Agriyltwral Engineering

RAINFALL AND /FROST MODELLING FOR SUGARCANE SYSTEM SYNCHRONIZATION IN PAKISTAN -

M. K. Ayub 7 . Regional Computer Center, Asian Institute of Technology

Bangkok, Thailand

A. C. Early

Irrigation Water Management Department, International Rice Research Institute, Manila, Philippines

ABSTRACT

Meteorological models for rainfall and frost were studied to syn- chronize the operational policies of the mill and crop harvesting poli- cies. For rainfall amounts, the weekly lag-one serial correlatior~ coeffi- cient was 0'52, significant at 99% level of probability; therefore a lag- lone Markov chain model with the principle of Monte Carlo technique was used for rainfall data generation. The rainfall data generated with technique when compared with historical data indicated close agree- ment for mean, standard deviation and skewness coefficient, although the model was not developed to preserve the skewness. The lag-one serial cornelation coefficient computed. from generated data were in good agreement with those computed from the historical data. For frost, the correlation coefficient was not computed. Due to heat stor- age processes of the atmosphere, it was assumed that frost data can also be generated with a lag-one Marltov chain. The data generated with this technique when compared with the historical data indicated con- sistency of mean and standard deviation, but it did not preserve the skewness coefficient. This study indicates that the Markov chain can be used effectively for rainfall and frost data generation, both elements having particular importance to sugarcane rendement and yield in irri- gated districts of Pakistan.

INTRODUCTION

The irrigated sugarcane districts of Pakistan have great production potential with year-round radiation, deep alluvial, medium textured soils and extensive irriga- tion facilities to support sugar cane growth (Ayub and ~ a r l y ~ 1. Several constraints, however, hinder sugarcane production and these include salinity accumulations, a water table approaching the soil surface, an extensively over-committed irriga-

767

768 AGRICULTURAL ENGINEE~ING

tion water supply, high potential evapotranspiration rates causing high irrigation demands, a low annual rainfall and frequent frost occurrence in the early calendar months of the year. The latter two meteorological elements, rainfall and frost, are two stochastic components of the environment of sugarcane production, the quantification of which directly influences tonnage yield as rainfall in monsoon season substitutes for irrigation and rendement and as frost influences maturity during the cool harvest season. Both of these elements can be incorporated in a larger, overall management model for a sugarcane district as per the work of Aste- ~onsman' and ~ a r l ~ ~ ,

Review of Past Modelling Research for Rainfall Events <' -%

caskey6 indicated that the rainfall probability distributions were closely approximated by a simple Markov chain model, consisting of a 1st order Markov chain and 2x2 transition matrix (wet and dry days). Comparison of pure random and Markov chain probabilities indicated that a Markov chain model was more reasonable than pure random assumption. Gabriel and ~ewmann' studying the succession of wet and dry days showed that a two state (wet and dry) Markov chain was a good model for representing this dichotomized process, indicating.\ the probability that a subsequent day (1+1) will be wet or dry and is clearly depen- dent upon the event which occurred on the previous day (t). wiser13 found that the f i t of a simple Markov chain model was unsatisfactory and proposed a modi- fied Markov probability model using four modified Urn models as special cases with the desired characteristics and reducible to a simple Markov chain model. The modi- fication was done by allowing the content of an Urn to vary in a specified way during a sequence of draws of balls of the same color from the Urn. When the se- quence is terminated by drawing a ball of opposite color, the content of the Urn is restored to i t s initial state. He concluded that modified models were superior to a simple Markov model, in some cases which reflects the conditions when prp- sistence plays a more important role than 'that assumed in the simple Markov chain model.

Hopkins and I3obillardl0 approximated the pattern of rainy days over a 45- year data for the six-month growing season (April-Sept.) by using a simple Markov chain model. First, they calculated the proportional frequencies of occurrence of dry and wet days according to previous one or two day events and found depen- dency of second order. Secondly, they used a two state transition matrix for each month and concluded that Markov chain models oversimplify conditions in two respects. First, the trends in transition probabilities guaranteeing the observed seasonal precipitation pattern must actually occur continuously rather than the discrete monthly steps; hence the system must be non-stationary even within the months. Secondly, the Markov chain model ignores the different origin of precipi- tation which were the cause of the errors of the model.

Feyerherm & ark^ considered the difference in the use of a first order

M. K. AYUB AND A. C. EARLY 769

Markov chain for estimating the probabilities for sequence of wet and dry days. These probabilities were found to have a dependence on the state-of the previous day, but only a slight dependence on the state of the day before. The probabili- ties of sequence of wet and dry as calculated from the observed data were com- pared with those of the independent, 1st order-dependent, and 2nd order-depend- ent probabilities. It was concluded that difference between them was negligible and

I that a 1st order Markov chain model was adequate for computing probabilities for short sequences.

Todovis & ~oolhiserl aimed at finding an explicit expression for the pro- bability distribution of the total amount of precipitation, Sn, during a period of n days. Under the hypothesis that total precipitation for K wet days in a period of n days long is independent regardless which of these K days were actually wet and which of the (n-k) days were dry, they showed that

where Nn is the number of wet days in an n-day period and sv is the total amount of precipitation for v wet days. P (sv < s) was evaluated assuming that P (S1 < s) = 1 - eq5, i.e., the amount of rainfall of wet day in exponentially distributed random variable. They further assumed that rainfall depth on different wet days were independent, therefore sv is the sum of v independent exponentially distri- buted random variables and thus have a Gamma distribution. P (Nn =v) was evaluated under two hypotheses: 1) there is no serial dependence in the sequence of wet and dry days, Nn is assumed binomially distributed and 2) the sequence of wet and dry days follows a two state Markov chain, hence the results of Gabriel and ~ewmann' are applicable. They found that Markov chain exponential model was superior to the binominal-exponential model, an indication that precipitation can not be treated as a succession of independent events. It appears that a new class of model i s a t hand i f one assumes that the dry wet condition of day i+l de- pends on conditions of day i and that the amount of precipitation on day i+l also depends on the measured depth on day i; the way to proceed is to divide the range of observation in n >, states rather than having only n = 2. An nxn transition matrix can then be estimated and the concept of Markov chain again applied.

Kharal & enrich' ' used the n state Markov chain to model the daily rainfall and considered the process to be stationary for each month, id. the year was di- vided in 12 seasons. The range for daily rainfall depth was divided into 14 intervals. Therefore, 13x14 = 182 transition probabilities must be estimated for each season. They did not, however, attempt to f i t the analytical distribution to conditional pro- babilities of the transition matrix. Whenever a state was reached, the mid-point of the corresponding interval was assigned as the general rainfall depth. They observed that any Markov chain approach suffers from the opposing effects of the need for a large number of states (to increase precision) and the explosion of the number of

the transition probabilities which must be estimated. Analytical distribution can not always be fitted to alleviate the problem.

Strictly speaking, the set of transformed observations does not constitute a random sample because of the persistence in the data and the lack of negative values. Therefore, the observed data which was not standardized and persistence was included by treating the series as a Markov chain model of lag one.

Frost Day Analysis and Generalization Techniques

The same principles that apply to wet and dry day analysis with the two state Markov chain is applied to the frost day and non-frost day. This is done by simply defining a frost as one with a minimum temperature less than O'C or 32'~. Several rainfall criteria defining wet days were used and the probabilities of Markov chain were considered to be constant for each month. The results were satisfactory for all defining criteria.

System Synchronization Applications

Guise and Wandg used quadratic programming to optimize the scheduling of production in Australian sugar mill with the aim to maximize the profit, subject to the limit of mill processing capacity, cane availability, sugar production in peak periods, fixed price and cost constraints. ~ste-~onsman' determined optimal harvest and irrigation policies for sugarcane production. The techniques used include a modification of the Simplex Algorithm of linear programming, utilized at three levels of planning at~d decision making. ~ a t - 1 ~ ~ developed a simulation mo- del using the principle of Markov chain with a crude Monte Carlo procedure to determine the effect of alternate irrigation policies, irrigation capacities and milling capacity constraints on the production of sugarcane in the Philippines.

Each application has a particular set of environmental circumstances that must be considered, but the application toward synchronization or provision of a steady cane supply from field to factory with optimization on net economic re- turns to the district i s the common goal of each research activity ( ~ y u b ~ ) .

MATERIALS AND METHODS

In designing a system, models are needed. It is not advisable to construct a model based solely on the historical data, as the system is designed for the future sequence of events, and not for the past. Moreover, the historical data available are generally not sufficient to represent true population of input. Therefore, the generation of data is required for system designing or system performance studies, so that the model can be used with satisfactory confidence in the analysis. For the sugarcane system, climatological models of rainfall and frost are needed. The models are required to be practical and representative of actual climatological

M. K . A Y U B A N D A . C. EARLY

events. The production models and mill performance models are also required to synchronize the field and milling operations. -

I Climatological Models

The actual models proposed are representative of those that are stochastic in nature. The aim of such models is to simulate as likely a sequence of events as possible, based on historical daia. The harvested sugarcane i s a perishable com- modity; storage is not permitted between weekly time periods. Moreover, the irriga- tion system in Punjab is "WARABANDI" which means a farmer gets irrigation wa- ter once a week. Therefore, the time unit selected is week to observe the effect of rainfall and frost on the tonnage and rendement yield of sugarcane type of data

I used.

An extensive meteorological record from 1914-1975 was available. Rainfall data from a 20-cm standard non-recording rain gauge were obtained. The data from 1914-1970 were recorded in inlday, while from 1971-1975 were recorded in cml day. The amount of daily rainfall was observed a t 08:OO a.m. For the previous 24- hour period, the data for temperature was obtained from a maximum and mini- mum thermometer, and was recorded daily in OF.

First order Markov chain was considered to be adequate to generate the data for rainfall and the frost. The first order or lag-one model for rainfall was justified by computing the auto-correlation coefficient, which is a measure of association within tfie series. The computed values of auto-correlation coefficient were ob- tained from 210 values (i.e. 4 years), as 50 to 300 values are suggested as an opti- mum number for computational purposes. The lag-one stationary transition pro- bability matrices were estimated from the historical record. As the data available were on daily basis, the daily rainfall was summed up into weekly rainfall. The accumulated weekly rainfall series were grouped into n+l states, providing an open class for the largest rainfall. The following example shows the limit of transition matrix. A (9x9) matrix was finally used.

I The class limit of the transition matrix were

Depth of rain Corresponding class limit

For frost, the correlation coefficient was not computed due to complex- i t ies involved. It was assumed that lag-one Markov chain model can also be used for

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frost data generation. The number ot frost days in each week were summed toge- ther, and the state of the transition 'matrix was determined by the number of the frost days. A (7x7) matrix exists if frost occurs continuously. The state presenting matrix of historical data was then subjected to frequency distribution. Frequency for each particular state for groups depended upon the states of the preceeding weeks' event. These probabilities when accumulated successively gave rise to con- tional transition pro,babilities matrices.

Procedure for Data Generation

1) The initial oonditions were given for time period TI from some dis- crete meteorological observation in the record.

2) The time counter was increment by one week T = T + 1

3) The rainfall amount class for the current period T was determined, so with the transition probabilities for rainfall states and frost states for the dis- crete Stationary Markov chain, as determined by enumeration from historical rain- fall and temperature record.

4) A random number between zero and one was generated. The class of the random event in the transition probability matrix of random event for the week of the rain year was determined by comparing the random number to the cumula- tive probability for the category of the previous week. Corresponding to period T, the position of random decimal number, and the class for the event T+l was determined. This method is in effect, the crude Monte Carlo procedure described by ~ u r a s ~ .

RESULTS AND DISCUSS1 ON

Rainfall Model

It i s assumed that weekly rainfall is a time series of accumulated rainfall of daily time series. The rainfall amount of a week is dependent on the rainfall of the week before. The assumption is justified by computing the auto-correlation co- efficient for different lags (Fig. 1 ). The graph of auto-correlation coefficient versus lag called a correlogram, clearly indicates that lag-one is significant a t 99% level of probability with an upper confidence limit of +0.152 and a lower confidence limit of -0.162. Other lag confidence limit tests were not made in this analysis. These values were "eglected with the assumptions that under the study, the highly si$ni- ficant results for lag-one were sufficient for the modelling of the general physical phenomenon. Moreover, the present event can affect the succeeding event, but i t s effect on the fourth area of subsequent events cannot be explained a t this time.

After selecting the Markov chain model, conditional probability transition

M. K. AYUB AND A. C. EARLY 773

matrices were estimated by using 61-year historical daily rainfall data which were assumed to weekly rainfall data. The transition matrices were a series of lag-one multistate matrices. The different states of the matrices correspand to the differ ent range of rainfall amount of the week. The threshold values to divide the states were chosen randomly, with the assumption of uriiform distribution.of rainfall within the week.

The numbe; of States of the matrices were determined by looking a t the per- formance of the model in generating rainfall sequences that should reproduce cer- tain statistical parameters of the historical data. By trial, 9 x 9 matrix was selected as the best one, The threshold values for different states of matrices are tabulated in Table 1.

p e r limit confidence ( Qgo/o ) I \

\ upper limit confidence Q ~ Q , ~ )

FIGURE 1. Correlogram of Historical Rainfall Data (212 readings), , "

In order to determine the. transition matrices, data were generated for the dif- ferent ~izes of transition matrices. The criteria to choose the best matrix was the mean and the standard deviation weekly rainfall amount as the statistical para-

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meters. These parameters were chosen because they were rather difficult properties of the series to reproduce.

The trial for selection of states of the matrix was started by 5 x 5 transition matrices. The above-mentioned statistical parameters were compared with historical series and the generated series. The number of states of the matrices were increased incrementally one by one up to (9 x 9) including the matrices. The last sets (9 x 9) were chosen because they reproduced the statistical properties closest to the his- torical data. An example matrix i s presented in Table 2.

TABLE 1. Threshold values of rain for different states

.". : Initial .- Final

State reading reading Range

Frost Model

The development of the model requires 3 assumptions. First, the daiiy frost series can be rega;ded as a sequence of two events, i.e., frost day and non-frost day. Secondly, the weekly frost series which is obtained by accumulating the frost days in each week has a significant first order persistence. Thirdly, a frost day is defined as a day for which the value of observed minimum temperature i s less than O'C.

As stated above, the daily frost series were converted into weekly frost series and the first order conditional probability transition matrices were investigated in the similar way as was done in rainfall analysis. A typical matrix from the chain is shown in Table 3.

The development of the model requires 3 assumptions. First, the daily frost series - - can be regarded as a sequence of two events, i.e., frost day and non-frost day. i

M, K. AYUB AND A. C.'EARLY 775

Secondly, the weekly frost series which is obtained by accumulating the frost days in each week has a significant first. order persistence. Thirdly, a frost day is defined as a day for which the value of observed minimum temperature is less than O'C.

As stated above, the daily frost series were convlrted into'weekly frost series and the first order conditional probability transition matrices were investigated in the similar way as was done in rainfall analysis. A typical matrix from the chain i s shown in Table 3.

TABLE 2. A conditional probability transition matrix

Period t 1 2 3 4 5 6 7 8 9

0.0- 0.47- 0.59- 0.78- 0.86- 0.89- 0.93- 0.97- - 0.46 0.58 0.77 0.85 0.88 0.92 0.96 1.00

0.0- 0.51- 0.68- - - - - - 0.50 0.67 1.00

3 0.33 0.33 0.33 :::;- ::::- - - -

0.0- 0.67 0.68- 0.90- - - - - - 0.67 0.67 0.89 1.00 - - - - I

0.0- 0.68- - - - - 0.67 0"67 0'67 1.00

I -

0.0 6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.68 0-67 0.67 0.67 0.67 0.67 - - I

0.67

8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 o.a 0.0

TABLE 3. A conditional probability transition matrix for frost, for week 15 to week 16 1

i

1 0.00 0.92- 0.94- 0.98- 0.91 0.93 0.97 1 .OO

2 0.00 - - - 1 .oo

3 0.00 0.51 - - 0.50 1 .OO

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Evaluation of Climatological Models

Clearly, the predicted values of given climatological elements will frequently differ to some degree from the actual observed values. The reliability of forecast and the basic requirements for a method of estimating this reliability are important. Such method should be objective; it should permit a comparison of accuracy of forecasts of various climatological phenomena. That requirement can be satisfied if the estimate of reliability is based on statistical methods. The basic statistical measures are mean, standard deviation, skewness coefficient and correlation co- efficient.

Rain in crns.

Historical Mean - Generated Mean - - - M~nirnurn Generated Mean

Maximum Generated Mean

2.30

2.20

2.10

2.00

1.90

1.70

, . WEEKS

FIGURE 2. Comparison of historical and generated mean.

M. K. AYUB AND A. C. EARLY 1 777

Rainfall in cms. 5001 I

Weeks

FIGURE 3. Comparison of standard deviation of rainfall

S k

10

9 i 8

7

6

5

4

3

2

I

0

Weeks

FIGURE 4. Comparison of skewness for rainfall r*$e

% "'i > 4%

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S 0 Coefficient - 0.9 r

1, /I- Generated I I I

~ ~ ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 -0 6

- 0.7 -08 - - 0.9 L J I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

2 4 6 8 10 12 14 16 18 20 Pi 24 26 28 31 32 34 36 38 40 42 44 46 48 50 52 54

Weeks

FIGURE 5. Comparison of serial correlation coefficient for rainfall

Days I Week Frost days

Standard deviation (frost) 28

Weeks Week

FIGURE 6.' Compariso'n of mean for frost F~GURE 7. Comparison of standard. deviation of frost

M. K. AYUB AND A. C. EARLY 779

For the purpose of comparison, the parameters from the historical record and randomly generated records were plotted on the graph. The average value of 28 sets of weekly generated data of rainfall and frost of the same equivalgnt length as the historical record were used on plotting. -, . Evaluation of Rainfall Model

Fig. 2 shows the comparison of the average weekly rainfall. I t shows a good agreement between the generated and historical data. It was observed that the varia- tion between the historical and general values increases with an increase in the mag- nitude of rainfall amount. During the wet period, the difference is as high as 17%. Since the model was constructed on a weekly basis, accuracy higher than this can- not be expected. The model on the whole reproduced this characteristic satisfac- torily.

The comparison of standard deviation (Fig. 3) indicates that model preserved this property satisfactorily. The maximum difference is observed during the wet period.

The comparison of skewness is shown in Fig. 4. Although the model was not constructed to preserve these characteristics, it holds this property quite satisfac- torily.

-- - -

Fig. 5 shows comparison of lag-one serial correlation coefficient. It i s clear that lag-one serial correlation coefficients which were significant in the historical data are in good agreement with the coefficients produced by the generated data.

Evaluation of Frost Model

The comparison of historical mean frost days per week and generated mean frost days per week shows a good agreement (Fig. 6). The frost model to some extent over estimates at the peak period. I t has significant agreement at earlier and latter periods.

Fig. 7 shows the comparison of standard deviation. The model reproduces desired characteristics satisfactorily. The model was not constructed to preserve the skewness.

SUMMARY AND CONCLUSIONS

Rainfall Model

Lag-one model correlation for weekly rainfall i s 0.52. The model i s also able to reproduce the standard deviation satisfactorily. Lag-one Markov chain did pre-

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serve the skewness. The comparison of serial correlation coefficient shows that they are dependent. Historical means plotted with generated means showed good agree- ment.

Frost Model

Lag-one Markov chain model can be bsed satisfactorily for frost data genera- tion, as the mean and standard deviation are in good agreement. However lag-one Markov chain model did not preserve skewness.

REFERENCES

1. Aste-Tonsman, J. (1973). Schedulinwnd programming the sugar cane crops in Peru. Unpublished doctoral dissertation, Cornell University.

2. Ayub, M.K. (1978). Development and refinements of component inputs for, simulation of sugar cane production and processing systems in Pakistan. Un- ' published M. Eng. Thesis, Asian Institute of Technology.

3. and A.C. Early (1980). Sugar cane systems synchronization oppor- tunities and production input models in Punjab of Pakistan. Proc. ISSCT,

(submitted, publication forthcoming).

4. Buras, N. e t al. (1973). Planning and updating farm irrigation schedules. I

ASCE J. Irr. & Drain. Divis, 99: 43-51.

Caskey, J.E. (1963). A Markov Chain model for the probability of precipita- tion occurrence.

Early, A.C. (1975). The influence of water management on operating policies for sugar cane districts in the Philippines. Unpublished doctoral dissertation, Cornell University.

Feyerherm, A.M. and L.D. Bark, (1967). Goodness of f i t of a Markov chain model for sequence of wet and dry 'days. J. Appl. Meteor, 6:770-773.

Gabriel, K.R. and J. Newmann, (1962). A Markov chainmodel for daily rain- fall occurrence at Tel aviv. Quart J Royal Meteor Soc. 88:90-95.

9. Guise, J.W.B. and C.J. Ryland, 1969). Production scheduling and allocation: a normatiye decision model for sugar milling. Australian J. Agr. Econ. 13: 8-24.

10. Hopkins, J.W. and P. Robillard, (1964). Some statistics of daily rainfall, occurrence from the Canadian prairie provinces. J. Appl. Meteor, 3:600-

M. K. AYUB AND A. C. EARLY 781

11. Kharal, N.N. and R.L. Henrich, (1974). A stochastic model Tor daily rainfall data generation. USDA Misc. Publ. 1275-1469-1485. .

12. Todovic, and Woolhiser, (1974).

13. Wiser, E.H. (1965). Modified Markov probability models of sequences of precipitation events. Mon Weath Rev. 93:511-516.

MODEL0 DE LLUVIA Y ESCARCHA PARA LA SINCRONIZACION DEL SISTEMA DEL AZUCAR EN PAKISTAN*

M. K. Ayub y A. C. Early**

ESUMEN

Modelos metereologicas de lluvia y de escarcha son reclueridas para sincronizar la politica de operaciones para cosechar la cosecha. Para cantidades de lluvia, la serie "lag-one" de correlacion de co- eficiente semanal era 0.52, significante al 99% de probabilidad; entonces, un modelo "lag-one" de cadena Markov con el principio de tecnica de Monte Carlo fu6 usado para la generacibn de datos de lluvia. Los datos de lluvia generados con la t6cnica cuando e compararon con 10s datos historicos indicaron conformidad en el promedio, desviacibn uniforme, y coeficiente oblCcuo (skewness coef- ficient). Aunque el modelo no fu4 dessarolado para el sesgo (skewness). La serie de "lag-one" de correlacibn de coeficiente cornputado de datos generados, congeniaban bien con las computados de datos historicos. Con respects a la escarcha, la correlaci6n de coeficiente no fub computada. Debido a1 procedimiento de calor en almacenaje de la atmosfera se tuvo como supuesto que el dato Be escarcha tmbien puede generar con un "lag-one" de cadena Markov. El dato gene- rado con esta tecnica, cuando se comparb con 10s datos historicas, indico consistencia de promedio y de desviacion uniforme, aunque no preverv6 el coeficiente de oblicualidad o de sesgo (skewness). Este estudio indica que la cadena Markov se puede usar eficazmente para la generacibn de datos de lluvia y de escarcha, ambos elementos teniendo particular importancia en el rendimiento de c a a dulce en distritos irrigados de Pakistan, * Papel preparado para ser presentado en el XV I I Congreso International de Tecnicos

de M a de Azbcar, PICC, Manila, Filipihas. +* lnvestigador Asociado, Centro Regional de Computer, lnstituto Asiatico de Tecnologr;?,

P.O. Box 2754, Bangkok, Taildndia lngeniero Agricultor de Irrigacibn de Aqba, lnstituto lnternacional de Estudios Cientificos de Arroz IIRRI), P.O. Box 933, Manila, Filipinas