output. The classification of forest area into different serai stages is intended to reflect (albeit in a very coarse fashion) the fact that the community composition and structural characteristics of a forest stand evolves as the stand ages. Most commonly, serai stages have been defined as age classes. The Biodiversity Guidebook provides nominal age-range definitions of early, mature, and old (and by inference, young, covering the ages between early and mature) serai stages. It is, however, the differences in structure (and hence habitat value) rather than the differences in age that we are trying to capture. Spies and Franklin (1988) note that "age alone indicates nothing about the ecological structure of the forest". This is an important distinction, since the structural features that we desire to preserve and develop can occur over a wide range of stand ages depending on site specific environmental conditions and disturbance history, and may in fact be accelerated or enhanced through the use of unevenaged management techniques. This makes the concept of stand age much less meaningful.  32  Particular emphasis is placed on the structural characteristics of old growth forests, since it is that component of the landscape mosaic that is being lost most rapidly due to forest harvesting. The Biodiversity Guidebook (BCMOF 1995b) recommends that definitions be eventually developed based on standlevel attributes, and suggests a scheme for defining the serai stage categories based on their structural state relative to the attributes of a typical "natural" old growth stand. The important defining attributes of a natural old growth stand are many and varied, and working definitions have not yet been developed for British Columbia forests. Thus I have adopted a simplified (but representative) model of the stand-level structural characteristics of old growth forests based on minimum values for the standing volume per hectare and the proportion of stand density resulting from (living) trees larger than a specified minimum diameter. To simplify the definition further for the purposes of this analysis, the stand density component of the old-growth definition has been based on the minimum standards for old growth Douglas-fir forests given in Spies and Franklin (1988). Their definition incorporates stand characteristics such as species mix, density of large living trees, canopy structure, snags, and coarse woody debris. Here I apply only the minimum standard for the density of large living trees of the dominant (Douglas-fir) species: 20 stems per hectare (8 stems per acre) greater than 81 centimeters diameter at breast height (32 inches DBH). Having thus defined the structural conditions which indicate the onset of "old-growthness", the Biodiversity Guidebook suggests that a stand be considered to meet the mature serai criteria when it has attained 70% of the minimum old growth standing volume and 70% of the minimum required density of large trees, and that a stand be classified as early serai while it contains less than 30% of the minimum old growth standing volume, regardless of the density of large trees (such as those left as standing trees during a past harvesting operation or natural disturbance) (BCMOF 1995b).  33  The Biodiversity Guidebook recognizes three levels of biodiversity management effort, designated as low-, intermediate-, and high-emphasis, and suggests acceptable ranges for the proportion of landscape area within each of the early, mature, and old serai stages, for each management emphasis option. As an example in the following development, consider the intermediateemphasis guidelines for the serai stage distribution for natural disturbance type 3 (NDT 3) landscapes in the Interior Cedar/ Hemlock (ICH) 12  biogeoclimatic zone (BCMOF, 1995b).  Table 1: Serai Stage Distribution Goals. Taken from Table 11, BCMOF 1995b.  Serai stage Early Mature & Old Old  % Forested area within landscape unit <46 >23 >14  The way in which serai habitats are distributed across the landscape is as important as the total proportion of the landscape within the different developmental stages (Spies and Franklin, 1988). This is reflected by the Biodiversity Guidebook in the form of acceptable ranges for the distribution of landscape area within small, medium, and large sized habitat patches, as summarized for NDT 3 landscapes in the ICH biogeoclimatic zone in the following table.  1 2  Natural disturbance type 3 is defined in the Biodiversity Guidebook (BCMOF 1995b) as encompassing those ecosystems subject to frequent stand initiating events.  34  Table 2: Patch Size Distribution Goals. Taken from Table 12, BCMOF 1995b.  % Forested area within landscape  Patch Size Class Small  !  20-30 25-40 30-50  <40  Medium j 40-80 Large j 80-250  These guidelines provide a set of quantifiable objectives (e.g. maintain between 25 and 40% of the forested area in patches between 40 and 80 hectares in size) that describe the desired state of the system from the perspective of managing for biodiversity. Actually attaining and maintaining these state-related objectives on a given landscape over the long term involves tradeoffs since they may be in conflict with other management objectives for landscape outputs, such as minimum desired wood volume extraction rates, or even with each other. Furthermore, the present state of the landscape may preclude achieving the objectives until many years into the future. The objectives for serai stage and patch size distributions, and commercial volume extraction rates can be represented within the generic MCDM control framework by first defining a set of spatially-integrated landscape system outputs for which objectives are to be set. 4 = proportion of forested area within small patches 4 = proportion of forested area within medium-sized patches 4 — proportion of forested area within large patches 4 proportion of forested area within early serai stage 4 - proportion of forested area within mature serai stage 4 = proportion of forested area within old serai stage 4 = Wood volume extracted (m) =  3  These variables provide an alternative state representation for the resource system. The values of these aggregate variables at a point in time are calculated  35  from the detailed stand-level state description developed in the previous chapter, and in turn they provide the feedback linkage between the detailed state dynamics model and the management control algorithm. The system state in terms of the aggregate variables is expressed mathematically as a vector containing the values of the individual output variables, and one such vector represents a unique point in state space. The specific objectives for each of these variables are the desired minimum and maximum values, or goals (either of which may be undefined, indicating an open interval). These goals define the desired operating region within the system state space, and our overall management objective is to have the system operate within, or as close as possible to, this region at all times. The present state of the landscape may well be outside the desired region. The multi-criteria feedback controller in Figure 1 is designed here to try to keep the trajectory traced by the system state within the desired operating region. Given a finite set of potential management actions (where each management action consists of a specific treatment to be applied to a specific unit of area in the landscape) that can feasibly be applied to the landscape in its present condition, each action will shift the system to a new state. The function of the controller is to rank the consequences of the available management actions (i.e. the set of possible next states) at every decision point, and then to add the highest ranked alternative to the list of activities scheduled for the current time step. When the current state is outside the target region, this means choosing management actions to move the current state toward the target region along the shortest possible path. While the current state is within the desired limits, management actions are chosen to keep the next state as far as possible from all borders of the operating region so as to reduce the possibility of subsequent disturbances (planned or natural) driving the system outside of the desired range. This  36  general scheme is developed into an heuristic procedure for ranking the next state alternatives in the following paragraphs. A set of Q scores can be assigned to each feasible management action (decision alternative) at a given point in time based on the position of the state that results from the application of the proposed action, relative to the target operating region (as defined by the goals associated with each objective). These scores represent the un-transformed criterion values, and the set of all feasible decision alternatives can then be ranked on the basis of these scores by applying standard MCDM techniques. The simplest situation is that of an output variable having a minimum, but no maximum, target level. For the t output variable, we define  Al =(/.-/.  .)  For a system output with both minimum and maximum desired levels, or one with only a specified maximum (which implies a minimum of 0), we define  i \ /..+/. \h - (, „ J; or /, < — - — f  Al  mi  (  , (.max -  +/.  / !\  r> i  h j; for i *  J,min  [  1  6  1  /,max  ~  l  The preceding definitions for A/ are such that Al will be negative when / is ;  t  ;  outside the desired range, and positive when / is inside the range. The shape (  of Al as a function of /, is illustrated by Figure 2, for minimum and maximum t  limits of 0.20 and 0.47, respectively.  37  Deviation from goals 0.20000  T  -0.60000 -  1  Next State, / ;  Figure 2: Performance criterion for a single system output variable with minimum and maximum desired levels  Defined in this way, each A/, is a general performance criterion (expressed as a benefit criterion for which larger numeric values indicate better performance) with which the consequences of a single decision alternative can be evaluated with respect to a single management objective. The individual criteria can be combined into an overall rank for a single decision alternative j byfirstapplying one of the scale transformation techniques used in MCDM analysis, and then performing the weighted summation operation of equation 13. For the present implementation, I have used the normal transformation technique (Howard, 1991), in which each performance criterion is divided by it's norm, calculated across all the decision alternatives.  38  1/2  c. - Al  where A/, . is the 1 untransformed performance criterion evaluated for th th  decision alternative, J is the number of decision alternatives, and c . is the i  transformed criterion evaluated for the 1 decision alternative. th  39  Chapter  5  EXPLORING THE MODEL  Introduction This chapter presents a qualitative exploration of the model developed in the preceding chapters in order to assess its strengths, weaknesses, and hence the suitability of the general framework for further development. This is a necessary prerequisite to applying any new modelling technique to real-world problem analyses. The model has been programmed for execution on 32-bit Windows platforms using the C++ language; this implementation has been 13  named "LandSim". In thefirstsection of the chapter, the behaviour of the landscape dynamics submodel is investigated for a tree species of economic and ecological importance in the British Columbia southern interior region (Bonnor, 1990; Vyse et al., 1990): Douglas-fir, Vseudotsuga men^esii var. glauca (Beissn.) Franco (Bums and Flonkalla, 1990). The second part of the chapter explores the management control submodel for a hypothetical landscape of Douglas-fir.  Evaluating the Landscape State submodel Several comparative runs of LandSim and FVS (also known as the Stand Prognosis Model) (Wykoff et al., 1982) were made for the purpose of  1 3  LandSim has been developed using the Borland is included in the appendices of this thesis.  C++©  40  compiler (version 4.52) The C + + source code  demonstrating that the stage class growth model framework is a viable approach to simulating the structural dynamics of the forest in each raster cell. In selecting the appropriate coefficients for the diameter increment and mortality model equations, the Pseudotsuga men^esii/Calamagrostis habitat type defined in Wykoff et al., 1982 and Wykoff, 1986 was chosen as representative of Douglas-fir forests in the Interior of British Columbia (Tisdale and McLean, 1957; Haeussler et al., 1990). Initial stand conditions for a selection of the comparative runs are given in Table 3. For each case in the table, plots were prepared to illustrate predicted basal area (square meters per hectare), mean DBH (centimeters), stand density (trees per hectare), and total volume (cubic meters per hectare) as functions of time. These results are presented in Figure 3 to Figure 6, and show that the stand growth model expressed by equations 3 and equation 12 produces similar qualitative patterns of stand development over a range of site, stand and disturbance conditions. There are clearly quantitative differences between the predictions made by the two models. For example, a comparison of Figure 3 (north aspect, even-aged, undisturbed) and Figure 4 (south aspect, even-aged, undisturbed) shows that the LandSim model predictions for stand basal area and volume are larger than the FVS predictions on south facing slopes. A further example of the differences is seen in comparing Figure 5 (north aspect, even-aged, light disturbance) and Figure 6 (north aspect, even-aged, and heavy disturbance). The predictions of the two models are more closely matched following a heavy disturbance than following a light one.  41  Table 3: Growth model comparison matrix and parameter values.  Case 1 2 3 4  Aspect North South North North  Stocking Even Even Even Even  Disturbance None None Light Thinning Heavy Thinning  Constants: Species Elevation Slope Duration Diameter Distribution Structure Natural regeneration  Douglas-fir 1219 (4000 feet) m.a.s.1. 20% 150 years 20 DBH classes, each spanning 5.08 cm (2 inches), starting at 0 None  Variable factors: Aspect:  North South Initial Stocking: Even Uneven  0 degrees' 180 degrees 1235 trees/ha (500 trees/acre) in the 1 class 1235 trees/ha (500 trees/acre) in the 1 class; 123.5 trees/ha (50 trees/acre) in the 5 class; 12.35 trees/ha (5 trees/acre) in the 10 class  st  st  th  th  Disturbance:  None Light Thinning  (80 % removal in each of the 5 smallest diameter classes) Heavy Thinning (80 % removal in each of the 16 largest diameter classes)  42  C a s e 1a 60  6000  5000  4000  — •  LandSim, Basal Area  — D - ^ F V S , Basal Area •-•  3000  - - LandSim, D B H  • - O - - FVS, DBH 2000  1000  Years  C a s e 1b  j 500 450 - 400 - 350  - • —  - 300 01 250 E  •  - - LandSim, Volume  o  •  - - F V S , Volume  200 150  >  •- 100 -- 50 -- 0 160  Figure 3: North aspect, even-aged stocking, no disturbance  43  LandSim, Density  • — F V S , Density  Case 2a 60  6000  5000  4000  - LandSim, Basal Area - F V S , Basal Area  3000  - - LandSim, D B H • - FVS, DBH  2000 +  1000  Years  Case 2b 1400  500  450  1200  400 1000  I  350  + 250  •  - - LandSim, Volume  200  •  - - F V S , Volume  150  400  100 200  50  40  60  80  100  120  140  160  Years  Figure 4: South aspect, even-aged stocking, no disturbance  44  LandSim, Density  •  ; 6oo  20  - • —  300  800  F V S , Density  The quantitative differences in model outputs relate to the differences in implementation between the two models. LandSim treats all trees in a DBH class as identical, whereas FVS simulates growth and survival processes for individual trees. FVS also predicts both diameter and height increments to simulate individual tree growth. In contrast, LandSim calculates only diameter increment, and calculates the height of all trees in each diameter class from the class midpoint DBH (see equation 7). Current versions of FVS make use of different increment prediction equations for small and large trees. To simplify the development of LandSim, only the large-tree diameter increment model has been implemented, and this is applied to all diameter classes. FVS determines a second estimate for mortality rate, in addition to the one used here (equation 10), and ultimately applies a mortality rate that is a weighted average of the two independent estimates. FVS also applies management actions at the start of each time period, prior to performing growth projections, while LandSim applies the management actions following the growth process in each simulation period. There are numerous other differences between the two implementations, however for the purposes of this thesis these differences are moot. Both models are an abstraction of true stand development over long time frames. The stand dynamics model presented in this thesis is a major simplification of the FVS model from which it has been derived, and does not include a realistic model of the natural regeneration process. No attempt has been made to calibrate it to any experimental stand development data, yet it produces very similar qualitative behaviour and thus is a viable approach to simulating the long term development of stand structure across landscapes.  45  Case 3b  Figure 5: North aspect, even-aged stocking, light thinning  46  Case 4a 60  •5^  + 50 40  c 30 n 20  LandSim, Basal Area  Q  -•  FVS, Basal Area  •  - - LandSim, DBH  •  - - FVS, DBH  E  10 1  50  150  100 Years  Case 4b  — LandSim, Density — FVS, Density • - LandSim, Volume - FVS, Volume  20  40  60  80  100  120  140  160  Years  Figure 6: North aspect, even-aged stocking, heavy thinning  47  Managing For Biodiversity And Timber: A Case Study This section examines the qualitative behaviour of the control portion of the modelling framework using a hypothetical landscape and a hypothetical management problem. Performance criteria were developed in the preceding chapter for commercial harvest and biodiversity conservation objectives. For this evaluation, commercial volume has been arbitrarily identified as all harvested trees with a DBH larger than 30.5 centimeters (12 inches). Topography and cutblock layouts for the hypothetical landscape have been borrowed from a portion of the Nehalliston creek watershed (3850 hectares) in the Kamloops forest region of British Columbia, Canada. The landscape was modelled on a raster grid using square 1 hectare cells. Elevation within the study area ranged from 323 to 1583 meters above sea level. There were 209 cutblocks ranging in area from 1 to 67 hectares, with an average area of 18.4 hectares. The landscape state sub-model was designed to represent a mosaic of monospecific stands, as well as multiple biogeoclimatic sub-zones and natural disturbance type zones. However, to simplify the analysis and interpretation of results, the forest cover has been constructed of pure Douglasfirstands, a single natural disturbance type zone, and a single biogeoclimatic sub-zone. The evaluation of model performance for a more complex and realistic landscape is left as a issue for future development of the model. Stand structure was represented using 16 diameter classes of 7.6 cm (3 inches) each . Initial 14  diameter distributions for all cells in the landscape were specified as follows. Each cutblock was randomly assigned an integer value between 1 and 4, in  1 4  Fewer classes were used than in the previous section of this chapter in order to reduce simulation times.  48  roughly equal proportions, representing the initial serai stage of all cells in that block (early^l, young=2, mature=3, old=4). Diameter distributions corresponding to each of the serai stages were then extracted from tables of typical diameter distributions maintained by the Ministry of Forests for the Kamloops forest region : these are illustrated in Figure 7. Each cell in the 15  landscape was then assigned the diameter distribution corresponding to the serai stage assigned to their parent cutblock. Figure 8 shows a map of the initial distribution of serai stages across the hypothetical landscape. The management control algorithm developed in the previous chapter is designed to select among multiple treatment options for a given cutblock (actually, any block for which there is at least one eligible action is automatically a candidate for two actions - do the eligible action, or do nothing) by associating each landscape cell with a certain treatment regime. A treatment regime in this sense is a set of predefined management actions that may be applied to landscape cells during the course of their structural development over time. This approach permits the modeling of different resource management zones, however for the purposes of this chapter a single treatment regime was assigned to all cutblocks. The treatments used are defined in Table 4 (the no treatment option is excluded from the table). For the purposes of this chapter, the eligibility criteria have been set so that all blocks are eligible for all treatments at all times. This was done in order to allow the control algorithm full latitude in selecting treatments for each candidate cutblock, since the focus of this analysis is to explore the qualitative behaviour of the treatment selection algorithm.  1 5  The 'typical' diameter distributions are derived from permanent and temporary sample plot data which have been aggregated to produce 'average' diameter distributions for each combination of inventory type group, age class height class and stocking class recognized in the B C Ministry of Forests inventory system. These average diameter distributions were obtained from Bob MacDonald, inventory and GIS Forester at the Kamloops Forest Region .  49  Simulations were carried out for a duration of 150 years using a period of 5 years. Numerous scenarios were executed to explore the effect of applying different weights to each of the landscape performance criteria; a subset of these are listed in Table 5. The weights given in the table represent the w  t  applied in the weighting summation procedure of equation 13.  Table 4: Treatment regime  % % % % % % % % % % % % % % % %  Name: Removal class 1 Removal class 2 Removal class 3 Removal class 4 Removal class 5 Removal class 6 Removal class 7 Removal class 8 Removal class 9 Removal class 10 Removal class 11 Removal class 12 Removal class 13 Removal class 14 Removal class 15 Removal class 16  LightThin  33 33 33 33 33 33 0 0 0 0 0 0 0 0 0 0  LightThin2 MediumThin HeavyThin ClearCut 0 99 0 66 99 0 0 66 0 99 66 0 99 0 66 0 99 0 66 0 0 99 0 66 66 99 0 33 66 99 0 33 99 33 66 o: 66 99 0 33 99 33 66 0 66 99 0 33 66 99 0 33 66 99 0 33 99 33 66 0 99 33 66 0  Table 5: Weights for M C D M Control test cases  Weights Scenario  Volume  Early  Mature  1  I  0  0  2 3  1 1  25000 8500  175000 8  Old  Small  0 0 1 0 0 5 8500 20000  50  Medium 0  0 80000  o  Large  ~ 0 20000  96 06 98 08  2 CD  W  CO  XJ  C  O •  o  OJ  •c to 5  a  15  £2 o E  CO  Q  o  o •a P J3  2  cu  0>|  CO  >-  15  H  cO o C co LU  9k  SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS/J  01o o  o o oo  o o  o o  CD  O  o m  O  o •<*•  O  o co  o o  CN  O O  -a •a •a 3.  Results Four plots were preparedforeach scenario listed in Table 5: three to show the time series of system output variables, and a fourth to show the area assigned in each period to each treatment from Table 4. The resulting plots are shown in Figure 9 to Figure 20. It is important to note that the figures showing the time seriesforthe serai stage and patch size system output variables include only the classes for which objectives were specified in Table 1 and Table 2. In other words, the serai stage figures do not show the "young" serai stage class and the patch sizefiguresdo not show the "extra-large" patch size class. The harvest volume time series in Figure 9 clearly illustrates the effect of the goal-seeking control scheme developed in the previous chapter. For scenario 1, harvest volume was the only system output considered by the control algorithm in deciding which actions to apply to the landscape, and the achieved commercial harvest volume tracks the midpoint between the minimum and maximum targets almost exactly. The time seriesforthe proportion of area in the early and old serai stages (Figure 10) also remain within the desired limits for the duration of the simulation, although this is just fortuitous in this case. The mature serai area time series remains outside of the desired range during most of the simulation, as do all three of the patch size time series (Figure 11). Figure 12 shows that,forthe most part, the volume objective was met by applying the clearcut treatment. Where the application of this treatment would have pushed the achieved harvest level outside of the desired range, other treatments were selected. Scenario 2 represents an attempt to find a weighting that would balance the objectives for both volume and serai stage distribution. The results are shown  53  gn  —  on  get OEI.  921 s  """"  $  s s  l  021  ou  got ooi ^^^^^p 06 98 08 SL  0Z 99 09  *  99 09  QP OP 9£  •St  eeje |e)0) jo uonjodojd  091  9H  on 9EI  oei. SZl OZl  su on 901001-  96 06 98 08 9i 0Z 99 09 99 09 9fr Ofr  9£ 0E 9Z 0Z  Slot 9 0  B3JE |E}OJ JO UOiyodOjd  Hectares untreated  8  o  8 8  8 091.  9n on  oet  0  sst O •  021 SU 0U  I •—  SOI  i  001  3|  Q  96 06  0)  E ro  E  98 08  2  s  .Q  CO CO  9Z  >"  0Z  33  99  §0  09 99 09 Sfr  -rasa  Ofr 9£  oe  21  ss oz 91 Ot  9 pajean sajepaH  1  E  in Figure 13 through Figure 16. The time series for each of the serai stage distribution variables remain with their desired ranges, except for slight undershoots of the minimum old serai area in years 60 and 75; however it was only possible to achieve this by significantly violating the volume goals in some periods of the simulation. Significantly more area was scheduled for management in most of the periods for which the volume goals were violated than in the previous scenarios. A greatly skewed set of weights was required to achieve these results, and this gives a good indication of how strongly the control algorithm is biased toward the achievement of the volume objectives. The results of the final scenario presented in Table 5 are shown in Figure 17 through Figure 20. This scenario was an attempt to satisfy all objectives simultaneously. The objectives for the serai stage distribution variables are met in every period of this simulation, except for a very slight underachievement of the minimum old serai area target in year 90. This was achieved using much smaller weights than in the previous case because, as discussed earlier, the serai stage and patch size variables are not independent. Each of the patch size distribution variables were within the target ranges during some of the simulation periods, however I was unable to find a set of weights for which these objectives were met throughout. Furthermore the weights applied are still heavily skewed, and the objectives for commercial volume harvest are significantly violated during most periods of the simulation. In comparing the results of scenarios 2 and 3, it is not immediately clear which is the more desired solution. Neither scenario succeeds in satisfying all of the objectives all of the time, as should be expected when the management objectives are in conflict. The individual patch size goals are achieved more often in scenario 2 than in scenario 3, although only in year 100 of the former are they achieved simultaneously. The volume output exceeds its maximum target less often in scenario 2 than in scenario 3; however it generally overshoots the target by a larger amount. There are also substantial  58  > •a  o >  4>  E  ON  4 — 1 co  o  I  ro X  JS CNI  o  sjd)dui Djqno  c o  a O >  in  a  o  0> D)  s co E v  -a  CO  o  e a i e |E}oj j o u o n j o d o j d  y/////// "SSSSS.  eaie |E}oj JO uoujodojd  'SSSSSSSSSSSSSSSSSSSSSSSSSSSJ  Hectares untreated  •  I Q  cu  E  cu CO  CN  CO CO  ID  g>  differences in the amount of area scheduled for management in each period, both within and between the two scenarios. The variability of the actively managed area across periods may have implications for access management, local employment levels, are other such issues.  Discussion The landscape dynamics component of the simulation framework, developed in Chapter 3, is able to produce predictions of the future development of stand structure that are qualitatively similar to those of FVS (Wykoff et al., 1982) for mono-specific stands. However, one of the short comings of the approach is that it is limited to mono-specific stands, and thus is unable to predict the successional development of stands. Getz and Haight (1989) apply their framework to mixed stands of white and redfirby defining separate diameter class state variables for each tree species. This is one approach that could be used here, although applying it to stands of several species would likely make the execution time of the model prohibitive on any landscapes of realistic size. As it is, the landscape management simulations above each take a little over an hour running on a 133MHz Pentium computer for a landscape of 3850 hectares. A better approach to expanding the applicability of their framework to mixed species stands would be to derive specific expressions for the progression, survival and regeneration rate parameters for each inventory type group in the study area using data obtained from the permanent and temporary sample plots that are maintained throughout the province by the Ministry of Forests. This approach would aggregate trees of different species together in the diameter classes tracked by the model. However, other state variables could be added to the model to predict the dynamics of the species mixture over time so that the tree population could be dis-aggregated where necessary. McTague and Stansfield (1995) developed explicit expressions for the predicition of species composition and the change in relative species abundance over time for mixed conifer forest types in the southwestern United States; a similar approach could be taken here.  63  ;1  OSI-  i i sn jj on set oei  3|  93U  OZl 9H-  •3:  OH  'M  SOL  0*  001  96  i — *  06 98  C  CO  E  108  3  It  "c5o  £ ra  a  « 9/  o >  >"  99  09 99  09 9fr  3. E Dfc  sjejeui DiqnD  0919n  on  sei oei. 921. 021  911 OH 901 001  -T3  96 06  c o  98 08  in  9Z  o 3  OZ  C/3  99  V  09  O  o >  SO  99 09  73  9fr Of 96  oe 92 02 9101 9 0  eaie |e)0) jo uoijjodojd  un  3. tin  esje |Bjot jo uonjodojd  Hectares untreated  8 in  8 9  8  8  8  o  IT)  o -+V,VSA>/////A\ OS I  8  31 o n 9£t  nan I  oei. 92t  •HI  oa  o •  x Q  c <u  3 -O  E  5  TO cij 4)  a  ID I?S«;>5W.I  8 CO  8  8  o  o  -+-  se  E  Perhaps the most striking feature of the landscape management simulations presented above is the enormous disparity in the weighting of management objectives required to produce scenarios 2 and 3. This aspect of the model behaviour is most probably due to a substantial difference in the ability of the control criterion expressed by equation 16 to discriminate between decision alternatives (a particular combination of cutblock and treatment) on the basis of the volume indicator as opposed to the biodiversity indicators. There is a large difference (five orders of magnitude) between the numerical value of the criteria as calculated from volume indicator versus the criteria calculated.from the per-cent area indicators for a given decision alternative. For example, when the volume indicator is exactly midway between its minimum and maximum goals, equation 16 evaluates to 12,743; when the mature serai area indicator is exactly midway between its minimum and maximum goals, equation 16 evaluates to 0.46. A critical step in the application of MCDM techniques is the transformation of the different criteria to commensurate scales. This step was implemented in the model development of the previous chapter using the normal transformation method (Howard, 1991), however the results of the simulations presented above indicate that either this method is inappropriate for the criteria used here, or that the use of equation 16 to calculate criterion values is inappropriate. One way to reduce the numerical differences between criterion values would be to scale the expressions in equation 16 by the difference between the maximum and minimum goals for each criterion, although this would not work in the case of a system output for which there is no specified maximum goal. This was attempted, and did in fact result in a increase in the sensitivity of the serai stage and patch size indicators to changes in their associated weights. Using a volume weight of 1, early and old serai weights of 10, and applying a weight of 600 to all other criteria, the results shown in Figure 21 to Figure 24 were  68  obtained. No attempt was made to find a schedule having better performance with respect to the patch size goals. Clearly, this modification of the criterion formulation is justified, although there remains a significant difference in the weights used to achieve the balance between volume and serai stage objectives. The ability of the control algorithm to identify management schedules that satisfy all objectives is constrained by several factors: the initial state of the landscape as expressed by the chosen system outputs; the specific configuration of the cutblocks; the management regime(s) defined for the simulation. The initial configuration of the landscape is a given, but the cutblock pattern and management regime taken in combination represent a finite set of actions available for controlling the system state through time. It is likely that the amount by which the volume objectives are violated in scenarios 2 and 3 could be reduced by using a different management regime than that defined in Table 4. A different cutblock pattern having more and smaller blocks, may well provide sufficient flexibility to satisfy the patch size distribution goals to a greater degree.  69  eaje lejoj to uonjodojd  •4-  Vr^^^^^ffflp^^^^^  091. Sfrl-  on 9£L  oei 9Zt 03 L 914  on901.  2  E  00 L 96 06 98 zzzzzzzzzzzzzzzzzzzz  08 9/ OL  99 09 99 09 9fr OV  9C  oe 9Z ijfrvvvwj JSS5SSS S S S S ^ v ^ v v v ^  OZ 91. 01 9 0  E3JE |B)0) j o  uorjjodojd  5  Hectares untreated  Chapter  6  CONCLUSION  This thesis has presented an analytical framework for developing spatially explicit simulations of forest landscape management in the presence of several conflicting management objectives. The framework combines the techniques of state space systems control and multiple criteria decision making. One advantage of adopting a state space representation of landscape dynamics is that the resulting model provides a detailed representation of system behaviour under a wide range of disturbance processes. The level of detail that can be represented with this framework permits an integration of stand level and landscape level structure within the same model, thus allowing the analysis to focus as much on the state of the living forest as on the harvest. Landscape attributes such as serai stage may be represented as functions of stand structure rather than simply stand age. This feature of the framework lends itself to application in studying the potential for accelerating the development of oldgrowth characteristics through appropriate management action. The framework also allows for the modelling of a full range of extractive management actions. Furthermore it is suitable for incorporating randomly occurring natural disturbances into the simulation of forest management activities, and although a study of the impacts of stochastic disturbance was not undertaken here, it was one of the primary motivating factors in pursuing this modelling approach. The framework provides a feedback linkage between the landscape state predictions and the management decision model such that the latter can employ rules, both simple and complex, that take into account the current state of the landscape, and of it's derived outputs.  74  To illustrate the application of the framework, a simulation model was developed to address the specific problem of managing a forested landscape for both timber production and the maintenance of stand and landscape level structural diversity, as outlined in the Biodiversity Guidebook (BCMOF 1995b) of the Forest Practices Code of British Columbia. Many simplifications of reality were made during the development of the model in order to limit the scope of the thesis, however this is true of any model. In spite of the simplifications, it was possible to represent a great deal of system detail within a relatively simple analytical framework, and the resulting simulation model was found to perform acceptably. The model framework as implemented here certainly has its shortcomings. For example, the landscape state dynamics component is only able to project structural conditions for stands composed of a single species, and thus neglects successional processes entirely. The MCDM based control component was found to be overly sensitive to the commercial volume harvest objective at the expense of the landscape structure objectives (desired serai stage and patch size distributions), and appears to be of limited effectiveness when dealing with more than a very few management objectives. Simulation times were substantial given that the hypothetical landscape used was not very large (3850 hectares) when compared to the 100,000 or more hectares of contiguous land that might normally be included in a study area. Potential remedies for the first two problems have already been discussed in the preceding chapter, and would be productive directions for future research and development. As for execution speed, there are undoubtedly efficiencies to be found in the way in which the model has been coded, and computer speeds are increasing substantially every year. Furthermore, the detailed model developed here is only one of an almost infinite number of ways to build upon the general framework for a specific problem situation. I feel, therefore, that the proposed  75  framework has merit as a methodology for integrating many of the contemporary factors relevant in the analysis of forest management scheduling problems. The simulation model developed in this thesis is only a first attempt at applying the state space control system framework to a greatly simplified forest management problem. The framework has great potential, however, for incorporating other aspects of the landscape and of management planning. For example, the state description could be expanded to incorporate the dynamics of other system attributes such as snags and coarse woody debris, vegetation biomass in the understory, and hydrological processes. The representation of disturbances could be extended to include non-extractive management actions such as planting and fertilization, as well as natural disturbances. Other objectives for management could easily be used as the basis for the system control, provided that the variables and functional relationships needed to evaluate the system state with respect to those objectives are incorporated into the framework. Some examples of other factors for which management objectives might be expressed are hydrologic yield, susceptibility to insect infestations, timber hauling distance (as a measure of transportation cost) and harvesting costs, and the employment potential associated with different management scenarios. Forest harvesting decisions are made continually, often in spite of scientific or economic analyses, and certainly without the aid of computer aided modelling. Nonetheless, given the obvious complexities of the problem, it is important to explore new methods for landscape simulation and analysis that are able to integrate the various facets of system behaviour deemed important for the management of a specific forest landscape. Frameworks for problem analysis will ultimately provide natural resource professionals with tools to help them  76  cope with some of the dynamic and spatial complexity of managing real forest landscapes, and to examine the inevitable trade-offs required to design sound, responsible, and equitable management plans. In closing, it is worth noting that analytical methodologies and the computer programs which are developed to automate them are not intended to replace decision makers, or to relieve them of their responsibility to make the decisions and be accountable for the consequences. Rather, such tools are only intended as aids to the decision making process, and should be used and interpreted with caution. Modelling is only one step in an iterative process of planning and decision making.  77  LITERATURE CITED  Alfaro, R.I., and Maclauchan, L.E. 1992. A method to calculate the losses caused by western spruce budworm in uneven-aged Douglasfirforests of British Columbia. Forest Ecology and Management 55: 295- 313. Arthaud, G.J., and Rose, D.W. 1996. A methodology for estimating production possibility frontiers for wildlife habitat and timber value at the landscape scale. Can. J. For. Res. 26: 2191-2200. Attiwill, P.M. 1994. 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Paper presented at the Forest Sector Conference, Vancouver, British Columbia, September 26-28. 11 p.  86  Wykoff, W.R. 1986. Supplement to the User's Guide for the Stand Prognosis Model-Version 5.0. USDA Forest Service, Intermountain Research Station General Technical Report INT-208. 36 p. Wykoff, W.R., Crookston, N.L., and Stage, A.R. 1982. User's Guide to the Stand Prognosis Model. USDA Forest Service, Intermountain Research Station General Technical Report INT-133. 113 p.  87  APPENDIX 1  Thefollowingequations are used to project the periodic diameter increment for trees in a given stage class. The stages are interpreted as diameter rat-breastheight (DBH) classes. All trees in a given DBH class are assumed to have DBH equal to the midpoint DBH for the class. For example, if the smallest DBH class covers the range 0 < DBH < 2 centimeters, then all trees in the class are assumed to have DBH=1 centimeter. Since equations 7, 8, and A l through A3 are applied in this case to trees of a given diameter class (rather than to individual trees, as in Prognosis), any terms containing or depending only on DBH are constant for a given tree species and diameter. This also applies to terms depending only on HT, since the expression for height of trees in a given diameter class involves only DBH (see equation 7). Therefore, the only terms which need to be calculated at each time step during a simulation are those which change as the diameter distribution of a given landscape cell changes (i.e. CR, BA, BAL, CCF, ADBH, and RELDBH). They are presented here (using the original symbology) for convenient reference; the reader should refer to Wykoff et al., 1982 and Wykoff, 1986 for a detailed discussion of the FVS model. PROB(a  + a DBH + a DBH  2  0  x  2  PROB(b DBH  bl  0  where CCF  ttee  ),DBH > 10'  XDBH < 10"  16  [Al]  is the single-tree crown competition factor; PROB is the number  of trees per acre associated with each record in an FVS sample tree list (and is set equal to 1 in the present case); DBH is the outside-bark diameter at breast  1 6  Tree crown competition factor (CCFtree) (equation 5, Wykoff, Crookston, and Stage, 1982)  88  height of the tree in inches; and a ,b are species specific constants (and are i  j  different from the a ,b introduced in equation 1 and equation 2). The reader i  i  should refer to Wykoff et al. (1982) for the relevant values of these constants. \n(CR) = HAB + b BAA + b BAA + b, \n(BAA) + 2  x  KCCF  2  +b CCF i  stmd  5  +b HCCF )  s Bd  6  sUnd  +  b DBH + b DBH + b \n(DBH) +  [A2]  2  7  s  b HT + b HT  2  l0  u  17  9  + b \n(HT) + u  b PCT + b \n(PCT) u  X4  where the b's are species specific constants (different from the b's defined in ;  ;  equations 6 or Al); CR is the crown ratio; HAB is a species / habitat type specific constant; BAA is the stand basal area per unit area, in ft2/acre , obtained from equation 6 21s BAA = b I A (where A is the stand (cell) area in N  acres); CCF „d is the stand crown competition factor, and is equal to sta  /^jCCF  tree  for all trees in the stand (cell); and PCT is the basal area percentile  rank of the tree (the percentage of total stand basal area represented by that tree and all trees that are the same size or smaller), where PCT is calculated for the 1 diameter class from the stand density vector (equation 6) TH  as PCT = 100(6. /b ) . T i e reader should refer to Wykoff et al. (1982) for the t  N  relevant values of the species-specific constants.  1 7  Crown Ratio (CR) (equation 13, Wykoff, Crookston, and Stage, 1982)  89  \n{dds) = HAB + LOC + b,SL cos(ASP) + b SL sin(ASP) + b SL + b SL + 2  2  3  4  b EL + b EL + b \n(DBH) + 2  5  6  7  b,CR + b CR + b (BAL/\00) + b BAL/(\00 • \n(DBH +1)) + 2  9  b DBH  2  n  m  u  +b CCF /100 l3  s  where dds is the squared inside-bark diameter increment per decade; LOC is a location constant intercept, dependent on species; HAB is a habitat type constant intercept, dependent on species; SL is the slope (%/100); EL is the elevation (100's of feet); ASP is the aspect (degrees); and BAL is the basal area in larger trees (ft /acre), calculated for the i'th diameter class from the stand 2  density vector (equation 6) as BAL = (b -b ) / A. The reader should refer t  N  t  to Wykoff (1986) for the relevant values of the various constants in equation A3.  1 8  Squared inside bark diameter increment per decade ( dds, inches /10 years) 2  (equation 4, Wykoff,  1986). This equation differs from the one given in Wykoff, 1986. There, the term multiplied by b u is not divided by 100. This is a misprint, however, as I found out after studying the F V S source code.  90  

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