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Forecasting Indian monsoon rainfall using regional circulation fields as predictors : an ensemble neural… Cannon, Alex Jason 2000

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FORECASTING INDIAN MONSOON RAINFALL USING REGIONAL CIRCULATION FIELDS AS PREDICTORS: AN ENSEMBLE NEURAL NETWORK APPROACH by Alex Jason Cannon B.Sc , University of British Columbia, 1995 Dipl . Met., University of British Columbia, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF Master of Science in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Geography) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A November 2000 © Alex Jason Cannon, 2000 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date Z S Aug^s-v 2.000 Abstract Pre-monsoon principal components of circulation fields covering the South Asian subcontinent were used as predictors for all-India summer monsoon rainfall (AISMR) during the period 1958-1998. Predictive skill and stationarity of non-linear ensemble neural network and linear multiple regression model relationships were assessed using a Monte Carlo-based resampling procedure. Monsoon precursor signals represented by the PCs were investigated using a model sensitivity analysis and comparisons were made with recent observational and general circulation modelling studies. Results suggest the presence of coherent, stable predictive relationships between A I S M R and the SLP field during November (median r=0.5, p=0.05), as well as between A I S M R and the 200 hPa geopotential height field during May (median r=0.67, p < 0.005). The latter relationship was essentially linear, and appears to be related to the winter-summer transition between a flow regime dominated by the subtropical westerly jet stream to one dominated by the tropical easterly jet stream, as well as to changes in strength and position of the upper tropospheric Tibetan anticyclone. The former relationship was nonlinear, appearing in essentially the same form at the 850 hPa level (median r=0.37, p=0.06). Possibly related to anomalous SSTs in the Arabian Sea, further work is required to determine physical mechanisms responsible for predictive skill at this lead-time. Weaker relationships were observed between summer monsoon rainfall and PCs of the SLP field during May (median r=0.47, p=0.10) and PCs of the 850 hPa geopotential height field during January (median r=0.40, p=0.10). May SLP-rainfall relationships were not significant until the early 1970s but remained relatively stable for the remainder of the record. Spatially, maximum i i correlations and model sensitivities were centred over northwest India and Pakistan, suggesting a link with pre-monsoon heating and development of the heat low over Pakistan. Relationships between January 850 hPa geopotential heights and rainfall were nonstationary, only showing sig-nificant correlations during portions of the record. Significant differences in skill between neural network and multiple linear regression models were present at this level and lead-time, consistent with indications of nonlinearity and interactions between inputs suggested by the sensitivity anal-ysis. Further work is required to determine whether these relationships have physical meaning or whether they are simply a statistical artifact. i i i Contents Abstract ii List of Tables vii List of Figures viii Chapter 1 Introduction 1 1.1 Research context 1 1.2 Historical work 4 1.2.1 Predictors for monsoon rainfall 4 1.2.2 Monsoon forecast models 8 1.3 Study outline 12 Chapter 2 Data and forecast models 15 2.1 Data 15 2.1.1 Sea-level pressure and geopotential height 15 2.1.2 All-India summer monsoon rainfall (AISMR) 18 2.2 Forecast models 19 2.2.1 Multiple linear regression (MLR) 19 2.2.2 Multi-layer perceptron neural network (MLP) 19 2.2.3 Ensemble averaging and early stopping 23 iv Chapter 3 Forecast results 26 3.1 Evaluating forecast performance 26 3.1.1 200 hPa geopotential height 27 3.1.2 500 hPa geopotential height 29 3.1.3 850 hPa geopotential height 29 3.1.4 SLP 29 3.2 Stationarity of circulation-AISMR relationships 33 3.2.1 May 33 3.2.2 April 42 3.2.3 March 42 3.2.4 February 42 3.2.5 January 42 3.2.6 December 43 3.2.7 November 43 3.2.8 October 43 Chapter 4 Interpreting circulation-AISMR relationships 44 4.1 Linear circulation-AISMR relationships 44 4.1.1 May 54 4.1.2 January 57 4.1.3 November 58 4.2 Nonlinear circulation-AISMR relationships 59 4.2.1 Neural network sensitivity analysis 59 4.2.2 Sensitivity analysis results 72 Chapter 5 Conclusion 92 5.1 Summary 92 5.2 Future research 94 v Bibliography 95 vi List of Tables 1.1 Summary of predictors for Indian summer monsoon rainfall. After Kumar et al. (1995) 5 2.1 Cumulative percent explained variance for PCs of gridded circulation data. N indi-cates the number of PCs retained for each atmospheric level 17 vi i List of Figures 1.1 Mean monthly precipitation and cumulative mean monthly precipitation totals for India, the largest area affected by the South Asian monsoon. Precipitation means are calculated from measurements during the 1871-1993 period 2 2.1 Plots of all-India monsoon rainfall (AISMR) and autocorrelations of the A I S M R data. 20 2.2 Example of a multi-layer perceptron neural network with four inputs, a single hidden layer with five nodes, and a single output 21 2.3 Plots of mean squared errors on training and out-of-bag data during neural network model optimization. When using the early stopping procedure to avoid overfitting, final model parameters are selected to minimize Eout~0f-bag 24 3.1 M L P and M L R split-sample validation results for 200 hPa geopotential height P C predictors 28 3.2 M L P and M L R split-sample validation results for 500 hPa geopotential height P C predictors 30 3.3 M L P and M L R split-sample validation results for 850 hPa geopotential height P C predictors 31 3.4 M L P and M L R split-sample validation results for SLP P C predictors 32 3.5 Sliding split-sample validation results for May P C predictors 34 3.6 Sliding split-sample validation results for Apr i l P C predictors 35 3.7 Sliding split-sample validation results for March P C predictors 36 vi i i 3.8 Sliding split-sample validation results for February P C predictors 37 3.9 Sliding split-sample validation results for January P C predictors 38 3.10 Sliding split-sample validation results for December P C predictors 39 3.11 Sliding split-sample validation results for November P C predictors 40 3.12 Sliding split-sample validation results for October P C predictors 41 4.1 P C loadings for May 200 hPa geopotential height data (top). Scatterplot showing the relationship between May 200 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve 45 4.1 (cont.) 46 4.2 P C loadings for May SLP data (top). Scatterplot showing the relationship between May SLP P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve. . 47 4.3 P C loadings for January 850 hPa geopotential height data (top). Scatterplot showing the relationship between January 850 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve 48 4.4 P C loadings for November 850 hPa geopotential height data (top). Scatterplot showing the relationship between November 850 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve 49 4.5 P C loadings for November SLP data (top). Scatterplot showing the relationship between November SLP P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally- weighted linear regression curve 50 ix 4.6 Correlations between grid-point values of May SLP and 200 hPa predictors and A I S M R 51 4.7 Correlations between grid-point values of January 850 hPa predictors and A I S M R . . 52 4.8 Correlations between grid-point values of November S L P and 850 hPa predictors and A I S M R 53 4.9 Composite 200 hPa geopotential height anomalies during excess and deficient sum-mer monsoon rainfall years 55 4.10 Graphical representation of the sensitivity analysis. A n example of the calculation procedure for the neural network sensitivities and partial derivatives is shown on the left-hand side. The resulting sensitivity plot is shown on the right-hand side 62 4.11 Example effects plots for inputs to the function in Equation 4.2 64 4.12 Example partial derivative plots for inputs to the function in Equation 4.2 67 4.13 Scatterplot showing quantitative measures of variable importance (abscissa) and variable interaction strength (ordinate) for inputs to the function in Equation 4.2. . 68 4.14 Stratified effects plots for inputs X\ and X2 (top) and X$ and X$ (bottom) to the function in Equation 4.2. Colours show ranges of the variables X2 and X3 (top) and X4 and X5 (bottom) respectively: black < 0.25, red > 0.25 and < 0.5, green > 0.5 and < 0.75, and blue > 0.75 69 4.15 Example effects plot for results from multiple neural network runs. Line segments plotted in black represent median values of effects and partial derivatives. Lower and upper line segments plotted in grey indicate lower and upper quartile values of effects and partial derivatives 70 4.16 Calculation of effects for (1) P C input variables and (2) for grid-points of P C input variables 71 4.17 Effects plots for May 200 hPa geopotential height P C predictors 73 4.18 Effects plots for May S L P P C predictors 74 4.19 Effects plots for January 850 hPa geopotential height P C predictors 75 x 4.20 Effects plots for November 850 hPa geopotential height P C predictors 76 4.21 Effects plots for November SLP P C predictors 77 4.22 Effects plots for May 200 hPa grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 81 4.23 Effects plots for May 200 hPa grid-point predictors in the ensemble M L P models. The top left plot corresponds to the grid-point centred on 34° N and 66° E . The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 82 4.24 Effects plots for May SLP grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 83 4.25 Effects plots for May SLP grid-point predictors in the ensemble M L P models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 84 4.26 Effects plots for January 850 hPa grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 85 4.27 Effects plots for January 850 hPa grid-point predictors in the ensemble M L P models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 86 4.28 Effects plots for November 850 hPa grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34° N and 66° E . The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 87 4.29 Effects plots for November 850 hPa grid-point predictors in the ensemble M L P mod-els. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 88 xi 4.30 Effects plots for November SLP grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 89 4.31 Effects plots for November S L P grid-point predictors in the ensemble M L P models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E 90 4.32 Stratified effects plots for P C I and PC2 of the January 850 hPa geopotential height field. Colours in each plot correspond to ranges in scores of the other P C : black < —1, red > —1 and < 0, green > 0 and < +1, and blue > +1 91 xi i Chapter 1 Introduction 1.1 Research context The South Asian monsoon is one of the most important large-scale weather systems in the world. Named for the Arabic for season, mausim, the monsoon circulation is characterized by winter-summer reversals in atmospheric flow over the eastern tropics and subtropics (Pant and Kumar, 1997). These changes are primarily driven by the seasonal cycle in solar insolation. Large gradients in pressure develop across the equator as the result of differential heating between ocean and land. During the northern hemisphere winter these pressure gradients drive dry winds out of continental Asia southwest across the equator; during the northern hemisphere summer, flow from the southern hemisphere ocean provides moisture for precipitation over the South Asian subcontinent. Precipi-tation totals during the winter monsoon season and the summer monsoon season differ by nearly an order of magnitude (Figure 1.1). Between 75-90 percent of the region's rainfall occurs during the four months of the summer monsoon (June-September); less than 25 percent occurs during all other months. The unequal seasonal distribution of rainfall makes the monsoon system vitally important to the Asian tropical and subtropical region. More than one billion people live in India, the largest and most populous area affected by the South Asian monsoon circulation. Of people in India, more than 75 percent of the country's workers are employed in agricultural production. Crops grown during the summer monsoon season 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC MONTH Figure 1.1: Mean monthly precipitation and cumulative mean monthly precipitation totals for India, the largest area affected by the South Asian monsoon. Precipitation means are calculated from measurements during the 1871-1993 period. 2 account for 60-70 percent of the country's total production of grain (Parthasarathy et al., 1992). Measures of summer monsoon rainfall have been shown to be highly correlated with seasonal foodgrain production (r = 0.878 for years 1966-1988) (Parthasarathy et al., 1992). As a result, consequences of rainfall-deficient summer monsoons can be severe. Meteorological droughts, years in which summer rainfall accounts for less than 75 percent of the long-term average at more than two-fifths of meteorological subdivisions in India, have occurred eight times during the period 1871-1984 (Parthasarathy et al., 1987). The major drought of 1877 resulted in a serious famine that caused more than five million deaths in Bombay, Madras, and Mysore (Dyson, 1989). As a result of this famine, the India Meteorological Department (IMD) developed a seasonal forecasting program to try and predict summer rainfall amounts prior to monsoon onset in June. The first experimental seasonal forecast, performed by H . H . Blanford in 1882, was based on an empirical relationship between extent and depth of Himalayan snow in spring and climate conditions in the northwest India (Blanford, 1884). Operational forecasts for monsoon rainfall in India and Burma were started in 1886. Since that time, seasonal predictions have remained the mandate of the I M D . In addition to large-scale boundary conditions such as Himalayan snow depth, predictors now include indicators of E l Nino-Southern Oscillation strength, indices related to surface pressure and temperature in India, regional upper-air circulation conditions, strength of cross-equatorial flow, and other hemispheric and global-scale atmospheric conditions (Kumar et al., 1995). Forecast models used for predictions have progressed in complexity from simple univariate linear regression models to full-scale atmospheric general circulation models. Because of the economic and societal importance of the summer monsoon in India, efforts to identify new predictors and models for seasonal forecasts of rainfall are ongoing. Even small increases in skill and lead time of these seasonal forecasts can be significant (Livezey, 1990). The next section summarizes results of previous work describing predictors and models for long-lead forecasts of Indian summer monsoon rainfall. This chapter concludes with an overview of goals for the current study and an outline of the remainder of this work. 3 1.2 Historical work 1.2.1 Predictors for monsoon rainfall While numerous, potential predictors for monsoon rainfall can be classified into four basic categories (Kumar et al., 1995). A summary of predictors and the strength of their association with summer monsoon rainfall is given in Table 1.1. This table is modified from the one given by Kumar et al. (1995) and has been updated to reflect new predictors described in the literature since the original date of publication. Brief descriptions of proposed causal relationships between these variables and monsoon rainfall are given in the following sections. E N S O indicators The development of predictors related to the El Nino-Southern Oscillation (ENSO) phenomenon began with the work of Walker (1924) who postulated a direct link between the Walker circulation and monsoon rainfall over India. Evidence suggests that strong monsoon years are associated with La Nina events and positive values of the Southern Oscillation index and that weak monsoon years are associated with E l Nino events and negative values of the Southern Oscillation index. However, because ENSO and the monsoon circulation develop in phase (or with the monsoon circulation leading ENSO), long-lead predictions based on ENSO parameters have largely been unsuccessful (Shukla and Paolino, 1983). Seasonal correlations between ENSO conditions and summer monsoon rainfall suggest a switch in sign of the relationships from positive in the preceding winter to negative in the spring. Thus, while precursory relationships between monsoon rainfall and the Southern Oscillation or SSTs in the equatorial Pacific Ocean during each individual season are poor predictors, evidence suggests that the pre-monsoon tendency of these conditions between winter and spring may be better indicators of monsoon rainfall (Parthasarathy et al., 1988). 4 Table 1.1: Summary of predictors for Indian summer monsoon rainfall. After Kumar et al. (1995). Class Predictor Per iod r Record Reference ENSO indicat. SLP Darwin M A M - D J F -0.63 1951-80 Parthasarathy et al . (1988) Tah i t i -Darwin M A M - D J F +0.43 1951-80 Parthasarathy et al . (1988) Plaisane M A M - D J F -0.40 1951-80 Parthasarathy et al . (1988) Santiago-Darwin M A M - D J F +0.52 1951-80 Parthasarathy et al . (1988) Adelaide M A M - D J F -0.36 1951-80 Parthasarathy et al . (1988) Cordoba M A M - D J F -0.40 1951-80 Parthasarathy et al . (1988) Buenes Aires M A M - D J F -0.40 1951-80 Parthasarathy et al . (1988) SST Pacific S S T Region III M A M - D J F -0.52 1951-80 Parthasarathy et al . (1988) Cross-equator. Agalega M A M - D J F -0.44 1951-80 Parthasarathy et al . (1988) Nouvelle-Agalega M A M - D J F +0.45 1951-80 Parthasarathy et al . (1988) Central Indian Ocean S S T S O N +0.87 1977-94 Cla rk et a l . (1999) S Indian Ocean O L R A p r . +0.75 1974-96 Prasad et a l . (2000) Bay of Bengal O L R M a y -0.66 1974-96 Prasad et al . (2000) A r a b i a n Sea S S T D J F +0.53 1945-94 Cla rk et a l . (1999) N W Aus t ra l i a S S T D J F +0.58 1945-94 Cla rk et a l . (1999) N Aus t . / Indon . S S T A p r . +0.55 1949-91 Nicholls (1995) Indian Ocean S S T M A M +0.51 1951-80 Parthasarathy et al . (1988) Global/hemis. Temp. N H sfc. air temp. J F +0.41 1965-94 V e r m a et al . (1985) De B i l t sfc. temp. J +0.66 1965-94 Dugam et al . (1993) Snow Eurasian snow-cover F -0.47 1973-92 Parthasarathy and Yang (1995) Eurasian snow-melt area May-Feb . -0.44 1966-85 K u m a r (1988) W Eurasian snow cover D J F M -0.63 1973-94 Bamza i and Shukla (1999) Central Siber ia snow cover N +0.76 1966-84 K r i p a l a n i and K u l k a r n i (1999) Moscow snow cover J -0.60 1966-84 K r i p a l a n i and K u l k a r n i (1999) QBO 10 h P a zonal wind (Balboa) J +0.52 1958-85 Bhalme et a l . (1987) Regional cond. SLP Bombay M A M - D J F -0.74 1951-80 Parthasarathy et al . (1988) Tr ivandrum M A M - D J F -0.63 1951-80 Parthasarathy et al . (1988) Min icoy M A M - D J F -0.49 1951-80 Parthasarathy et al . (1988) W central India M A M -0.63 1951-80 Parthasarathy et al . (1992) Jhodpur M a y -0.69 1965-94 Bha lme et al . (1986) Heat low at Pakis tan M a y -0.57 1948-92 Singh et al . (1995) Sfc. temp. West central India (mean) M A M +0.60 1951-80 Parthasarathy et al . (1990) E peninsular India (min.) M a r . +0.69 1951-80 K u m a r et a l . (1997) W central India (min.) M a y +0.67 1951-80 K u m a r et a l . (1997) Circulation 500 h P a ridge location A p r . +0.71 1939-84 Mooley et al . (1986) 500 h P a ridge disp. A p r . - M a r . +0.73 1967-87 K u m a r et a l . (1992) A h m e d 850 h P a hgt. M a y -0.34 1965-94 Bha lme et al . (1986) Ca lcu t t a 850 h P a hgt. M a y -0.47 1965-94 Bha lme et al . (1986) N W India 200 h P a wind M a y -0.59 1965-94 V e r m a and Kamte (1980) 200 h P a merid. wind M a y -0.57 1965-94 Parthasarathy et a l . (1991) 5 Cross-equatorial flow and moisture flux Moisture flux from the Indian Ocean and Arabian Sea drives monsoon rainfall during the summer season. While work has suggested that cross-equatorial flow during the onset of the summer monsoon is correlated with monsoon rainfall (Parthasarathy et al., 1988), parameters related to this mechanism have not found use in empirical forecast models (Kumar et al., 1995). Recent work by Prasad et al. (2000), however, suggests that outgoing long-wave radiation in the Indian Ocean and the Bay of Bengal, indicators of SST and areas of convection over the ocean, may be good predictors of monsoon rainfall in India. At longer lead times, Clark et al. (1999) found that SSTs in the central Indian Ocean, Arabian Sea, and northwest of Australia during preceding fall and winter months were highly correlated with monsoon rainfall. Because these relationships did not appear to persist through the spring, Clark et al. (1999) hypothesized that they were most likely due to a delayed coupled ocean-atmosphere mode of variability and were not directly related to moisture supply and evaporation. Global/hemispheric conditions Three main aspects of conditions measured on global and hemispheric spatial scales have been linked to the summer monsoon circulation in India. Firstly, northern hemisphere temperatures appear to be positively correlated with monsoon rainfall (Verma et al., 1985, Dugam et al., 1993) and are significant in winter, a full season prior to monsoon onset. Representing a slowly varying boundary condition for the development of the monsoon circulation, snow-cover in Eurasia during winter (Kumar, 1988, Parthasarathy and Yang, 1995, Bamzai and Shukla, 1999, Kripalani and Kulkarni, 1999) has also been linked with monsoon rainfall amounts during summer. Measures of hemispheric temperature and snow-cover, however, are likely related and are not independent pre-dictors of monsoon rainfall amounts. Lastly, the quasi-biennial oscillation in tropical stratospheric winds has been correlated with the strength of the summer monsoon over India, possibly through modulation of the frequency of storms during the monsoon season (Bhalme et al., 1987). Taking 6 into consideration the downward progression in the reversal of zonal stratospheric winds, Bhalme et al. (1987) found significant correlations between winds at the 10 hPa level during January and summer monsoon rainfall. Regional conditions Of the four main categories, recent work suggests that regional conditions over India form the most important set of predictors for summer monsoon rainfall (Bhalme et al., 1987, Parthasarathy et al., 1988, Kumar et al., 1997). Two primary mechanisms have been identified as reliable indicators of monsoon performance in India: 1) development of the surface heat low over Pakistan and northwest India, and 2) the transition between westerly and easterly upper-level flow regimes. Bhalme et al. (1986), Parthasarathy et al. (1988), Parthasarathy et al. (1990), Parthasarathy et al. (1992), Singh et al. (1995), and Kumar et al. (1997) each identified significant correlations between summer monsoon rainfall and indices related to surface level pressures and temperatures over India during spring. Strongest correlations were located over northwestern India and Pakistan, the centre of a strong surface heat low. Related to the development of land-sea thermal gradients, sea-level pressures typically exhibited negative correlations with summer rainfall and temperatures exhibited positive correlations. The establishment of the monsoon circulation over India also involves the transition from an upper-level flow regime dominated by the westerly subtropical jet stream to one dominated by the tropical easterly jet stream (Pant and Kumar, 1997). Reflecting seasonal variations in maximum heating and in location and intensity of thermal gradients, upper-level flow is strongly influenced by topography such as the Himalayas and the Tibetan plateau which act as mechanical barriers to flow and as high-level heat sources. Verma and Kamte (1980) and Parthasarathy et al. (1991) investigated correlations between the meridional component of May winds at 200 hPa and summer monsoon rainfall. Southerly meridional winds were typically associated with weak monsoon rainfall, reflecting the equatorward intrusion of subtropical westerlies to the west of India resulting in large-scale failure of the monsoon circulation (Joseph et al., 1981). 7 In addition to development of the heat low and characteristics of the upper-level flow, location of the 500 hPa ridge over India during spring has also been identified as one of the strongest predictors of summer monsoon rainfall (Mooley et al., 1986, Kumar et al., 1992). North-south positioning of the ridge reflects variations in solar heating and the seasonal evolution of the mid-tropospheric circulation. More southerly ridge positions are typically associated with the persistence of colder tropospheric conditions into June and decreased summer monsoon rainfall (Mooley et al., 1986). A l l of the studies discussed above identified correlations between regional circulation indices and Indian summer monsoon rainfall. Recent observational and modelling evidence suggests that coherent patterns in the large-scale pre-monsoon circulation exist and that these patterns can be used as predictors of monsoon strength in India (Yang et al., 1996). Principal components or empirical orthogonal functions (EOFs) of circulation fields may be an effective means of capturing this type of variability and could be less sensitive to small shifts in location than point-based circulation indices. However, predictors that integrate regional circulation conditions have not yet been used to forecast summer monsoon rainfall. Prasad and Sikka (1982) studied the association between summer monsoon rainfall and empirical orthogonal functions of the 700 hPa geopotential height field over India. Circulation-rainfall relationships were only studied during summer months; precursor patterns were not investigated. Similarly, Singh and Kripalani (1986) used extended EOFs to track the evolution of sea-level pressure and 700 hPa height fields over India during summer months. Relationships between the extended EOFs and rainfall were then identified. In both studies, intraseasonal variations in the EOFs were linked with changes in summer rainfall. 1.2.2 Monsoon forecast models While predictors for monsoon rainfall have evolved substantially since the initial forecasts by Blan-ford (1884), forecast techniques have remained relatively constant for much of the past century. The majority of studies and operational forecasts for monsoon rainfall use some form of empirical or statistical model. Of these efforts, those using linear regression are most common. Nonlinear 8 models for monsoon prediction have only recently been studied in detail. Dynamic models are still not commonly used for producing operational seasonal forecasts, but have been widely used for investigating physical mechanisms driving the South Asian monsoon circulation. Linear regression models Linear regression models have been used for monsoon prediction since the early twentieth century and account for the majority of empirical forecast models presented to date. A l l of the studies listed in Table 1.1 used simple linear correlation or regression analyses. In each, linear relationships were identified between one or more predictors and a single predictand. While it is possible to account for nonlinear and interactive relationships using this methodology, form of the relationships must be known a priori so that appropriate transformations can be applied to the predictors. Because of the large number of potential predictors, many of which are correlated with one another, operational models typically use some form of step-wise selection procedure to remove redundant and irrelevant inputs (Kumar et al., 1995). Recently, a number of studies have focused on using principal component analysis (PCA) to reduce the dimensionality of the predictor set used for monsoon prediction. Kumar et al. (1995) found that the first principal component (PC) of 19 predictors accounted for more than 46 percent of variance in the full dataset. Because the length of record available for building monsoon forecast models is typically less than 50 years, data reduction techniques are crucial for producing models that are capable of forecasting well. Overfitting can be difficult to avoid, especially when the number of potential predictors is large relative to the dataset size. As a result, producing accurate forecast models can be difficult, even when using a simple model such as linear regression. Small sample sizes also hinder proper model performance evaluations. Kumar et al. (1995) suggests that validation procedures employed in many monsoon forecasting studies are not sufficient for estimating true forecast performance or for comparing the performance of different models. 9 Nonlinear regression models As an alternative to linear models, empirical models capable of representing complex nonlinear and interactive input-output relationships have also been used in the seasonal prediction of monsoon rainfall. Hsieh and Tang (1998) and Gardner and Dorling (1998) summarize the application of one such model, the neural network, to prediction and data analysis problems in the atmospheric and oceanographic sciences. Two excellent introductions to neural network models can be found in Bishop (1995) and Reed and Marks (1999). Neural networks form a class of computational model loosely based on the biological ner-vous system. Typically organized in connected layers of simple processing nodes, these models are capable of performing a variety of statistical tasks including classification, clustering, and regres-sion (Sarle, 1994). The multi-layer preceptron (MLP) neural network, one of the most common architectures, is capable of representing nonlinear mappings between input and output variables. Nonlinearity, as well as interactions between variables, can be modelled without prior specifica-tion. Neural network models can have arbitrary numbers of inputs and outputs and the internal complexity of the models can be easily adjusted by the user. Because of their inherent flexibility, neural networks are well-suited for use in a variety of forecasting and data analysis problems, especially in cases where input-output relationships are complex and are not easily diagnosed prior to model building. For example, neural network models have been successfully used for the short-term prediction of variables related to air pollution (Boznar et al., 1993, Lee, 1995, Pasini and Potesta, 1995, Y i and Prybutok, 1996, Comrie, 1997, Cannon and Lord, 2000) and severe weather (Marzban and Stumpf, 1996, Hall et al., 1999), among others. Neural networks have also been used to produce longer-term seasonal forecasts of climatic and oceanographic variables. Hastenrath et al. (1995) predicted rainfall during December-February over the Highveld of South Africa using the Southern Oscillation index, quasi-biennial oscillation index, Indian Ocean equatorial winds, and Indian Ocean sea-surface temperatures as predictors. 10 Explained variance more than doubled relative to linear models using the same set of predictors (62 percent compared to 30 percent). More recently, Tangang et al. (1997), Tangang et al. (1998), and Tangang et al. (1998) predicted sea-surface temperatures in the equatorial Pacific Ocean at lead-times up to 15 months using EOFs of sea-level pressure and wind-stress fields. Results were typically better than those from linear models, particularly as lead-time was increased. In monsoon forecasting, Navone and Ceccatto (1994) used a neural network model to represent the relationship between 500 hPa ridge position over India in Apr i l and summer monsoon rainfall. Results indicated substantial improvement over a linear model that used the same predictor. However, further work was identified as necessary to demonstrate that this model is capable of generating accurate forecasts of summer monsoon rainfall (Kumar et al., 1995). Goswami and Srividya (1996) developed a neural network-based forecast system to predict monsoon rainfall from previous values of the rainfall series. Like Navone and Ceccatto (1994), results were encouraging. However, out of sample testing was limited and used relatively small, fixed test sets (maximum of 15 years). Despite their strengths, Hsieh and Tang (1998) identified three main areas of difficulty in using neural networks to forecast aspects of the climate system. Firstly, climatic records are typically short relative to the timescale of interest; nonlinear instability may result in the search for the optimum set of weights on a small dataset. Secondly, meteorological data are typically measured at numerous grid points; using each relevant grid-point as an individual predictor in the model may lead to overfitting, especially when coupled with a short period of record. Finally, relationships identified using neural networks may be difficult to interpret; nonlinearity and interconnectivity of neural network architectures make individual parameters almost incomprehensible. Solutions to each of these three problems have been addressed and adopted by neural net-work practitioners in the climatic sciences (see Tangang et al., 1998, for example). Use of ensemble neural network models, nonconvergent training methods, pre-processing with principal component analysis, and visualization and interpretation techniques are all described by Hsieh and Tang (1998). However, these types of methods have not been widely adopted by monsoon forecasters. 11 General circulation models Large-scale dynamical models or general circulation models (GCMs) have not been widely used to prepare seasonal forecasts for monsoon rainfall (Kumar et al., 1995). Because of their complexity and computational requirements, real-time use of GCMs for forecasting is inherently difficult and forecast accuracy is still relatively poor. In a recent review of the prospects for monsoon pre-dictability on seasonal time scales, Webster et al. (1998) concluded that dynamic models still lag behind empirical models in terms of skill. Recent monsoon modelling exercises do, however, suggest that GCMs are useful for inves-tigating physical mechanisms driving monsoon variability. For example, Soman and Slingo (1995) found that a G C M was capable of simulating strong and weak monsoon conditions when forced with different patterns of prescribed SSTs in the tropical Pacific and Indian Oceans. Vernekar et al. (1995) successfully simulated weak summer monsoon conditions in a G C M forced with anomalously high Eurasian snow cover. Yang et al. (1996) found that large-scale precursory signals associated with interannual variability of summer monsoon strength were well simulated in a ten year run of the Goddard Laboratory for Atmospheres GCM. While not yet suitable for real-time seasonal monsoon forecasting, dynamical modelling is a viable method for verifying the physical meaning of empirically derived forecast rules. 1.3 Study outline After reviewing prior studies on seasonal forecasting of monsoon rainfall, two specific areas of re-search appear warranted. Firstly, while specific circulation indices have been identified as important predictors for monsoon rainfall, no study has tried to systematically relate circulation conditions over the entire Indian region to the strength of the summer monsoon. Principal components of regional circulation fields have been studied in relation to monsoon rainfall, but only as a means of tracking evolution of the monsoon during the summer months, never in a seasonal forecasting context. Secondly, past investigations have focused almost exclusively on linear relationships be-12 tween predictors and monsoon rainfall. Studies that employ nonlinear techniques, such as neural networks, have typically used predictors identified using linear correlation or regression analysis. Modern neural network training techniques have not been widely adopted and comparisons be-tween linear and nonlinear methods have been relatively weak and have not been able to clearly demonstrate the advantages of more complex model types. In studies that do claim superiority of nonlinear models, little has been done to interpret the predictor-predictand relationships. The current study attempts to examine, in a systematic fashion, linear and nonlinear rela-tionships between regional circulation conditions over India and summer monsoon rainfall. Specific goals of this study are: 1. the identification of circulation-based predictors for summer monsoon rainfall in India 2. the evaluation and comparison of multiple linear regression and neural network forecast mod-els 3. the examination and interpretation of relationships between circulation conditions and sum-mer monsoon rainfall. Four aspects of the approach used are novel in the context of monsoon rainfall prediction. Firstly, principal components of gridded circulation fields from several atmospheric levels are used as predictors for monsoon rainfall. Secondly, ensemble neural networks are adopted to improve the quality of the forecasts. Thirdly, predictive skill of the neural network and linear regression models and stationarity of the circulation-rainfall relationships are evaluated using a Monte-Carlo based split-sample resampling technique. Lastly, a graphical sensitivity analysis is used to investigate relationships between the circulation predictors and summer monsoon rainfall. A preliminary study applying neural networks and circulation PCs to monsoon forecasting was published by Cannon and McKendry (1999). Similarities and differences in methodology and results between the published study and the current work will be identified throughout the remainder of the text. Circulation data and empirical forecast methods are described in Chapter 13 2. Model forecast results are described in Chapter 3. Linear and nonlinear relationships between circulation predictors and summer monsoon rainfall are discussed in Chapter 4. Finally, a summary of study results and suggestions for future research are presented in Chapter 5. 14 Chapter 2 Data and forecast models 2.1 Data 2.1.1 Sea-level pressure and geopotential height In this study, relationships between Indian monsoon rainfall and gridded circulation data from four atmospheric levels over the South Asian subcontinent were investigated. Daily averaged SLP, 850 hPa, 500 hPa, and 200 hPa geopotential height data (2.5° by 2.5° resolution) were obtained from N C E P / N C A R reanalysis output (Kalnay et al., 1996) provided by the Climate Diagnostics Center (NOAA) . A subset covering the region from 62.5°E to 95°E and 7.5°N to 35°N was extracted and monthly averages for pre-summer monsoon months (October-May) were computed for the 41 year period from 1958 to the end of 1998. This extends the analysis conducted by Cannon and McKendry (1999) which only considered the period 1958-1994 and months from December to May. Gridded data were averaged in space to a resolution of 5° by 5°. Standardized SLP and geopotential height anomalies were then computed by subtracting the climatological monthly mean from each of the monthly average values and dividing by the monthly standard deviation. These eight series of standardized monthly anomaly fields (36 grid points) were used as indicators of pre-monsoon circulation at each atmospheric level. Because grid-point values tend to be highly correlated in space, P C A was used to reduce the dimensionality of the original data set and provide uncorrelated representations of the circulation data. For each series and level, 15 an S-mode P C A (Green, 1978) was performed on the correlation matrix R of the pre-monsoon circulation anomalies R=±x'.x. (2.1) where N is the number of years and Xs is the matrix of standardized circulation anomalies under consideration. The correlation matrix was used rather than the covariance matrix to ensure that all grid points were weighted equally, irrespective of their variance. For each monthly series of circulation data, the correlation matrix was decomposed into a matrix of eigenvectors U and a diagonal matrix of eigenvalues D. P C loadings F and standardized P C scores Zs were then computed F = UD05 (2.2) Zs = XaUD-°-5. (2.3) This method of P C A pre-processing differs from that conducted by Cannon and McKendry (1999). In the current study, P C A was applied to each level and month separately. In Cannon and McKendry (1999), P C A was applied to all pre-monsoon months at a given level simultaneously. P C scores for each pre-monsoon month were then extracted from the full set of scores. While this increased the effective size of each data series in the P C A , ability to interpret circulation-rainfall relationships was reduced because data for all months shared the same set of P C loadings. Applying P C A to each month individually allowed spatial relationships to be investigated on a monthly time scale. Barnston and Livezey (1987) suggest that the improved interpretability of single month rotated P C A solutions makes up for the loss of sample size that comes with not pooling data from adjacent months. 16 Table 2.1: Cumulative percent explained variance for PCs of gridded circulation data. N indicates the number of PCs retained for each atmospheric level. Level N O C T N O V D E C J A N F E B M A R A P R M A Y SLP 3 93.0 88.5 91.9 92.4 93.0 90.3 89.0 88.7 850 hPa 3 95.5 93.6 95.6 94.9 93.4 93.7 93.4 93.5 500 hPa 4 96.5 95.6 97.5 97.2 97.2 97.3 96.5 96.4 200 hPa 4 97.7 97.2 98.1 97.5 97.7 97.1 96.9 97.4 The number of PCs to retain as predictors was determined by taking the average of three truncation criteria. Rule-N is a Monte-Carlo testing procedure developed for use with PCs of gridded atmospheric data (Overland and Preisendorfer, 1982). A large sample of independent variables is generated from a Gaussian distribution of random numbers. These variables are formed into matrices of the same dimension as the data being analyzed and the correlation matrix for each set is computed. Principal components are then calculated for each correlation matrix and the percent variance accounted for by each component is determined. PCs of the actual data are retained if the percent variance accounted for by a given component exceeds the percent variance accounted for by the 95*^ percentile of the corresponding resampled component distribution. The eigenvalue-one rule (Kaiser, 1959) retains components with eigenvalues exceeding unity. As a result, this rule retains PCs that account for more variance than the average explained by a single original variable. Lastly, the number of PCs accounting for 90 percent of the total variance in the original dataset was used as a criterion for truncation. Table 2.1 shows the cumulative percent variance accounted for by the retained components at each level and month. In all cases cumulative explained variance exceeded 88 percent. The number of PCs retained using this combined procedure was typically greater than the number recommended by the rule-N test and less than the number recommended by the eigenvalue-one rule. A recent study by Livezey and Smith (1999) suggests that the rule-N test is conservative and that information contained in higher PCs may be useful in maximizing forecast performance. Green (1978) suggests that the eigenvalue-one rule may retain more components than indicated by subjective rules or criteria that take into account sampling variability, such as rule-N. Combining 17 these tests provided a truncation criterion that was neither too conservative nor too liberal. Preliminary inspection of the retained component loadings showed patterns characteristic of those found by Buell (1975). As interpretation of the developed models was one of the goals of this study, the PCs were rotated using Kaiser's varimax criterion (Kaiser, 1958). Enhanced mete-orological and climatic interpretability of rotated PC solutions has been documented by Richman (1986) and Barnston and Livezey (1987), among others. Varimax rotation seeks simple structure of the PC loadings while maintaining uncorrelated PC scores. The original loadings matrix F is rotated such that the new loadings matrix G contains columns with as few non-zero entries as possible. Standardized scores Z8V of the rotated components Zsv = XsG(G'G)~l (2.4) were used as predictors in the empirical forecast models described in Section 2.2. 2.1.2 All-India summer monsoon rainfall ( A I S M R ) The AISMR data set consists of area-weighted averages of summer (June-September) rainfall totals from 306 district rain gauge stations. Approximately ninety percent of the country is accounted for, with a number of hilly regions removed from the analysis. Originally developed and described in Parthasarathy et al. (1987) for the period 1871-1984, the data set was recently extended in Parthasarathy et al. (1994) to include years through 1993. Data from 1994-1998 were obtained from the Indian Institute of Tropical Meteorology. To match the available circulation data record, AISMR totals for years from 1958 to 1998 were extracted from the series. Plots of the time series and its autocorrelations over the period (Figure 2.1) show the sig-nificant interannual variability and lack of persistence in AISMR. While this lack of persistence has hindered attempts at pure time-series forecasts for AISMR (Kumar et al., 1995), it does facilitate significance testing for the presence of circulation-AISMR relationships. Tests on time series re-quire the number of degrees of freedom to be lowered to take into account serial correlations in the 18 data (Quenouille, 1952). As a result of low autocorrelations in the AISMR series, magnitudes of the corrections for data considered in the current study were small. For simplicity, all correlation coefficients reported subsequently have not been corrected for the presence of serial correlations in the series. For use in the forecast models, the mean value over the 1958-1998 subset was removed from each summer's reported total. These anomalies were then standardized using the standard deviation over the subset data period. For use in subsequent sections, years with AISMR values greater than +1 standard deviation or less than -1 standard deviation from the subset mean were defined as having had excess or deficient rainfall totals respectively. 2.2 Forecast models 2 . 2 . 1 Mult iple linear regression ( M L R ) Multiple linear regression (MLR) was used to construct models relating the pre-monsoon circulation PCs to AISMR. MLR models were of the form where Y is the model output, b is an intercept parameter, mi are slope parameters, and Xi are model inputs. Slope and intercept parameters were set using standard least-squares minimization techniques. 2 . 2 . 2 Multi-layer perceptron neural network ( M L P ) An example of the MLP neural network architecture used to forecast AISMR is shown in Figure 2.2. Model outputs are generated by feeding input data through the model from left to right. Input variables Xi are first presented at the input layer and are multiplied by the hidden-layer weights lWij, the main parameters in the neural network model. Output from each hidden-layer node hj (2.5) 19 u. ci O < -r— 15 10 Lag Figure 2.1: Plots of all-India monsoon rainfall (AISMR) and autocorrelations of the A I S M R data. 20 Hidden layer Figure 2.2: Example of a multi-layer perceptron neural network with four inputs, a single hidden layer with five nodes, and a single output. is obtained by taking the hyperbolic tangent of the sum of weighted inputs plus the hidden-layer bias 1bj, a parameter analogous to the intercept term in a linear regression model hj = tanh (^T X{ lWij + . (2.6) Use of the hyperbolic tangent, a sigmoid-shaped function, in the hidden-layer nodes allows the model to represent nonlinear relationships. Output from the network Y is a linear function of the weighted hidden-layer outputs 21 Y = J2hj 2wj + 2 b (2-7) j where 2Wj are the output layer weights and 2b is the output layer bias. Training of the neural network involves minimizing the mean squared error E cases over the set of N observed input / output cases. Training cases are repeatedly presented to the network and model weights and biases are adjusted to minimize E. Training can be accomplished using a variety of nonlinear multivariate optimization algorithms, for example gradient descent (commonly referred to as backpropagation in the neural network literature) or conjugate gradients. Resilient backpropagation (RPROP) (Riedmiller and Braun, 1993) was used as the training algorithm for all neural networks in the current study. R P R O P is a heuristic modification of backpropagation that uses adjustable learning rates for each weight and bias, rather than a single, fixed learning rate for all weights and biases. In the R P R O P algorithm, learning rates are modified during training based upon the sign of the error gradient. If the sign of the gradient is constant between iterations, the learning rate is increased. If the sign switches, the learning rate is decreased. R P R O P is insensitive to values of the control parameters and is one of the fastest and most efficient neural network training algorithms (Reed and Marks, 1999). Default control parameters, described in Riedmiller and Braun (1993), were adopted in the current study. Initial values for weights and biases were set to random values ranging from -0.5 to 0.5. The number of nodes in the hidden-layer was fixed at five for all training runs. Although this yielded a relatively large number of adjustable model parameters (26 for models with three inputs and 31 for models with four inputs), the effective number of parameters in the models was controlled by averaging the results of an ensemble of networks and by stopping the training of each model prior to reaching the minimum value of E. As discussed below, this procedure prevented the model from overfitting the training data. 22 2.2.3 Ensemble averaging and early stopping As suggested by Hsieh and Tang (1998), means of avoiding nonlinear instability and overfitting of the training data are necessary when applying neural networks to typical climate forecasting problems. In the current study, an ensemble averaging procedure was combined with nonconvergent training, both methods recommended by Hsieh and Tang (1998), to help accomplish these goals. Bootstrap aggregation (bagging), a procedure developed by Breiman (1996) for improving the stability and accuracy of unstable models, was used to reduce overfitting and improve perfor-mance of the neural networks developed in the current study. In bagging, a number of training data sets are generated by sampling with replacement from the available pool of data, models are trained on the resampled data sets, and results from the individual ensemble members are averaged to yield final output values. Ensemble averaging reduces variance of the outputs due to the random initialization of weights and biases and the random selection of training set cases. Each resampled training set is selected to be equal in size as the original data set. As resampling is done with replacement, approximately 37 percent of training cases are not represented in each of the bagged sets. These out-of-bag cases can be used as a means of estimating the generalization error of the model and thereby stopping training prior to overfitting on the training data. This procedure, known as early stopping, is a simple method for ensuring optimum model performance on data not used in the training process (Finnoff et al., 1993). Rather than choosing weights and biases that maximize performance on the training set, the final neural network weights and biases are chosen to maximize model performance on data held out of the training dataset. This process is illustrated in Figure 2.3. During the optimization process, network weights and biases are stored after each presenta-tion of training cases and a value of E is calculated using these data ( E t r a i n ) . The out-of-bag cases are then presented to the model and another value of E is calculated using the stored weights and biases (Eout-0f-bag)- This provides an estimate of the model's performance on data not included in the training set. Following convergence on the training dataset, weights and biases corresponding 23 1.20 -T o.oo I : M : : i i : i : i i i i i ! : i ! i i : i : i : : : : i : i : : : : : : : ! M : : : : : : : : : i i ! : : : : : i : I 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 Training iteration Figure 2.3: Plots of mean squared errors on training and out-of-bag data during neural network model optimization. When using the early stopping procedure to avoid overfitting, final model parameters are selected to minimize Eout-0f-bag-to the minimum value of Eout-0f-bag a r e retrieved and are used as final parameters in the trained model. The combined use of bagging and early stopping offers an attractive means of preventing individual networks from overfitting while still allowing the ensemble model to make use of all available data (Perrone and Cooper, 1993). Results from recent studies also indicate that ensemble neural network models using early stopping are less sensitive to the number of hidden nodes used than individual neural networks (Perrone and Cooper, 1993, Tetko et al., 1995). For this reason, five hidden-layer nodes were used in the current study. While this provided sufficient model complexity 24 to overfit the training data, the early stopping / bagging procedure ensured adequate performance on out-of-sample data. Each ensemble model in the current study consisted of 20 individual neural networks. 25 Chapter 3 Forecast results 3.1 Evaluating forecast performance Given the small sample size available (41 years), a Monte-Carlo resampling procedure was used to estimate the true forecast skill of the trained ensemble neural network and M L R models. The procedure used in the current study differs from the one used in Cannon and McKendry (1999). In Cannon and McKendry (1999), a bootstrap-based procedure (LeBaron and Weigend, 1998) was used to evaluate the performance of 100 individual M L R and neural network models. Out-of-bootstrap results from the individual models were then combined and leave-one-out bootstrap estimates of error for the resulting bagged models (Rao and Tibshirani, 1997) were used as indicators of ensemble model performance. In the initial study, comparisons between the individual models and the ensemble models suggested that ensemble averaging of the neural network models improved skill but averaging of the M L R models did not, a result consistent with findings of Breiman (1996). Based on this result, bagged ensemble neural network models and individual M L R models were used exclusively in the current study. To estimate forecast performances of the M L R and ensemble neural networks, a Monte-Carlo resampling procedure based on the repeated application of split-sample validation (Weiss and Kulikowski, 1991) was used. Model parameters were set using three-quarters of the available cases (30 years) and the resulting models were used to predict output values for the 26 remaining cases (11 years). Predictive skill on test set cases was evaluated using the Pearson product-moment correlation coefficient (r). The split and test procedure was repeated 200 times, each trial using different randomly selected training and testing sets. Empirical distributions of r for the 200 trials were used to estimate the significance of the circulation-AISMR relationships. Because models in each split-sample validation trial used the same training and testing sets, the significance of differences in model performance was also estimated. Split-sample validation results for the MLR and ensemble MLP neural network model fore-casts are shown in Figures 3.1-3.4 for 200 hPa, 500 hPa, 850 hPa, and SLP circulation predictors. Boxplots show empirical distributions of r for the two model types as well as empirical distributions of differences in r between the MLR and ensemble neural network models. The solid line within each box indicates the median r value obtained for the 200 test sets. The upper and lower edges of the box represent the upper and lower quartiles. Whiskers extend to 1.5 times the interquartile range; outliers are marked individually. For each atmospheric level, median values of r and differ-ences in r between models are given below the boxplots. Also shown are probability values for the significance of r and differences in r obtained from the split-sample validation trials. The level of significance was set at p = 0.10. 3.1.1 200 hPa geopotential height Of the predictor fields and lead times, PCs of 200 hPa geopotential height over India in May exhib-ited the strongest correlations with summer monsoon rainfall (Figure 3.1). Split-sample validation correlation coefficients equaled or exceeded 0.65 for MLR (median r = 0.67, p < 0.005) and ensem-ble MLP models (median r = 0.65, p < 0.005). No significant differences in skill between model types was noted at any lead-time. While a maximum in skill was evident in November, results were not statistically significant. 27 200 hPa MLR o o in o o OCT NOV DEC JAN FEB MAR APR MAY Median -0.1 0.31 -0.15 0.02 0.12 0.15 0.02 0.67 p-value 0.62 0.16 0.66 0.45 0.38 0.32 0.48 0 200 hPa MLP OCT NOV DEC JAN FEB MAR APR MAY Median -0.16 0.25 -0.15 -0.09 0.1 0.04 0.05 0.65 p-value 0.68 0.22 0.66 0.61 0.36 0.44 0.42 0 C 5 N 200 hPa MLP-MLR cc 3T £. o CC OCT NOV DEC JAN Median -0.02 -0.06 0 -0.05 p-value 0.55 0.69 0.5 0.74 FEB MAR APR MAY 0 -0.05 0.04 -0.01 0.5 0.72 0.38 0.68 Figure 3.1: M L P and M L R split-sample validation results for 200 hPa geopotential height P C predictors. 28 3.1.2 500 hPa geopotential height Split-sample validation results for models using 500 hPa geopotential height PCs as predictors are shown in Figure 3.2. Like SLP and 850 hPa geopotential height predictors, PCs during May showed a maximum in predictive skill for A I S M R over the study period. For both M L R (median r = 0.41, p = 0.12) and ensemble M L P models (median r = 0.43, p = 0.14), however, results were not significant at the p = 0.10 level. Significant positive skill was not evident at any other lead-time at this level. 3.1.3 850 hPa geopotential height For PCs of 850 hPa geopotential height, a maximum in skill was evident in November, although significant correlations were only noted for the ensemble neural network models (median r — 0.37, p = 0.06) and not for the M L R models (median r = 0.27, p = 0.13). However, the difference in skill between models was not statistically significant at this lead-time. While positive, correlations during May were also not significant for either model type. Interestingly, level of predictive skill for the ensemble neural network models was high during January (median r = 0.40, p = 0.10), significantly higher than for the M L R models (median difference in r = 0.25, p = 0.09). 3.1.4 SLP Split-sample validation results for models using SLP PCs as predictors are shown in Figure 3.4. For linear models, PCs during November showed significant correlations with A I S M R (median r — 0.39, p = 0.09). Outputs from ensemble neural network models also exhibited significant correlations with A I S M R in November (median r = 0.5, p = 0.05), as well as May (median r = 0.47, p — 0.10), the latter a month when M L R models also showed strong correlations with A I S M R (median r = 0.39, p — 0.12). At no lead time, however, were significant differences between the linear and nonlinear models noted. 29 500 hPa MLR o d in d o OCT NOV DEC JAN FEB MAR APR MAY Median -0.28 0.12 -0.16 0.13 0.15 -0.05 -0.2 0.41 p-value 0.88 0.33 0.64 0.34 0.36 0.55 0.76 0.12 500 hPa MLP o o OCT NOV DEC JAN FEB MAR APR MAY Median -0.16 0.14 -0.23 0.07 0.15 -0.09 -0.18 0.43 p-value 0.74 0.34 0.74 0.43 0.34 0.6 0.72 0.14 500 hPa MLP-MLR I D ? \ OCT NOV DEC JAN FEB MAR APR MAY Median 0.11 0.01 -0.02 -0.05 0 -0.01 0.01 0.02 p-value 0.29 0.48 0.56 0.64 0.49 0.52 0.47 0.42 Figure 3.2: MLP and MLR split-sample validation results for 500 hPa geopotential height PC predictors. 30 \ 850 hPa MLR OCT NOV DEC JAN FEB MAR APR MAY Median 0.06 0.27 -0.14 0.12 -0.08 -0.25 0.01 0.28 p-value 0.42 0.13 0.65 0.32 0.6 0.87 0.48 0.2 850 hPa MLP in d in d o OCT NOV DEC JAN FEB MAR APR MAY Median -0.03 0.37 -0.07 0.4 -0.18 -0.02 -0.04 0.31 p-value 0.56 0.06 0.57 0.1 0.71 0.54 0.56 0.16 850 hPa MLP-MLR OCT NOV DEC JAN FEB MAR APR MAY Median -0.07 0.11 0.06 0.25 -0.05 0.18 -0.03 0.04 p-value 0.64 0.2 0.4 0.09 0.62 0.2 0.57 0.38 Figure 3.3: MLP and MLR split-sample validation results for 850 hPa geopotential height PC predictors. 31 SLP MLR m o OCT NOV DEC JAN FEB MAR APR MAY Median 0.03 0.39 0.12 0.15 -0.32 -0.17 0.06 0.39 p-value 0.46 0.09 0.35 0.29 0.82 0.72 0.45 0.12 SLP MLP OCT NOV DEC JAN FEB MAR APR MAY Median -0.06 0.5 0.12 0.2 -0.14 -0.02 -0.1 0.47 p-value 0.6 0.05 0.39 0.24 0.64 0.53 0.6 0.1 SLP MLP-MLR in in OCT NOV DEC JAN FEB MAR APR MAY Median -0.04 0.11 0.01 0.07 0.19 0.16 -0.08 0.05 p-value 0.62 0.23 0.46 0.38 0.29 0.27 0.74 0.33 ure 3.4: M L P and M L R split-sample validation results for SLP P C predictors. 32 3.2 Stationarity of circulation-AISMR relationships Split-sample validation results from Section 3.1 were evaluated using training and test sets selected from the entire 41 year period of record. As previous studies have shown that the strength of predictor-AISMR relationships may change on interdecadal time scales (Parthasarathy et al., 1991, Hastenrath and Greischar, 1993), some means of evaluating stationarity of the circulation-AISMR relationships was also needed. In Cannon and McKendry (1999), multiple correlation coefficients between PCs and A I S M R were calculated for 11 year and 15 year sliding windows. This method did not, however, accurately describe out of sample forecast performance and only measured the strength of linear relationships between circulation PCs and AISMR. In the current study, split-sample validation was used to evaluate stationarity of both M L R and ensemble neural network forecast skill between 1958 and 1998. As described in the previous section, 30 years of data were used for model training and 11 years of data were used to test out-of-sample model performance. Training and test sets were not, however, selected randomly from the 41 year period of record. Test sets were instead taken from an 11 year sliding window and training data from the remaining 30 years. To gauge variability in ensemble neural network results, ten ensembles were trained and evaluated for each 11 year window. Results are shown in Figures 3.5-3.12. Bars indicate correlation coefficients for M L R models and lines indicate the range in correlation coefficients for the ten ensemble neural networks. The black line indicates the median correlation coefficient for the ten ensemble neural network models. Results are plotted for the centre year in each 11 year window. Stationarity of model relationships during each month wil l be discussed in turn. 3.2.1 May Of the relationships identified in the previous section, only PCs of the May 200 hPa geopotential height field exhibited long-periods of significant, positive skill for both linear and nonlinear forecast models (Figure 3.5). Skill for both model types exceeded r = 0.8 through the middle of the 1970s. 33 May S L P May 850 hPa hgt + MLP • MLR — p = 0.05 Centre year of 11 year sliding window - - p = 0.10 + MLP • MLR — p = 0.05 Centre year of 11 year sliding window -- p = o.io May 500 hPa hgt May 200 hPa hgt 1.00 0.80 0.60 0.40 0.20 rro.oo -0.20 -0.40 -0.60 -0.80 -1.00 II n + MLP B MLR — p = 0.05 Centreyearof 11 year sliding window -- p = o.io 1.00 0.80 0.60 0.40 0.20 OCO.OO -0.20 -0.40 -0.60 -0.80 -1.00 + MLP S MLR — p = 0.05 Centre year of 11 year sliding window -- p = o.io Figure 3.5: Sliding split-sample validation results for May PC predictors. 34 Figure 3.6: Sliding split-sample validation results for April PC predictors. 35 Figure 3.7: Sliding split-sample validation results for March PC predictors. 36 Figure 3.8: Sliding split-sample validation results for February PC predictors. 37 Figure 3.9: Sliding split-sample validation results for January PC predictors. 38 Figure 3.10: Sliding split-sample validation results for December PC predictors. 39 Nov S L P Nov 850 hPa hgt 1.00 0.80 0.60 0.40 0.20 OCO.OO -0.20 -0.40 -0.60 -0.80 -1.00 4r [j C O O i O U L O C O - ' - ^ - r ^ t D c o r * . r - - r > . c o c o a o o j o o o i o i o i C B O ) + MLP • MLR — p = 0.05 Centre year of 11 year sliding window - - p = 0.1 o 1.00 0.80 0.60 0.40 0.20 OCO.OO -0.20 -0.40 -0.60 -0.80 -1.00 "te •/ir 1 t In c o c D O i c v j i o c o T - r i - r ^ o c o J X I C D C D r - - h * r ^ C O O O C O C T ) O i r j ) 0 ) c n o > C T ) c n o > 0 5 0 J C T ) 0 5 + MLP B MLR — p = 0.05 Centre year of 11 year sliding window -- p -o .10 Figure 3.11: Sliding split-sample validation results for November PC predictors. 40 Figure 3.12: Sliding split-sample validation results for October PC predictors. 41 In the 1980s, however, skill dropped to the p = 0.05 level. Values below the p = 0.10 significance level were noted for windows centred between 1989-1991. Results for the 500 hPa level show a similar pattern, with significant, positive skill values through the mid 1970s and sharp declines in the 1980s and early 1990s. Like 200 hPa geopotential heights, however, windows centred on the final two years (1992 and 1993) exhibited increased r values. For S L P and 850 hPa geopotential height predictors during May, skill levels were significant from 1983 onward, with a secondary period of positive skills in the early to mid 1970s. Models performed very poorly during the early portion of the study period. 3.2.2 April No stable periods of positive model skill were noted over the study period at this lead-time (Fig-ure 3.6). 3.2.3 March No stable periods of positive model skill were noted over the study period at this lead-time (Fig-ure 3.7). 3.2.4 February No stable periods of positive model skill were noted over the study period at this lead-time (Fig-ure 3.8). 3.2.5 January For models using PCs of circulation fields during January as predictors, only 850 hPa geopotential heights showed any prolonged periods of significant skill (Figure 3.9). At this level and lead-time, skill of the neural network models was consistently higher than that of the linear models. Centred on the mid to late 1970s, ensemble neural networks showed positive skill above the p = 0.10 level for five consecutive sliding window periods. Skill during the 1980s and 1990s declined to near zero. 42 Positive skill was, however, evident in results for SLP PCs over this latter period of record, nearing the p = 0.10 level from 1982 onward. 3.2.6 December No stable periods of positive model skill were noted over the study period at this lead-time (Fig-ure 3.10). 3.2.7 November Ensemble neural networks using November PCs of SLP and 850 hPa geopotential height fields as predictors exhibited positive skill levels over the entire period of record (Figure 3.11). For SLP predictors, models showed positive skill at the p = 0.05 level since the early 1980s, with a dip below the p = 0.10 level during the mid 1990s. Prior to 1981, skill was positive, but not significant at the p = 0.10 level. For 850 hPa geopotential height predictors, significant skill levels were noted during the early 1980s. For both atmospheric levels, M L R models tended to perform less well than the corresponding neural network models. No periods of significant skill were evident in results for the 500 hPa geopotential height data. While significant correlation coefficients for 200 hPa geopotential height predictors were noted in the early portion of the record, models performed poorly after 1977, showing almost no skill over the last few decades. 3.2.8 October No stable periods of positive model skill were noted over the study period at this lead-time (Fig-ure 3.12). 43 Chapter 4 Interpret ing c i r cu l a t i on -A ISMR relationships 4.1 Linear circulation-AISMR relationships Results presented in the previous chapter suggest that significant relationships between pre-monsoon circulation conditions over India and rainfall during the summer season were present during the 1958-1998 period. In this chapter, the nature of these relationships are examined in detail, starting with linear relationships between pre-monsoon circulation fields and AISMR. Figures 4.1-4.5 show rotated P C loadings for each of the levels and lead-times reporting statistically significant circulation-AISMR relationships (May 200 hPa geopotential height and SLP fields, the January 850 hPa geopotential height field, and November 850 hPa geopotential height and SLP fields). Contour values show the magnitude of linear correlations between P C scores and the standardized anomaly series at each grid point. Scatterplots between P C scores and AISMR, as well as correlation coefficients between the circulation PCs and AISMR, are also given, as are best-fit linear regression and locally-weighted linear regression lines (Cleveland and Loader, 1996). For comparison, correlation coefficients between A I S M R and circulation anomalies at individual grid-points are given in Figures 4.6-4.8. Salient features of these plots wil l be discussed for each month in turn. 44 M A Y 200 h P a hgt (rot. PC1 40.86%) M A Y 200 h P a hgt (rot. P C 2 21.24%) MAY 200 hPa hgt (PC 1) r=0.16 (p=0.31) MAY 200 hPa hgt (PC2) r=-0.20 (p=0.21) 600 700 800 900 1000 600 700 800 900 1000 AISMR (mm) AISMR (mm) Figure 4.1: P C loadings for May 200 hPa geopotential height data (top). Scatterplot showing the relationship between May 200 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve. 45 M A Y 200 h P a hgt (rot. P C 3 20.57%) M A Y 200 h P a hgt (rot. P C 4 14.76%) MAY 200 hPa hgt (PC3) r=0.45 (p=0.003) MAY 200 hPa hgt (PC4) r=0.47 (p=0.002) 600 700 800 900 1000 600 700 800 900 1000 AISMR (mm) AISMR (mm) Figure 4.1: (cont.) 46 MAY SLP (rot. PC1 38.4%) MAY SLP (rot. PC2 34.34%) MAY SLP (rot. PC3 15.92%) MAYSLP(PCI) r=-0.36 (p=0.02) MAYSLP(PC2) r=-0.32 (p=0.04) MAYSLP(PC3) r=0.10 (p=0.55) -2 • * -2 -2 I , , , , 1 . 3 J , , , , 1 I , , , ,— 600 700 800 900 1000 600 700 800 900 1000 600 700 800 900 1000 AISMR (mm) AISMR (mm) AISMR (mm) Figure 4.2: P C loadings for May SLP data (top). Scatterplot showing the relationship between May S L P P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve. 47I JAN 850 hPa hgt (rot. PC1 45.18%) JAN 850 hPa hgt (rot. PC2 34.79%) JAN 850 hPa hgt (rot. PC3 14.94%) 66° E71° E76° E81 0 E86° E91'° E 66° E 71° E 76° E 81° E 86° E 910 E 66° E 71° E 76° E 81° E 86° E91° E JAN 850 hPa hgt (PCI) r=0.12(p=0.44) V'-. 700 800 900 AISMR (mm) JAN 850 hPa hgt (PC2) 2 r=-0.28 (p-0.08) JAN 850 hPa hgt (PC3) r-0.06 (p-0.69) 700 800 900 AISMR (mm) 700 800 900 AISMR (mm) Figure 4.3: P C loadings for January 850 hPa geopotential height data (top). Scatterplot showing! the relationship between January 850 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve. 48 NOV 850 hPa hgt (rot. PC1 34.78%) NOV 850 hPa hgt (rot. PC2 31.81%) NOV 850 hPa hgt (rot. PC3 27.02%) NOV 850 hPahgt(PC1) r=0.03 (p-0.87) NOV 850 hPa hgt (PC2) r=0.05 (p-0.77) NOV 850 hPa hgt (PC3) r=-0.40 (p-0.01 > 600 700 800 900 1000 600 700 800 900 1000 600 700 800 900 1000 AISMR (mm) AISMR (mm) AISMR (mm) Figure 4.4: P C loadings for November 850 hPa geopotential height data (top). Scatterplot showing: the relationship between November 850 hPa geopotential height P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression line. The dashed line shows the locally-weighted linear regression curve. 49 NOV SLP (rot. PC1 31.34%) NOV SLP (rot. PC2 29.82%) NOV SLP (rot. PC3 27.34%) 34° N 29° N 1° E 7 6 ° E 8 1 ° E 8 6 ° E 9 1 ° E NOV SLP (PC1) r=0.04 (p-0.83) 1° E 76° E 81 0 E 86° E 91° E " !>60 E 71 0 E 76° E 81 0 E 86° E 91 0 E r-0.14 (p=0.37) NOV SLP (PC3) r=-0.46 (p=0.003) 600 700 800 900 1000 AISMR (mm) 600 700 800 900 1000 AISMR (mm) 600 700 800 900 1000 AISMR (mm) Figure 4.5: P C loadings for November SLP data (top). Scatterplot showing the relationship between November SLP P C scores and A I S M R (bottom). The solid line shows the best-fit linear regression! line. The dashed line shows the locally- weighted linear regression curve. 50 MAY SLP correl. with AISMR 34° N 29° N 24° N 19° N 14° N 9° N MAY 200 hPa hgt correl. with AISMR 34° N 29° N 14° N - l 0' 9° N 0.^  66° E71° E76° E81° E 86° E91° E 66° E 71° E 76° E 81° E 86° E 91° E jure 4.6: Correlations between grid-point values of May SLP and 200 hPa predictors and AISMR. 51 JAN 850 hPa hgt correl. with AISMR Figure 4.7: Correlations between grid-point values of January 850 hPa predictors and AISMR. 52 NOV SLP correl. with AISMR NOV 850 hPa hgt correl. with AISMR Figure 4.8: Correlations between grid-point values of November SLP and 850 hPa predictors and AISMR. 53 4.1.1 May Relationships between rotated PCs of 200 hPa geopotential height anomalies during May and A I S M R are shown in Figure 4.1. Scores of PC3 and PC4 in May were strongly and significantly correlated (r = 0.45, p = 0.003 and r = 0.47, p = 0.002 respectively) with summer monsoon rainfall. Correlations between A I S M R and P C I and PC2 were not significant. Best-fit linear and locally-weighted regression lines for May 200 hPa PCs do not suggest the presence of nonlinear relationships between A I S M R and these components, a finding consistent with model forecasts presented in the previous chapter. Significant differences in skill between the ensemble neural network and M L R models were not detected by the split-sample validation procedure. Inspection of the rotated P C loadings and the scatterplots shown in Figure 4.1 suggests that the two most important pre-monsoon circulation characteristics were 1) the magnitude and sign of 200 hPa geopotential height anomalies over Pakistan in the northwestern portion of the domain (PC3), and 2) the magnitude and sign of the 200 hPa geopotential height anomalies across central India (PC4). May months with positive (negative) height anomalies centered over Pakistan were generally associated with positive (negative) A I S M R anomalies. May months with positive (negative) height anomalies over the west-central India were associated with positive (negative) A I S M R anomalies (PC4). Composite maps of 200 hPa geopotential height anomalies in May for excess and deficient A I S M R years, as defined in Section 2.1.2, are given in Figure 4.9. Observations during these extreme monsoon years confirm the two relationships outlined above. Excess A I S M R years tended to have positive height anomalies over much of India, with a maximum centered over the northwestern corner of the domain and a band of positive anomalies oriented across central India. Conversely, deficient years were associated with negative height anomalies over much of the region, with a strong anomaly center positioned in the northwest of the study region. Correlations between A I S M R and grid-point anomalies of 200 hPa geopotential height are shown in the right panel of Figure 4.6 and show a similar spatial pattern. 54 MAY 200 hPa hgt anomaly (excess) MAY 200 hPa hgt anomaly (deficient) Figure 4.9: Composite 200 hPa geopotential height anomalies during excess and deficient summer monsoon rainfall years. 55 Yang et al. (1996), in a recent observational and G C M study, considered the response of the Indian summer monsoon to spring geopotential height, wind, and temperature anomalies at the 850 hPa and 200 hPa levels. They found evidence of precursor relationships similar to those shown here and consistent with the upper level circulation features identified in previous studies (Verma and Kamte, 1980, Parthasarathy et al., 1991). Strong monsoon years were associated with positive 200 hPa height anomalies over the entire Indian region, while weak monsoon years were associated with negative anomalies. Easterly wind anomalies in spring were associated with strong monsoon years, while enhanced upper level westerlies were associated with weak monsoon years. Prior to strong monsoon years an anomalous upper-level anticyclonic circulation was present over the Tibetan plateau, while an anomalous cyclonic circulation existed before weak monsoon years. These monsoon precursors appear to be related to the nature of the winter-summer transition in upper level circulation regimes from one dominated by the subtropical westerly jet stream to one dominated by the tropical easterly jet stream. Early onset or increased strength of the tropical easterly jet stream could provide dynamic forcing capable of enhancing summer rainfall over the Indian region (Webster, 1987). Variations in intensity and position of the upper-level Tibetan plateau anticyclone have also been linked with variability in summer monsoon rainfall over India (Pant and Kumar, 1997). Strength of relationships between the S L P anomaly field during May and A I S M R (Fig-ure 4.2) were weaker than for 200 hPa geopotential heights. Significant negative correlations were present between the first and second rotated PCs (r = —0.36, p = 0.02 and r = —0.32, p = 0.04 respectively). Again, scatterplots show little indication of nonlinearity in circulation-AISMR rela-tionships. Loadings for P C I show positive correlations along a corridor oriented from the northwest to the southeast of the domain, with maxima located over Pakistan and the Bay of Bengal. Pos-itive (negative) SLP anomalies in these regions were generally associated with negative (positive) A I S M R anomalies. Loadings for PC2 show positive correlations centered over the Arabian Sea and Indian Ocean in the southwest of the domain, extending northward through central India. Positive (negative) anomalies over this region were generally associated with negative (positive) A I S M R 56 anomalies. Absolute magnitudes of correlations between AISMR and May SLP anomalies (left panel of Figure 4.6) were highest along a corridor stretching from Pakistan south into Sri Lanka (r < —0.4). Within this corridor, maximum correlations were centered over Rajasthan in the northwest of India, extending into southeastern Pakistan (r < —0.5). While less strong, significant correlations were also present over the Arabian Sea, Bay of Bengal, and into eastern India and Nepal. The relationships found between the PCs of the May SLP anomaly field and AISMR are consistent with findings previously reported by Bhalme et al. (1986), Parthasarathy et al. (1992), and Singh et al. (1995), and most likely reflect the strength of pre-monsoon heating and the development of the heat low over Pakistan and west-central India. Negative correlations between SLP anomalies over the Bay of Bengal are consistent with findings of Prasad et al. (2000), who found a significant relationship between outgoing longwave radiation in the same area during May, and Clark et al. (1999), who found moderate correlations between Bay of Bengal SSTs during spring and AISMR. 4.1.2 January PC loadings for the 850 hPa geopotential height anomaly field during January and scatterplots between PC scores and AISMR are given in Figure 4.3. Correlations were typically low, with the strongest relationship observed between rainfall and rotated PC2 (r = —0.28, p = 0.08). Loadings for PC2 (Figure 4.3) and correlations between 850 hPa geopotential height anomalies and AISMR (Figure 4.7) were highest over the northern portion of the domain and tended to decrease southward toward peninsular India. As results from the previous chapter indicate significant differences between MLR and neural network models using 850 hPa geopotential height PCs as predictors, linear correlations may not represent the true strength of the circulation-rainfall relationship at this level and lead-time. The scatterplot and best-fit lines between PC scores and AISMR do not, however, seem to indicate the presence of a simple nonlinear relationship between PC2 and AISMR. Closer inspection does 57 suggest that the linear relationship between PC2 and AISMR may have been stronger when scores of PC2 were positive and less strong when scores were negative. After stratifying by the sign of PC2, r — —0.63 (p = 0.001, N = 24) when scores were greater than or equal to zero and r = —0.19 (p = 0.47, N = 17) when scores were less than zero. A physical mechanism linking 850 hPa geopotential height anomalies in January with AISMR has not been suggested in the monsoon forecasting literature. With strongest correla-tions found over the northern portion of the domain, it is possible that the response at 850 hPa is related to snow cover in the Himalayas, a boundary condition thought to be relevant to subsequent development of the summer monsoon circulation (Dey and Kumar, 1983). While recent observa-tional and modelling studies have linked regional snow cover and snow depths at various points in Eurasia with AISMR (Parthasarathy and Yang, 1995, Bamzai and Shukla, 1999, Kripalani and Kulkarni, 1999), Bamzai and Shukla (1999) found no significant relationship between Himalayan snow cover and summer monsoon rainfall. Interestingly, Sankar-Rao et al. (1996) found that the winter snow-summer rainfall relationship was strongest when excluding E l Nino years from the analysis. In a different context, Power et al. (1999) found that ENSO effects in Australia were not apparent during the warm phase of the interdecadal Pacific Oscillation. A similar effect might account for the absence of a correlation in the current analysis during years with anomalously low heights over the northern portion of the study domain. 4.1.3 November Relationships between AISMR and rotated PCs of 850 hPa geopotential height and SLP anomalies during November are shown in Figures 4.4 and 4.5. For both fields, significant relationships were observed between the third rotated PC and AISMR (r = —0.40, p = 0.01 for 850 hPa geopotential height and r = —0.46, p = 0.003 for SLP). Loadings were similar for each field, exhibiting maximum correlations over the Arabian Sea in the southeast portion of the domain. This spatial pattern is also reflected in the correlation fields between AISMR and the individual grid-point anomalies (Figure 4.8). 58 As with the 850 hPa geopotential height field in January, performance of ensemble neural networks using November circulation PCs as predictors exceeded that of the corresponding linear models. While not statistically significant, differences were notable. Again, however, inspection of the scatterplots and best-fit lines did not suggest the presence of simple forms of nonlinearity. Significant regional circulation-AISMR relationships occurring at a lead-time six months prior to monsoon onset have not previously been identified in other studies on the monsoon circu-lation. It is possible that correlations between A I S M R and November surface circulation over the Arabian Sea are related to anomalous SSTs in this region. In a recent study by Clark et al. (1999), significant positive correlations between A I S M R and SSTs in the Indian Ocean and Arabian Sea were noted in the fall and winter preceding the summer monsoon. Correlations of the opposite sign would be expected for SLP and geopotential heights. 4.2 Nonlinear circulation-AISMR relationships 4.2.1 Neural network sensitivity analysis In the previous section, linear relationships between pre-monsoon circulation conditions over India and summer monsoon rainfall were identified and investigated. While useful for investigating con-ditions immediately prior to monsoon onset when relationships appeared most linear, identifying possible nonlinear relationships from the scatterplots of A I S M R and P C scores was not easy. As results from Chapter 3 suggest the presence of differences in skill between linear regression and neural network models at longer lead-times (particularly during January and November), some means of interpreting outputs from the neural network models was needed. Given the interconnec-tivity and nonlinearity of neural networks, however, relationships learned by these models can be difficult to decipher (Sarle, 1998). Most studies accept results from neural networks as being from a "black box", acceptable for use but impenetrable to attempts at interpretation. Few studies in the atmospheric sciences have attempted to interpret input-output mappings learned by such models. 59 Hewitson and Crane (1994) developed a neural network model relating PCs of synoptic-scale circulation conditions to local precipitation in southern Mexico. One of their main objectives was to identify the relative importance of circulation variables at the surface and at 500 hPa to variability in precipitation at Chiapas. To achieve their goal, they measured the sensitivity of the neural network to small perturbations in P C scores on each day in the record. Vector lengths of sensitivities for SLP and 500 hPa geopotential height PCs were used as measures of the importance of each atmospheric level to precipitation on a given day. To avoid small or zero lengths resulting from sensitivities of opposite sign, vector lengths were computed using absolute values of the individual sensitivities. As this approach only considers the sensitivity of the network to small changes in input, irrespective of the magnitude of the output variable, the same sensitivity value can have very different meanings depending on the value of the output. Hewitson and Crane (1994) recognized this problem and scaled the sensitivities by a logarithmic function of precipitation, thereby giving more weight to sensitivities associated with low precipitation amounts. Time series plots of sensitivities were then constructed to gain insight into the relative importance of SLP and 500 hPa conditions to precipitation during different seasons. Spatial sensitivity maps were also plotted, showing the influence of different regions during different periods of time. Novel in this approach was the mapping of the P C sensitivities back to the original grid-point locations. In a different study, Tangang et al. (1998) applied a similar form of sensitivity analysis to neural network models used to predict SST anomalies in the equatorial Pacific Ocean. Using lagged SST and extended PCs of gridded SLP as inputs, they assessed the sensitivity of neural networks to removal or pruning of input variables. Unlike the analysis performed by Hewitson and Crane (1994), this method considers the net importance of a given input, not sensitivities of individual cases. Root mean-squared errors and correlation coefficients between outputs from the original network and the pruned networks were computed and used as indicators of input variable importance. Based on these measures of importance, an improved model was constructed containing four of the eight original inputs. While powerful and capable of estimating the influence of different input variables on the 60 output, these methods do not provide any insight into the types of relationships between input variables and the output. Trends, nonlinearities, and interactions cannot easily be diagnosed. Recently, Plate et al. (1997) and Plate et al. (1998) proposed a modified form of sensitivity analysis for visualizing and interpreting input-output mappings from neural network models. Combining aspects of the sensitivity analyses proposed by Hewitson and Crane (1994) and Tangang et al. (1998), insight into the importance of inputs, nonlinearity of modelled relationships, interactions between inputs, and trends in the effects of inputs can be readily identified using this method. This form of sensitivity analysis was used in the current study to help interpret relationships between pre-monsoon circulation fields and AISMR. Because this type of analysis has not been widely used in the atmospheric sciences, a rela-tively detailed description is given here. Similar to the exposition in Plate et al. (1997) and Plate et al. (1998) this discussion also compares the method with those of Hewitson and Crane (1994) and Tangang et al. (1998) and adds details relevant for use in climatological applications. A schematic representation of the sensitivity analysis procedure is given in Figure 4.10 to complement the text description. Examples of sensitivity effects plots are given in Figure 4.11 for reference. In the sensitivity analysis, effects of changing model inputs X{ from some arbitrary baseline values bi to their original values are calculated for a given set of cases. These output effects A ; are defined as &i = Y(X) - Y(XU Xi-U bh X i + 1 , X k ) (4.1) where k is the index of the last input variable. For standardized inputs, values of bi are typically set to zero, the mean of each input. By definition A , equals zero when Xi equals the baseline value. As in the study conducted by Tangang et al. (1998), the effect of removing input variables from the network forms the basis for the sensitivity analysis. Unlike in Tangang et al. (1998), however, values of the effects are examined for individual cases. This allows the function computed by the neural network to be plotted graphically, giving more detailed insight into the nature of 61 6.0 5.0 Y(X) 4.0 Y(X,..,bh.,XJ 3.0 2.0 1.0 Y(X) versus Xt (X -SX,Y(X ,.,X -5X,X )) 1 sx (x 1.+fflf,y(x„.,x I.+«,x ] k)) 1.0 A(Xt) versusXt slops= nx,,.,x;+<fr,.,x,)-HX„.,A:-&,.,x,) h 2.0 h 1.0 0.0 -1.0 A{xt) L-2.0 -3.0 Figure 4.10: Graphical representation of the sensitivity analysis. A n example of the calculation procedure for the neural network sensitivities and partial derivatives is shown on the left-hand side. The resulting sensitivity plot is shown on the right-hand side. 62 the modelled input-output relationships. Graphs are constructed by plotting values of variable Xi along the abscissa and variable effects Aj along the ordinate. For complicated relationships involving both nonlinearities and interactions, standard scat-terplots of the effects can become difficult to interpret. To enhance interpretability, effects are instead plotted as short line segments, where slopes of the segments are partial derivatives of net-work output with respect Xi evaluated at X. By plotting line segments instead of dots, trends in effects and types of nonlinearities are easier to identify. Calculation of the partial derivatives can be done analytically (Egmont-Petersen et al., 1994) or approximated using finite differences (Bishop, 1995). In the sensitivity analysis conducted by Hewitson and Crane (1994) the latter approach was used; partial derivatives of neural network outputs were approximated using finite differences. In the current study, as indicated in Figure 4.10, slopes were approximated using centred differences. For climatological applications, partial derivatives can be used to construct time-series plots and maps of sensitivities or sensitivity vector lengths in the manner described by Hewitson and Crane (1994). However, more detailed investigations of neural network mappings can be achieved by analysing the partial derivatives in conjunction with values of A j . As shown by Plate et al. (1998), effects plots can be used to gain insight into the nonlinearity of modelled relationships, trends in the effects of input variables, importance of variables, and the presence of interactions between variables. Trends and nonlinearity in effects plots relate directly to trends and nonlinearity in effects of a particular variable on model output. Overall vertical range indicates variable importance and vertical spread at points along the abscissa indicates interactions between the plotted input and at least one other input variable. Variables with no effect appear as straight horizontal lines on effects plots. Additive variables plot as individual lines or curves. To illustrate these characteristics, effects plots for a simple, synthetic function of six variables Y = 5 sin(10 XiX2) + 20 (X3 - 0.5)2 - 10 X4 + 20 XbX6 (4.2) are given in Figure 4.11. For this particular function, variables X\ and X2 interact with one another 63 0.00 0.25 0.50 0.75 1.00 Xi 12.00 8.00 4.00 0.00 -4.00 -8.00 -12.00 X < I I I . i i i i i t ->\ ll > A "*' / ' , i i i i i 0.00 0.25 0.50 0.75 1.00 12.00 8.00 4.00 0.00 -4.00 -8.00 -12.00 1 , 1 , i I i 1 i - - u. 'v ' -', » V i 1 i 1 i 1 1 1 1 > 0.00 0.25 0.50 0.75 1.00 12.00 8.00 4.00 0.00 -4.00 -8.00 -12.00 0.00 0.25 0.50 0.75 1.00 X 6 Figure 4.11: Example effects plots for inputs to the function in Equation 4.2. 64 and have a nonlinear effect on Y. X$ is not involved in any interactions, instead having an additive nonlinear effect on Y. X4 is also additive, and has a negative linear effect on Y. Variables X 5 and XQ exhibit a simple multiplicative relationship. A l l of these relationships can be seen in the effects plots shown in Figure 4.11. Vertical spread in the effects for X\ and X2 indicates the presence of an interaction. The sinusoidal form of the nonlinearity for these variables can be seen by following the trace of the line segment slopes in their effects plots. Similarly, the parabolic relationship is clearly evident in the effects plot for X 3 ; absence of vertical spread suggests an additive contribution for this variable. The slope of effects for X4 are negative, suggesting that increases in this variable are associated with decreases in Y. The effects lie along a straight line, indicating that the relationship between X4 and Y is linear and additive. Vertical spread in effects for X 5 and XQ suggests the presence of interactions. The linear trace of the slopes in effects suggests a simple multiplicative interaction. In terms of variable importance, roughly measured by the vertical range in effects, variables X\ and X2 appear most important, followed by X5 and XQ, then X4, and finally X3. Interaction strength is more difficult to diagnose graphically but, judging from the degree of vertical spread, X\ and X2 appear less additive than X5 and X§. In addition to graphical means of estimating variable importance and interaction strength, Plate et al. (1998) also introduced objective measures for these quantities. Variable importance is measured by computing the variance of effects for Xi. This is essentially the root M S E measure adopted by Tangang et al. (1998). Variable interaction strength or degree of non-additivity is measured by taking the M S E of a smooth fit to a scatterplot of the partial derivatives. Partial derivative scatterplots for Equation 4.2 are given in Figure 4.12. Additive variables, such as X 3 and X4, plot as lines or curves and are well approximated by a smooth curve. Variables involved in interactions, such as X\, X2, X5, and X$, plot as clouds of points and are poorly approximated by a smooth curve. Low error therefore indicates near additivity and high error indicates non-additivity. Curves can be drawn using any of a number of smoothing techniques (Cleveland and Loader, 1996). Locally-weighted linear regression was chosen for this particular example. Figure 4.13 shows plots 65 of variable importance versus interaction strength for variables X\ to XQ. The objective measures are consistent with impressions gained through visual inspection of the effects plots. In Figure 4.11, determining which variables were involved in a given interaction was rela-tively straightforward. In practice, however, relationships may be complex and involve multiple variables. In these situations, simple effects plots are not sufficient for identifying which variables are interacting with one another. Instead, as shown by Plate et al. (1998), stratified effects plots may be used to help identify specific interactions. In stratified effects plots, line segments for a given variable are coloured according to ranges of values of another variable. If the two variables are involved in an interaction, colours should appear in distinct bands or areas of the effects plot. If the two variables are not involved in an interaction, colours should be distributed randomly across the plot. Examples of stratified effects plots for variables X\ and X% and X 5 and X$ are given in Figure 4.14. For X\ and X2 colours show ranges of variables X2 and X 3 ; for X 5 and X% colours show ranges of X4 and X 5 . The plot indicates the presence of an interaction between X\ and X2 and between X§ and X$, but not between X2 and X3 or between X4 and X 5 . Modifications to the basic sensitivity analysis procedure were suggested following the appli-cation of this procedure to two problems in the atmospheric and hydrologic sciences. Cannon and Lord (2000) investigated predictors in a neural network forecast model for surface-level ozone con-centrations. In Cannon and Whitfield (1999) and Cannon and Whitfield (2000), effects plots were used to visualize relationships between precipitation and antecedent climate conditions in models for stream pH. In the three studies, model performance was evaluated using cross-validation. Due to differences in initial conditions and training data, variations in the variable effects were noted between members of the cross-validation trials. Based on these results, a modification to the sen-sitivity analysis was suggested for use in situations where multiple models are constructed for a given problem. Rather than choose a single model to represent the input-output relationships, effects plots instead use information from all available models. Conceptually, this is similar to the bootstrapping procedure used by Baxt and White (1995). To represent average conditions over the models, median values of effects and partial derivatives are plotted. To show variability, lower 66 69.40 50.00 > 30.60 M •g 11.20 <o •o - -8.20 CO ra-27.60 Q. -47.00 -66.40 _ J I I I I I I I L_ 0.000.200.400.600.801.00 68.40 49.00 | 29.60 H •c 10.20 0) T3 - -9-20 CO ra-28.60 Q. -48.00 -67.40 0.000.200.400.600.801.00 X, -20.00 -28.00 0.000.200.400.600.801.00 X 3 0.000.200.400.600.801.00 -8.60 -9.00 | -9.40-| ~ -9.80 o 2-10.20 CO m-10.60 -11.00 -11.40 _i i i i i_ 0.000.200.400.600.801.00 X 4 24.00 20.00 CD £16 .00 -12.00 CD T3 - 8.00 CC a 4.00 a. 0.00 -4.00 _i i i i i i i i i_ •: • i • "" • 'I'; :.' ' 0.000.200.400.600.801.00 XR Figure 4.12: Example partial derivative plots for inputs to the function in Equation 4.2. 67 X h-^ m O co ^ O LU O CO < GC LU 10 9 8 7 6 5 4 3 2 1 0 X1 • X2 • ) <5 • • X6 1 X3 -• X4 I r « i i I 1 I 6 8 10 12 14 16 18 20 EFFECTS VARIANCE (VARIABLE IMPORTANCE) Figure 4.13: Scatterplot showing quantitative measures of variable importance (abscissa) and vari-able interaction strength (ordinate) for inputs to the function in Equation 4.2. 68 12.00 8.00 4.00 0.00 -4.00 -8.00 -12.00 _i i i i i i_ 0.00 0.25 0.50 0.75 1.00 12.00 8.00 4.00 H * 0.00 CM < -4.00 -8.00 -12.00 , ,;,"\ 0.00 0.25 0.50 0.75 1.00 X 2 12.00 8.00 4.00 * o.oo H -4.00-I -8.00 -12.00 0.00 0.25 0.50 0.75 1.00 Xc 0.00 0.25 0.50 0.75 1.00 XK Figure 4.14: Stratified effects plots for inputs X\ and X2 (top) and X 5 and XQ (bottom) to the function in Equation 4.2. Colours show ranges of the variables X2 and X 3 (top) and X4 and X 5 (bottom) respectively: black < 0.25, red > 0.25 and < 0.5, green > 0.5 and < 0.75, and blue > 0.75. 69 120 80 40 -0 --40 -80 4 -0.93 -120 -2.36 -0.84 0.69 2.21 PC 3 (S.D.) Figure 4.15: Example effects plot for results from multiple neural network runs. Line segments plotted in black represent median values of effects and partial derivatives. Lower and upper line segments plotted in grey indicate lower and upper quartile values of effects and partial derivatives. and upper quartile values of effects and partial derivatives are also plotted on the same graph. A n example is given in Figure 4.15. Empirical distributions of objective measures for variable importance and interaction strength can also be computed and used to compare effects of different variables in the model. Modifications to the sensitivity analysis procedure may also be required in climatological applications using P C A to reduce dataset dimension. In cases where gridded circulation data are represented using PCs, interpreting sensitivities in terms of the original grid-point locations may be difficult. While P C loadings can help in the spatial interpretation of the effects plots (eg. Figures 4.1-4.5), the presence of nonlinear and interactive effects make this difficult. One alternative is to use a procedure similar to that suggested by Hewitson and Crane (1994) to map the 70 1. Model 1. Perturb PCs Ensemble neural network I 2. Perturb gridded inputs 2. Model Figure 4.16: Calculation of effects for (1) PC input variables and (2) for grid-points of PC input variables. 71 P C sensitivities back to the original grid-points. As shown in Figure 4.16, the model is expanded to include the projection of the grid-point data onto the PCs. Instead of perturbing the PCs, the grid-point data are perturbed, the perturbed data are then projected onto the PCs, and the resulting scores are entered into the model. Plate (1998) applied a similar procedure to a model relating PCs of spectroscopy data to fat content of meat. Effects plots for the original spectral components were derived from a model built using the PCs as inputs. 4.2.2 Sensitivity analysis results The sensitivity analysis procedure described above was applied to M L R and ensemble M L P models for A I S M R . Results for P C predictors derived from May 200 hPa geopotential height and SLP fields, the January 850 hPa geopotential height field, and November 850 hPa geopotential height and SLP fields are given in Figures 4.17-4.21. Because the total number of P C predictors was relatively small, quantitative measures of variable importance and interaction strength were not calculated. Sensitivity plots for individual grid-point anomalies are given in Figures 4.22-4.31. In each case, results from the ensemble M L P neural network are plotted along with those from the corresponding M L R model. To facilitate visual comparison, the ordinate of each plot is scaled to the same range for all levels and lead-times. M a y For May predictors, sensitivity effects plots for the linear and nonlinear models each showed the same basic trends and magnitudes of variable effects (Figures 4.17-4.18). Consistent with results from the previous section, P C I , PC3 , and PC4 of the May 200 hPa geopotential height field showed positive linear trends, with PC2 exhibiting a negative linear trend. Little evidence of interactions between variables was present. Effects were strongest for PC3 and PC4, stronger than for any other P C predictor, agreeing with results of the linear correlation analysis. Median and upper and lower quartiles of effects and partial derivatives were very consistent, showing little variability across the 200 split-sample validation trials. 72 120-80 -40 -0 --40 --80 --120--2.14 -0.43 1.27 2.97 PC 1 (S.D.) -2.14 -0.43 1.27 2.97 PC 1 (S.D.) 120-80 -40 -E E 0 -< -40 -•80 --120-120 - 1.40 -8 0 - 0.93-40 - 0.47 -E. 0 -Q co 0.00-o -40-< -0.47 --80- -0.93 --120- -1.40 --2.17 -0.58 1.02 2.61 PC 2 (S.D.) -2.17 -0.58 1.02 2.61 PC 2 (S.D.) 120 -| 1 40 -80 - 0 93 -40 - 0 47 -E d 0 - CO 0 00 -< < -40 - 0 47 --80- 0 93 -120- 1 4 0 --2.22 -t).50 1.22 2.94 PC 3 (S.D.) 120 -| 1.40-80 - 0.93 -4 0 - 0.47 -E d E 0- CO 0.00 -< -40- 0.47 --80 - 0.93 -•120- 1.40 --2.22 -0.50 1.22 2.94 PC 3 (S.D.) 120 80 40 "E E 0 < -40 -80 •120 1.40 0.93 0.47 Q co 0.00 < •0.47 -0.93 •1.40 May 200 hPa MLP 120 -, 1.40-80 - 0 .93-40 - 0.47 -E 0 -< -40-d co 0.00--0.47 --80 - -0.93 --120- -1 .40-May 200 hPa MLR ; * / -2.57 -0.94 0.69 2.33 PC 4 (S.D.) -2.57 -0.94 0.69 2.33 PC 4 (S.D.) ;ure 4.17: Effects plots for May 200 hPa geopotential height PC predictors. 73 120 80-| 40 £ 0 -40 H -80 -120 120 -| 1.40 -80- 0.93 -40- 0.47 -(mm) 0- (S.D.) 0.00 -< -40 -< 0.47 --80- 0.93--120- 1.40--2.04 -0.59 0.85 2.29 PC 1 (S.D.) -2.04 -0.59 0.85 2.29 PC 1 (S.D.) 120 80 40 B 0--40--80--120 Q co 0.00 120 n 1.40 -80- 0.93 -40-£ 0-< -40-0.47 -Q co 0.00-<i -0.47 --80- -0.93 --120- -1.40 --2.39 -0.83 0.73 2.29 PC 2 (S.D.) -2.39 -0.83 0.73 2.29 PC 2 (S.D.) 120 80-| 40 0 -40-| -80 -120 120 - 1.40 -80- 0.93-40-£ 0-< -40-0.47 -S 0-00 -< -0.47 --80- -0.93--120- -1.40--2.01 -0.66 0.68 2.03 PC 3 (S.D.) -2.01 -0.66 0.68 2.03 PC 3 (S.D.) Figure 4.18: Effects plots for May SLP PC predictors. 74 -2.70 -1.'l7 0.36 1.90 PC 1 (S.D.) 120- 1.40 -January 850 NPa MLR 80 - 0.93 -40 - 0.47 -- — -E Q E. 0- w 0.00-< < -40 - -0.47 --80- -0.93--120 - -1.40 - i i i i i -2.70 -1.17 0.36 1.90 PC 1 (S.D.) -40--80--120-120- 1.40 -80- 0.93 -40- 0.47 -e o - d c^ O.OO -< -40-< -0.47--80 - -0.93--120 - -1.40--2.25 -0.84 0.58 1.99 PC 2 (S.D.) -2.25 -0.84 0.58 1.99 PC 2 (S.D.) 120 - 1.40 -80- 0.93-40-B E 0-< -40-0.47 -ci 0.00-< -0.47 --80- -0.93 --120- -1.40--1.79 -0.21 1.37 2.95 PC 3 (S.D.) -1.79 -0.21 1.37 2.95 PC 3 (S.D.) Figure 4.19: Effects plots for January 850 hPa geopotential height PC predictors. 75 120 80 40-| £ 0 -40 -80 -120 -J 120 1.40 -80- 0.93 -40- 0.47 -E 0- ^0.00-< -40-< -0.47 --80- -0.93--120- -1.40 --1.97 -0.57 0.82 2.22 PC 1 (S.D.) -1.97 -0.57 0.82 2.22 PC 1 (S.D.) 120 80 40 -| E 0 < -40 -80 -120 120 - 1.40 -80- 0.93 -40- 0.47 -E" Q w 0.00-< -40-•a -0.47 --80- -0.93--120- -1.40 --2.31 -0.95 0.40 1.76 PC 2 (S.D.) -2.31 -0.95 0.40 1.76 PC 2 (S.D.) 120 80 40 1 ° H -40 -80 -120 O co 0.00 n 1 1 1 1 ' 1 r--2.60 -1.14 0.31 1.76 PC 3 (S.D.) 120 n 1.40 -| 80- 0.93 -40-E" E, 0-< -40-0.47-d co 0.00-< -0.47--80- -0.93--120- -1.40 --2.60 -1.14 0.31 1.76 PC 3 (S.D.) Figure 4.20: Effects plots for November 850 hPa geopotential height PC predictors. 76 -40 -80 -120-1 120 n 1.40 -80- 0.93-40- 0.47-(mm) 0- (S.D.) 0.00 -< -40-< 0.47 --80- 0.93 --120- 1.40 --1.84 -0.52 0.81 2.13 PC 1 (S.D.) -1.84 -0.52 0.81 2.13 PC 1 (S.D.) -40 -] -80 -120 -> 1.40 0.93 0.47 d -0.47 -0.93 -1.40 November SUP MLP -2J3 -0.84 0.44 1.73 PC 2 (S.D.) 120 80 40 _E 0 -40 -80 -120 -2.13 -0.84 0.44 1.73 PC 2 (S.D.) Q co 0.00 -2.36 -0.84 0.69 2.21 PC 3 (S.D.) 120 1.40-November SLP MLR 80- 0.93 -40-£ 0-< -40-0.47 -d co 0.00--0.47 --80- -0.93 --120- -1.40 --2.36 -0.84 0.69 2.21 PC 3 (S.D.) Figure 4.21: Effects plots for November SLP PC predictors. 77 Similarly, effects plots for May SLP PCs agreed with results of the linear correlation analysis; both P C I and PC2 exhibited negative relationships with A I S M R . PC3 showed a small positive effect in both M L R and M L P models. Relationships were linear, although plots for the neural network model suggest that low levels of interaction may have been present between PCs. Variability amongst the split-sample validation trials was greater than for the 200 hPa PCs. Spatial patterns in effects were obtained by applying the sensitivity analysis to individual grid-points (Figure 4.22-4.25). For the 200 hPa geopotential height field, largest effects were present in the central northwestern (positive trend) and northeastern (negative trend) portions of the study domain. Again, differences between linear and neural network models were small. Comparing results with the correlation map presented in Figure 4.6, trends in model sensitivity were generally consistent with magnitudes and signs of correlation coefficients between grid point anomalies and AISMR. Magnitudes of negative effects in the northeastern portion of the domain were, however, much larger than might be expected from the correlation analysis. Given that grid-point anomalies were projected onto a set PCs which account for only a portion of variance in the circulation field, such discrepancies are not unusual. Correlations between A I S M R and circulation anomalies reconstructed using the subset of PCs (not shown) do indicate stronger negative anomalies in northeastern India. Spatial patterns in effects for the SLP field in May were also consistent with correlation patterns presented in Figure 4.6. Small negative effects were present along an axis oriented from the northwest to the southeast of the domain. Like 200 hPa geopotential height, however, anomalous effects were noted in the northeast. January Effects plots for January 850 hPa PCs are shown in Figure 4.19. Unlike results for May predictors, distinct differences between linear and neural network models were present at this level and lead-time. Plots for the M L R model suggest a small positive effect for P C I , a larger negative effect for PC2, and almost no effect for PC3 . Ensemble neural network results were similar to those for PC3 in the linear model, but also indicated the presence of interactions between P C I and PC2 and 78 a nonlinear relationship between PC2 and A I S M R . Sensitivities from the split-sample validation trials were more variable than those reported for the May predictors. The possibility of a nonlinear relationship between PC2 and A I S M R was suggested in Sec-tion 4.1.2, where positive values of PC2 were found to be strongly associated with A I S M R but negative values of PC2 showed almost no relationship with summer rainfall. The effects plot for PC2 in the M L P model confirms this relationship. Perturbing negative scores of PC2 had almost no effect on the model while perturbing positive scores resulted in a significant negative trend in model output. Partial derivatives suggest that sensitivity of the model was greatest when positive values of PG2 were low to moderate (less than one standard deviation unit) and decreased with higher values of PC2 (greater than one standard deviation unit). Effects plots for the ensemble M L P model also suggest the presence of an interaction between P C I and PC2. Stratified effects plots for these two variables are shown in Figure 4.32. Unlike previous plots which showed effects for observed input cases, line segments were instead plotted at 200 random points spanning the input space. This allowed sparsely populated areas of the input space to be visualized and facilitated interpretation of the interaction. Results show a multiplicative relationship between P C I and PC2. Negative trends in effects of PC2 were strongest when values of P C I were high and weaker when values of P C I were low. Conversely, effects of P C I were positive when values of PC2 were lowest, and near zero or negative when values of PC2 were moderate to high. Mapping P C sensitivities onto grid-points (Figures 4.26-4.27), substantial differences be-tween M L R and ensemble neural network results were present at this level and lead-time. The nonlinear relationship identified between A I S M R and PC2 (Figure 4.19) was present in results for the M L P model and was confined to the northern and central eastern portions of the domain, areas where loadings for PC2 were highest (Figure 4.3). Results for the M L R model suggest negative effects at these points. Small positive effects were present in the south, most likely the result of the weak relationship identified between A I S M R and P C I . 79 November Effects plots for PCs of 850 hPa geopotential height and SLP fields in November (Figures 4.20-4.21) shared the same basic features. Similarity in effects was expected as PCs (Figure 4.3) and correlation maps (Figure 4.7) were very similar for these fields. For both S L P and 850 hPa geopo-tential heights, P C I and PC2 had little effect on model output, while PC3 had a strong negative effect, almost equal in range to effects for PC3 and PC4 of May 200 hPa geopotential heights. P C I and PC2 each exhibited approximately linear effects, with PC2 of the S L P field showing a small negative trend. Median and upper quartile effects for P C I of both fields and PC2 of the 850 hPa geopotential height field exhibited small positive trends, although lower quartiles of effects suggest that these trends were not significant. While weak interactions were present between the three PCs, differences in effects of the linear and neural network models were most evident for PC3. Trends in effects for M L R and M L P models were both negative, but slopes of effects for the neural network model decreased with absolute magnitude of PC3. This sigmoidal nonlinearity was present in results for both circulation fields. Performance statistics for neural network models using November 850 hPa geopotential height and SLP PCs as predictors (Figure 3.3 and 3.4) were better than corresponding linear models, although significant differences were not identified. The slight nonlinearity in the relationship between the PC3 and A I S M R might explain the small difference in skill present between the model types. As expected, spatial patterns in effects (Figures 4.30-4.31) were consistent with the P C loadings and linear correlation maps. Large negative effects were present at grid-points centred over the Arabian Sea in the southwest portion of the domain. At these grid-points, the sigmoidal nonlinearity was not clear, instead appearing as an interaction in which trends of effects were either near zero or negative. This may be an indication of year to year shifts in the spatial centre of the P C 2 - A I S M R relationship or could be the result of a weak interactions between the three circulation PCs. 80 Figure 4.22: Effects plots for May 200 hPa grid-point predictors in the MLR models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 81 Figure 4.23: Effects plots for May 200 hPa grid-point predictors in the ensemble M L P models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to" the grid-point centred on 9°N and 91°E. 82 Figure 4.24: Effects plots for May SLP grid-point predictors in the M L R models. The top left plot corresponds to the grid-point centred on 34° N and 66° E . The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 83 Figure 4.25: Effects plots for May SLP grid-point predictors in the ensemble MLP models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 84 Figure 4.26: Effects plots for January 850 hPa grid-point predictors in the MLR models. The top left plot corresponds to the grid-point centred on 34° N and 66° E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 85 Figure 4.27: Effects plots for January 850 hPa grid-point predictors in the ensemble MLP models. The top left plot corresponds to the grid-point centred on 34° N and 66° E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 86 Figure 4.28: Effects plots for November 850 hPa grid-point predictors in the MLR models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 87 Figure 4.29: Effects plots for November 850 hPa grid-point predictors in the ensemble MLP models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 88 Figure 4.30: Effects plots for November SLP grid-point predictors in the MLR models. The top left plot corresponds to the grid-point centred on 34°N and 66°E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 89 Figure 4.31: Effects plots for November SLP grid-point predictors in the ensemble MLP models. The top left plot corresponds to the grid-point centred on 34° N and 66° E. The bottom right plot corresponds to the grid-point centred on 9°N and 91°E. 90 120 80 40 I 04 < -40 -80 -120 -J 1.40 0.93 0.47 S o.oo < -0.47 -0.93 -1.40 i i i ' i i i _ Jan 850 hPa MLP — - - -120 80 40 I "A < -40-| -80 -120 -J -2.00 -0.70 0.60 1.90 PC 1 (S.D.) -1.96 -0.64' 0.67 1.99 PC 2 (S.D.) Figure 4.32: Stratified effects plots for P C I and PC2 of the January 850 hPa geopotential height field. Colours in each plot correspond to ranges in scores of the other P C : black < — 1, red > — 1 and < 0, green > 0 and < +1, and blue > +1. 91 Chapter 5 Conclus ion 5.1 Summary Pre-monsoon principal components (PCs) of 200 hPa, 500 hPa, and 850 hPa geopotential height and sea-level pressure (SLP) fields covering the South Asian subcontinent were used as predictors for all-India summer monsoon rainfall (AISMR) during the period 1958-1998. Predictive skill and stationarity of non-linear ensemble neural network and linear multiple regression model relation-ships were assessed using a Monte Carlo-based resampling procedure. Monsoon precursor signals represented by the PCs were investigated using a model sensitivity analysis and comparisons were made with recent observational and general circulation modeling studies. Pre-monsoon PCs of the 200 hPa geopotential height field in May formed a compact, in-terpretable, and significant set of predictors for A I S M R (median r = 0.67, p < 0.005). Skills of 500 hPa and SLP PCs were also strong (median r = 0.41, p = 0.12 and r = 0.47, p = 0.10 re-spectively). In terms of stationarity, predictive skills of 500 hPa and 200 hPa PCs were strongest in earlier decades but have declined slightly since the late 1970's. SLP PCs, however, have only shown significant skill levels since the early 1970's. Differences in performance between the neural network and multiple regression models during May were not significant and sensitivity analysis results suggest linear circulation-rainfall relationships. Precursor signals at 200 hPa during May were most likely related to the nature of the winter-92 summer transition between easterly and westerly jet stream circulation regimes and to variations in intensity and position of the upper-level Tibetan plateau anticyclone. Signals were similar to those identified by Yang et al. (1996) in a G C M modelling and observational study. SLP precursors during May were consistent with results of Bhalme et al. (1986), Parthasarathy et al. (1992), Singh et al. (1995), Clark et al. (1999), and Prasad et al. (2000) and reflect pre-monsoon heating and development of the heat low over Pakistan. At longer lead-times, neural network results tended to perform better than multiple linear regression, although differences were only significant for January 850 hPa PCs (median r = 0.4, p = 0.10; median difference r = 0.25, p = 0.09). At other lead-times prior to monsoon onset in June, only November SLP (median r = 0.5, p = 0.05) and 850 hPa PCs (median r = 0.37, p = 0.06) were significantly correlated with AISMR. The Monte-Carlo, resampling procedure suggests that circulation-monsoon relationships for January 850 hPa PCs were not stationary during the study period. Skill was positive during early portions of the record but little skill was evident in latter portions of the record. Results from the sensitivity analysis indicate that the trend in the relationship between January 850 hPa PCs and A I S M R was negative when values of the 2nd P C was positive, but was near zero when the 2nd P C was negative. This may be an indication that the precursor relationship between the regional circulation during January and A I S M R has been modulated by another source of variability, similar to the modulation of ENSO impacts in Australia by inter-decadal variability in SST over the Pacific Ocean (Power et al., 1999). Interactions between P C I and PC2 were also present, contributing to the increased skill of the neural network model. Strength of the relationship between the SLP field during November and A I S M R was rel-atively high during much of the record, remaining positive over the past two decades, a period when many predictors have showed declines in skill (Parthasarathy et al., 1991, Hastenrath and Greischar, 1993). Strength of the relationship between the 850 hPa geopotential height field and A I S M R was also relatively high, but has declined during the last decade. Slight nonlinearity in the relationship between the 3rd P C and A I S M R was evident, explaining the small difference in skills between the linear and nonlinear models. The same sigmoidal nonlinearity was also present 93 between the 3rd P C of the 850 hPa geopotential height field during November. Given that the rotated P C loadings for SLP and 850 hPa geopotential heights were similar, this suggests that the nonlinear circulation-AISMR relationship was a coherent near-surface precursor signal of monsoon strength. It is possible that this signal was related to anomalous SSTs in the Arabian Sea (Clark et al., 1999). 5.2 Future research The combined use of P C A , neural network models, and sensitivity analyses allowed relationships between regional circulation conditions over South Asia and Indian summer monsoon rainfall to be investigated in detail. Further work is required to validate proposed physical mechanisms responsible for the empirical relationships identified in the current study. Empirical comparisons between PCs exhibiting significant correlations with A I S M R and indices identified by other workers might be a simple means of starting this process. A more complex dynamical analysis, similar to the one performed by Yang et al. (1996), might be another viable approach. As previous work has shown that predictors for A I S M R are highly inter-correlated, an integrated assessment of the utility of circulation fields in an operational forecast setting is also necessary. Combining circulation predictors from different lead-times with ENSO, cross-equatorial, global/hemispheric, and other regional predictors would indicate whether or not the PCs identified in the current study are capable of generating real increases in predictive skill. 94 Bib l iography Bamzai, A . S. and J . Shukla (1999). 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