Rainfall persistence: Detection, modelling, costs and value of probability information

AgriculturalSystems6 (198~81) 285 302

RAINFALL PERSISTENCE: DETECTION, MODELLING, COSTS AND VALUE OF PROBABILITY INFORMATION

S. R. HARRISON

University of Queensland, St. Lucia, Queensland, 4067, Australia

SUMMARY

The basic hypothesis discussed in this paper is that the presence of persistence or bunchiness in rainfall sequences reduces Jarm profitability. After a review of past invest igat ions into rainfall persistence, sta t ist ical tests are sho wn to indicate posit ire inter-temporal correlations in excess of seasonality jor weekly and monthly precipitation data from three recording stations in the northern Brigalow region of Queensland. A relatively simple procedure is advanced .for generating artificial rainfall sequences containing persistence. Simulation experiments with a grazingproperty model incorporating this synthesis procedure are used to test the basic hypothesis. Results indicate substantial reductions in both income and the rate of asset accumulation due to persistence in weekh' rainfidl. Finally, provision of meteorological information on persistence in the form of conditional rainfall probabilities is proposed, and a simulation and decision theol3' approach is suggested /or determining the potential value of such information.

l.

INTRODUCTION

The nature of rainfall series over time at point locations has been a matter of controversy. Patterns of various kinds--trend, cycles, autocorrelations and bunchiness, in addition to normal seasonal fluctuations within years- have been "detected'. These observations go back as far as biblical times when seven years of plenty followed by seven years of famine were experienced in Egypt (Genesis, 41 ). Regular movements of more than a dozen frequencies ranging from seven days to 2600 years have been identified by L a m b (1973). Other researchers believe that while particular weather conditions may be 'persistent' there are no regular cycles at work (e.g. Troup, 1965). Rainfall persistence may be defined loosely as the tendency for an 285 Agricultural Systems 0308-521X/81/0006-0285/$02" 50 '~ A p p l i e d Science Publishers Ltd, E n g l a n d . 198 l Printed in G r e a t Britain

286

s. R, HARRISON

amount of rain in a given time period (e.g. a week or a year) to be similar to that of one of more immediate past periods. No regular periodicity or cycle is implied. Under persistence, if the current period is wetter than normal (however defined) then the next period has an increased probability of being wet also; if the current period is dry then the next period has an increased probability of being dry. In statistical terms the conditional probability of a wet period following a wet period is greater than that of a wet period following a dry period, and rainfall amounts in successive time periods are positively correlated. The effect of persistence is that frequencies of the longer runs of favourable (and adverse) weather exceed those of an inter-temporally independent process. For short time periods such as the week and month this would imply an increased frequency of 'good" and 'poor" seasons and fewer "normal" seasons than would otherwise be expected. Persistence in annual rainfall would imply the bunching of good and of poor years. Runs of wet and of dry weather may have considerable impact on plant and animal production and hence on the economic performance of the farm business. A probable effect of within-year persistence is increased variability of annual incomes. Another less obvious effect may arise in extensive grazing regions due to asymmetric economic impact of wet and dry years: the former are associated with relatively small increases above average production while the latter involve major expense, e.g. for purchased fodder, agistment fees and forced sales or stock losses. This would reduce the absolute level of income as averaged over a number of years, with consequent effects on rate of asset accumulation and farm viability. In this paper the phenomenon of rainfall persistence is examined from a farmmanagement systems viewpoint. Statistical analyses for rainfall persistence are reviewed and several tests are applied to data from the northern Brigalow region of Queensland, Australia. A relatively simple rainfall synthesis procedure which incorporates persistence in weekly rainfall series is outlined. This procedure has been included in a whole-farm model of a cattle breeding and fattening property in the Brigalow region. Simulation experiments carried out with the overall model allow an evaluation of farm business performance under persistent and random weekly rainfall regimes.

2,

DETECTION OF RAINFALL PERSISTENCE

Whether or not persistence is detected in a given set of rainfall records will depend on a number of factors, including the location of the recording station: time interval of aggregation: nature of statistical tests applied, and length of record. In the lower rainfall areas the likelihood of runs of good and poor seasons appears to increase (Campbell, 1966), but relative variability typically is also higher {Pattison, 1964), and this may make detection difficult.

RAINFALL PERSISTENCE

287

From an agricultural systems viewpoint, the time unit of interest in rainfall modelling is usually the day or week, and occasionally the month. While it is often impossible to detect non-randomness in records of annual rainfall, no one would seriously doubt that hourly and daily rainfall are persistent. Frequently observed runs of wet (and dry) days associating with prevailing atmospheric pressure conditions would have very low probability of occurrence if daily rainfall were independent. Given that daily rainfall amounts are positively correlated, one might expect weekly rainfall totals to be related also, since each pair of successive weeks contains adjoining days. Following this reasoning, as the length of the time interval is increased the carryover effect is attenuated, persistence is weakened and the likelihood of detection falls. However, it is to be recognised that climatological factors other than short-term carryover effects are also associated with nonrandomness in rainfall patterns over time. The existence of cycles or runs is often asserted from a subjective appraisal of rainfall records, but mere inspection of the data can be misleading and statistical tests are desirable to place confidence levels on such conclusions. There appears to be no consensus as to the most appropriate statistical procedures to apply, and it has even been suggested that existing tests are inadequate (Campbell, 1966; Byerlee, 1968). The possibility that rainfall series are composed of a number of cycles, each of a fixed duration, superimposed on one another, may be investigated through spectral analysis. Using this technique, O'Mahoney (1961), Bowen (1967) and Coughlan (1975) have found evidence of two to three year and seven year cycles in rainfall for widely separated Australian cities; these are apparently related to fluctuations in atmospheric pressures between the Pacific and Indian Oceans known as the Southern Oscillation. A plethora of tests have been applied for persistence of a not-necessarily cyclical nature. These include both parametric and non-parametric methods, the latter seeming to offer greatest promise due to the particular shapes of rainfall distributions. While annual rainfall totals frequently approximate a normal distribution, for shorter intervals simple transformations such as logarithms and square roots often fail to produce a variate following any well known distribution (Goodspeed & Pierrehumbert, 1975). Some of the attempts to detect rainfall persistence are summarised in Table 1, in which findings are classified as P (persistence detected), N (no persistence), and I (results inconclusive). Probably the most commonly used procedures are Chi-squared tests on distributions of run lengths about the mean (Leeper, 1953: Greve et al., 1960: Bostwick, 1962) and the median (Maher, 1967) and autocorrelation tests. The latter group have included the Wald Wolfowitz test (Pattison, 1964) and R. L. Anderson's circular autocorrelation coefficient (Phillips, 1969).5" While the results of these studies are conflicting, in some cases it is possible to explain the negative or "I" Other possible autocorrelation tests include the non-Neuman ratio, T. W. Anderson's non-circular autocorrelation coefficient, and the Durbin Watson and Morin tests on residuals. A useful account of these tests is provided by Filan (1972).

288

s.R. HARRISON TABLE

1

SUMMARY OF TESTS FOR RAINFALL PERSISTENCE

Count O'

USA

l~l'pe of test

Coefficient of variability sequence Run lengths Persistence of departure from normal in consecutive months Autocorrelation test

Australia

Coefficient of variability sequence Run lengths Persistence ratio Persistence of departure from normal in consecutive months

Annual rainJbll

Shorter time intervals

Clawson (1937) P Hildreth (1959) 1 Greve et al. (1960) P Bostwick (1962) P Pattison (1964) N Pattison (1964) N Rutherford (1950) P keeper (1953) N

Body (1966) P Maher (1967) P Verhagen & Hirst (1961) I

P = persistence detected, N = no persistence, 1 = results inconclusive. inconclusive finding. For example, tests carried out by Verhagen & Hirst ( 1961 ) and Pattison (1964) applied to transitions above and below the mean for individual pairs of months, rather than to the rainfall record as a whole, and thus probably lacked statistical detection power. The study of Phillips (1969) is interesting in that monthly rainfall at Armidale, N S W , was significantly autocorrelated for various time lags. However, the coefficients were positive for months close together or separated by nearly a full year, and negative for intermediate lags, reflecting the normal intra-year seasonal pattern. This leads us to a more precise statistical definition o f persistence, viz. that part o f serial correlation not explained by seasonal effects. In order to test for persistence in rainfall for intervals of less than one year it is first necessary to remove seasonality. The appropriate procedure for doing this will depend on the nature o f the seasonal effect, e.g. whether it forms an additive or multiplicative c o m p o n e n t in the series. An attempt to eliminate seasonality before testing for persistence is discussed in the next section. The power of statistical tests and the likelihood o f detecting persistence is in general restricted by the period of rainfall records available. For example, relatively few Australian recording stations have more than 100 years of recorded data and none more than 150 years. In contrast, the conclusions of L a m b (1973) reported above are based on over 250 years of rainfall records and on inferences for earlier centuries drawn from tree-ring series and oxygen isotope measurements in snow deposits. As a general conclusion, objective tests carried out to date strongly suggest that Jbr m a n y locations rainfall a m o u n t s for periods up to one m o n t h are persistent;

289

RAINFALL PERSISTENCE

TABLE 2 EXPECTED AND OBSERVEDDISTRIBUTIONSOF RUNS BELOW THE MEDIAN FOR MONTHLY RAINFALL

Length of run (monthsl

1 2 3 4 5 6 7 or more Total ~ Chi squared statistic b

Dingo

Expected frequencI'

Observed frequency

Expected Jrequency

107"0 53"5 26.7 13.4 6.7 3-3 3.4

115 49 13 21 4 6 6

86.5 43.3 21.6 10"8 5-4 2-7 2.7

214.0 (222)

214

17.60"*

Twin Hills

Bombandy

Observed Expected J?equency frequency 85 41 18 10 9 2 8

173-0 (192)

173

Observed frequency

110.5 55'3 27"6 13"8 6.9 3-5 3.6

118 42 25 13 9 4 10

221.0 (243)

221

13-79"

16.09*

Numbers in parentheses are total frequencies of runs before scaling. h Critical values at the 5 and 1 °,o significance levels are 12-59 and 16.81, respectively.

further, some bunching of annual rainfall amounts which cannot be explained by chance probably takes place also.

3.

RAINFALL PERSISTENCE IN THE NORTHERN BRIGALOW REGION

A number of tests of persistence have been applied to rainfall data from the northern Brigalow region of Queensland, an area located north of the Tropic of Capricorn between Rockhampton and Mackay and on the western side of the coastal ranges. The region is characterised by predominantly summer rainfall and dry winters (e.g. mean January rainfall is approximately 13 cm compared with July and August means of less than 2 cm). Average annual rainfall varies between about 60 and 70 cm, being lowest in the north-west. Daily and monthly rainfall records for Dingo (in the south), Bombandy (central) and Twin Hills (in the north-west) have been obtained from the Australian Bureau of Meteorology. These recording stations were chosen because of their length of records and spatial distribution.t The daily figures have been aggregated into standard week totals,~ and a number of tests applied where appropriate to monthly rainfall, weekly rainfall or both. Examination of run length distributions Following Body (as reported by Maher, 1967), observed frequencies of runs below monthly medians of various lengths are compared with frequencies expected under the hypothesis of independence. As indicated in Table 2, Chi-squared statistics t The periods of monthly records are: Dingo, 74 years: Bombandy, 64 years: Twin Hills, 81 years. Daily records are somewhat shorter: Dingo, 73 years; Bombandy, 53 years; Twin Hills, 67 years. Short gaps early in the recording periods were judged to have no practical effect on the tests. :~ The last week of February in leap years and the last week of each July contain eight days.

290

s.R. HARRISON

exceed the 5 ",, critical value for each of the three centres. The frequencies of runs of three months or less duration are less than expected while the frequencies of long runs (of six months or more) exceed expectations. Tests on the total numbers of runs of monthly rainfall, both above and below the medians, have also been performed using the standard normal distribution (see Freund & Williams, 1959. 277 88). Positive persistence is indicated for the lower-rainfall centres of Bombandy (standard normal variate = 2.74) and Twin Hills (2.78) while the test statistic for Dingo (1 "28) achieves significance at the 10 "o level. Runs tests were not applied to weekly rainfall because many of the medians were zero.

E.vamination of regression residuals In order to measure and subsequently remove seasonal variation a linear model of the following form was postulated for monthly rainfall. 12

Rij = rio + >'j fl,nX., + .c;i,i m=2 where R u is rainfall in month j of year i: X,, are dummy variables for February through to December (1 in the particular month and zero otherwise): flo is a constant' /;,, are month effects: and ~;u is random variation. The dummy regressor for January was excluded to prevent linear dependence among explanatory variables, estimated fl coefficients measuring differences in means relative to that of January (i.e. month effects are assumed to be additive}. For each month, the residuals 12

Ci,j = R,j -/}o

m=2

provide a measure of random variation. These residuals were found to be largest in both positive and negative directions for the summer months for which both means and variances of rainfall amounts are greatest, suggesting that the seasonal component is not adequately removed in an additive model.

A utocorrelation coeJflcients of deseasonalised data An alternative method of removing seasonal variation which assumes a multiplicative model was devised. Recorded rainfall amounts were multiplied by the ratio of mean rainfall for all months over all years R.. to the mean level for the particular month R.~, i.e.

X u = (R../R.j)Rij

291

RAINFALL PERSISTENCE TABLE 3 AUTOCORRELATIONCOEFFICIENTSFOR ADJUSTEDRAINFALLLAGGEDONE TO SIX PERIODS

Lag (periods')

Monthly rainfall Dingo

l 2 3 4 5 6 Critical values

5~.o significance level significance level

Weekly rainfall

(74 years)

Bombandy Twin Hills (63 years) (81 years)

(73 years)

Dingo

Bombandy Twin Hills (53)'ears)

(67),ears)

0"0679* 0.0500 0.0338 -0-0115 -0.0246 -0.0246

0"0720* 0-0458 0.0412 0.0150 0.0144 -0.0093

0.0542* 0.0446 0.0527* -0.0064 0-0067 -0.0472

0.0481"* 0.0255 0.0143 0"0000 0.0128 -0.0082

0-0843** 0-0037 0.0242 0.0038 0'0092 -0.0001

0"0830** 0'0055 0"0282* 0-0104 0"0207 0"0143

0.0541

0-0583

0.0517

0-0264

0.0310

0.0276

0.0792

0-0828

0.0736

0-0375

0.0439

0,0391

where Xij is the adjusted rainfall for month j of year i. This transformation inflates figures for the low rainfall winter months and scales down values for the wetter summer months, an adjustment being made to the variances as well as the levels of rainfall records. While seasonality may not have been completely eliminated, especially in regard to skewness and higher moments, it is felt that a sufficient reduction was made to allow meaningful inter-temporal comparisons. A similar transformation has been applied to weekly rainfall. Autocorrelation coefficients (circular definition) of monthly and weekly rainfall at the three centres for lags of one to six periods are presented in Table 3. In each case rainfall in successive periods is significantly correlated. Coefficients for greater lags are consistently non-significant (the only exception being Twin Hill monthly and weekly rainfall lagged three periods) and tend towards zero as the length of lag increases. Briefly, these tests show that run length distributions of monthly rainfall and autocorrelation coefficients for 'deseasonalised' monthly and weekly data consistently indicate that monthly and weekly rainfall are persistent in the northern Brigalow region.

4.

SYNTHESIS OF RAINFALL SERIES

Background The generation of artificial series of rainfall data pla~cs an important role in agricultural systems studies and in water resources research. While historical series are sometimes used (e.g. see Wright, 1970), these are often too short, are cumbersome for modelling purposes and in any case constitute just one (perhaps unrepresentative) sample from an infinite population. As little progress has been made in

292

S. R. HARRISON

modelling the physical process resulting in rainfall, the procedure usually adopted is to devise a statistical sampling routine to generate rainfall series containing the most important characteristics of the historical trace, such as means, variances, skewness and persistence. Agricultural systems studies have frequently overlooked the latter property.t The literature on rainfall synthesis is reviewed by Goodspeed & Pierrehumbert (1975). Also, considerable attention has been paid to the modelling of persistence by hydrologists in the study of catchment runoff, and recent reviews of this field are provided by McMahon (1977) and Srikanthan & McMahon (1978a).

The method A relatively simple method of rainfall synthesis has been devised for use with a property development model for ballot blocks in Area 3 of the Fitzroy Basin Land Development Scheme. The overall model and rainfall module are described elsewhere (Harrison, 1976; Harrison & Longworth, 1977) and only a brief outline will be provided here. Weekly rainfall is modelled as a one-stage Markov chain process, with rainfall states based on decile levels to form a 10 × 10 transition matrix. The general element of this matrix Pij is the probability of a rainfall amount in decile class j in week w conditional upon an amount in decile class i in week w - 1. If rainfall were highly persistent from week to week, in the sense of remaining in the same decile class, then the transition matrix would approach an identity matrix. On the other hand, if rainfall in successive weeks were independent, all p;j would be approximately 0.1, the conditional distributions for each week w being constant regardless of the state in week w - 1. The extent of persistence, then, is indicated by the degree of concentration of high values (Pii > 0.l) around the principal diagonal and low probabilities furthest from this diagonal. The frequencies of transitions for each pair of weeks throughout the year were determined for each of the three centres. These frequencies were next pooled on a "seasonal' basis, seasons adopted being summer (weeks 1 to 18 inclusive), winter (weeks 19 to 35) and spring (weeks 36 to 52). Transition matrices for each season and for the year as a whole were calculated using the maximum likelihood method, that is: 10

p~j = / ; j / L / ; ~ j

1

where pi i is the estimated probability, andJlj is the observed frequency, of transitions from state i to statej.~ To facilitate calculations, a multiple of 10 years has been See, for example, Longworth (1969), Bravo (1970), Chudleigh (1971), Dumsday (1971) and Trebeck (1971). In Dumsday's model weeklythough not daily rainfall was assumed to be independent. :~Where records contained a high proportion of weeks without rain the frequencies of transitions between these weeks were distributed uniformly over all decile classes containing zeroes.

RAINFALL PERSISTENCE

293

TABLE 4 SUMMER TRANSITION MATRIX FOR WEEKLY RAINFALL, DINGO

Week w

t

0"1182 0"1182 0"1181 0.1085 0.0892 0-1007 0.0929 0.0798 0.0845 0.0900

0.1182 0.1182 0.1181 0.1085 0.0892 0.1007 0.0929 0.0798 0.0845 0.0900

0.1152 0.1152 0.1152 0.1139 0.0941 0.1059 0.0999 0.0752 0.0808 0.0845

0.1039 0-1039 0.1128 0.1004 0.1116 0.0957 0.1274 0-0773 0.0719 0.0951

0.1091 0.1091 0.1060 0.1037 0-1052 0-0763 0.1217 0.0828 0-0954 0-0907

0.1021 0.1021 0.1051 0.1077 0.0883 0.1154 0.1100 0.1061 0.0679 0.0953

0.0862 0.0862 0-0826 0.1116 0.1200 0.1341 0.0454 0.1208 0.1199 0.0932

0.0955 0.0955 0.0910 0.1071 0.1149 0.0587 0.1096 0.1261 0.1008 0.1008

0-0825 0.0825 0.0770 0.0725 O.0928 0"0986 0.1157 0.1092 0-1429 0.1262

0.0690 0.0690 0.0743 0.0661 0.0946 0'1139 0.0845 0.1429 0.1513 0.1345

0.0879 0-0879 0.0886 0.0924 0.0847 0.0946 0.1051 0.1062 0.1181 0.1347

0.0828 0.0828 0.0842 0.0838 0.0852 0-0919 0.0979 0-1194 0.1627 0.1092

TABLE 5 TRANSITIONMATRIXFOR WEEKLYRAINFALLIN ALL SEASONS,DINGO

Week w

I

0.1117 0.1117 0.1108 0.1038 0.1014 0-0986 0.0965 0-0901 0.0834 0.0921

0-1117 0.1117 0.1108 0.1038 0.1014 0.0986 0.0965 0-0901 0.0834 0.0921

0.1097 0.1097 0.1089 0.1053 0-1052 0.1001 0.0985 0.0880 0.0847 0-0899

0.1073 0.1073 0.1095 0.1003 0.1050 0.0989 0.1099 0.0883 0.0783 0.0980

0.0986 0-0986 0-0972 0.1023 0.1100 0.0925 0.1175 0.0885 0.0994 0-0928

0.1059 0.1059 0.1063 0.1104 0.0958 0.1189 0.0875 0.1037 0.0838 0-0872

0-0911 0.0911 0.093 0.1049 0-1122 0-1073 0.0931 0-1037 0.0973 0.1044

0.0935 0.0935 0.0944 0-0930 0-0992 0.0986 0.0974 0.1219 0.1089 0.0997

for e a c h c e n t r e , t a n d a c o m p u t e r p r o g r a m was w r i t t e n to d e r i v e t h e m a t r i c e s . T a b l e s 4 a n d 5 c o n t a i n the t r a n s i t i o n m a t r i c e s for s u m m e r a n d a l l - y e a r at D i n g o . A l t h o u g h s o m e i r r e g u l a r i t i e s are p r e s e n t , e s t i m a t e d p r o b a b i l i t i e s t e n d to be g r e a t e s t a r o u n d the p r i n c i p a l d i a g o n a l s . included

T h e t r a n s i t i o n m a t r i x for s p r i n g was f o u n d to be s i m i l a r to t h a t for s u m m e r . t h o u g h the p r o b a b i l i t i e s for w i n t e r w e r e less stable. Since the p a t t e r n o f persistence is a p p r o x i m a t e l y the s a m e t h r o u g h o u t the year, t r a n s i t i o n m a t r i c e s for e a c h season h a v e been p o o l e d (as in T a b l e 5) to f u r t h e r r e m o v e i r r e g u l a r i t i e s a n d p r o v i d e m o r e reliable probability estimates. T r a n s i t i o n m a t r i c e s for the o t h e r t w o c e n t r e s w e r e s i m i l a r in f o r m to t h a t for D i n g o , b u t w i t h m o r e i r r e g u l a r i t i e s d u e to the l o w e r n u m b e r s o f o b s e r v a t i o n s . T h e p o s s i b i l i t y t h e r e f o r e exists f o r spatial p o o l i n g o f t r a n s i t i o n m a t r i c e s to a l l o w f u r t h e r s m o o t h i n g o f p r o b a b i l i t i e s . H o w e v e r , in the a b s e n c e o f a m o r e d e t a i l e d c o m p a r i s o n o f m a t r i c e s a n d i n c l u s i o n o f a d d i t i o n a l r e c o r d i n g s t a t i o n s this was n o t c a r r i e d out.

Operation o/" the rainJall m o d u l e T h e rainfall g e n e r a t i o n m o d e l for e a c h c e n t r e is c o n s t r u c t e d as follows. First. t Seventy years weekly rainfall were included for Dingo, 40 for Bombandy and 60 for Twin Hills.

294

s.R. HARRISON

historical data for each standard week over all years are grouped into frequency distributions. The distributions for each four successive weeks (i.e. week 1 to 4, 5 to 8 . . . . 49 to 52) are then pooled on the assumption that they do not differ appreciably within each standard month. Each of the resulting 13 distributions o f weekly rainfall is then expressed in cumulative form, smoothed and divided into percentiles.t Also, each of the conditional distributions (i.e. each row) o f the transition matrix is expressed in cumulative form, i.e. Pij values are replaced by J c;~ = ) k

p~j

for every i

1

The sampling procedure consists of two stages, viz. selection of a decile class and interpolation within that class. In the first stage, given that rainfall in ,,,leek w - 1 is in decile class i, a uniform r a n d o m number r between 0 and 1 is obtained, and decile state j for week w is selected where g.J ~ - r < c ~ , i The second sampling stage follows the bracket median approach suggested by Phillips ( 1971) in which two r a n d o m numbers are selected, the first indicating the 0.01 probability interval and the second being used for a linear interpolation within this interval. Comments

on rainJall m o d u l e

Several features of this synthesis method warrant further comment. Division of states according to decile values rather than fixed absolute levels (as in the models of Pattison, 1969 and Haan et al., 1976) maintains equal frequencies in all classes for different times of the year. Thus frequencies of transitions between various states is of the same order t h r o u g h o u t the year, making estimated probability values more stable. Also, the probability estimates are independent of seasonality, allowing pooling where appropriate to reduce irregularities. Even so, a large number of probabilities have to be estimated, and it may be desirable to further reduce irregularities in these probabilities by limiting the transition matrix to say five states, particularly if the rainfall record is short. The second feature concerns the sampling procedure within decile states. Haan el a/. (1976) found that the assumption of a uniform distribution over broad states results in upward bias in rainfall amounts. ::l: Further division of decile states into percentiles restricts the uniform distributions to narrow ranges and largely eliminates this bias. + A computer program developed by Buchin (1965) and made available by the University of New England was used to fit a series of osculating quadratic functions to each distribution and to determine medians of each percentile class. ++The Haan et al. (1976)model has only seven states; a uniform distribution is assumed within the first six and an exponential distribution in the upper state.

295

RAINFALL PERSISTENCE

t~

,¢ 0

r~

o ~b

G eq

~O-trsr~

~ '

~

~._.o

-

=

==

~ ~

%

m~

.__."5-

z=

296

s. R. HARRISON

One weakness of this and other similar models may be noted. While short-term persistence is maintained in the generated rainfall sequences, the long-term persistence as evidenced by the Hurst effect (Klemes, 1974) is not. Although procedures for incorporating long-term persistence have been devised,t they are not widely used because of their complexity and excessive computer time demands (Srikanthan & McMahon, 1978b). The ability of the above model to generate synthetic data with similar statistical properties as the historical trace was assessed using a computer validation package developed by Harrison & Fick (1978). Results presented in Table 6 confirm that the upward bias in mean simulated rainfall level as reported by Haan et al. has been largely overcome and that the variances and lag-one autocorrelation coefficients are adequately preserved. Chi-squared statistics and tests of the cumulative distributions also indicate that the model is adequate.

5.

THE COST OF PERSISTENCE

The economic impact of rainfall persistence has been examined by way of a computer simulation experiment using the whole-firm model of a cattle grazing property in the Brigalow region. In the overall model the rainfall synthesis module is linked through a soil water-budgeting routine to a pasture condition submodel which in turn drives a livestock performance submodel. In this way rainfall indirectly determines calving rates, liveweight gains, timing and numbers of stock sold and supplementary feeding requirements. Each encounter with the model simulates the physical and financial operation of the farm business over a 15-year planning horizon. The simulation experiment consisted of six treatments, i.e. three locations by two rainfall regimes. Rainfall data were generated by using the weekly rainfall distributions and pooled all-year transition matrix for each of the three centres in turn. Persistent series were generated in the manner outlined above, and to obtain independent series each transition matrix was replaced by one with uniform conditional distributions (i.e. with all elements set at 0.1). Each treatment was replicated 30 times. The analysis assumes that the overall model is applicable to each location in the sense that production relationships, cost levels and other parameters do not vary. The same property development program is followed for all encounters with the model and identical stochastic series of beef prices and inflation rates are employed for corresponding replicates of each treatment. For each centre, then, the difference in financial performance of the farm firm is attributable solely to the different rainfall pattern. t These include fractional Gaussian noise, broken line and mixed auto-regressive moving average models (McMahon, 1977).

297

RAINFALL PERSISTENCE

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"~" ~ ~ ~ i-~ ,~I-

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z <

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R A I N F A L L PERSISTENCE

299

Table 7 presents mean net incomes before tax and mean terminal net worths over all replicates. The income series are also graphed as Fig. 1. Income differences are somewhat variable over time and between locations. However, after the property is fully developed pre-tax income is consistently reduced, the fall being of the order of $5000 per a n n u m . t Also, net worth at the end of 15 years declines by approximately $15000. As postulated, rainfall persistence results in a considerable cost to the grazier, the reduction in pre-tax income being of the order of five per cent in the last five years of the planning horizon.

6.

VALUE T O M E T E O R O L O G I C A L I N F O R M A T I O N

An attempt is now made to indicate what implications the above findings may have for services of meteorological bureaux and to suggest how benefits of any additional services in this area may be evaluated. This section of the paper is of necessity largely speculative and is designed to provoke further thought on the topic. In the field of farm management there are two main users of meteorological services, viz. the researcher (especially in the agricultural systems area) and the farm operator. The former makes extensive demands for a variety of data, on precipitation and also temperature, evaporation, solar radiation, wind-speed, frost etc. Here primary data are readily available, but some further analysis (e.g. correlations, conditional probabilities) and development of statistical modelling procedures to take account of factors such as persistence could be of value. The benefits of such services would not be immediate, and would be dependent on a wider acceptance of systems models as routine management aids. On the applied level, use of historical rainfall probabilities has often been advocated as an aid to farm decision making, Queensland examples being provided by Robinson & Mawson (1975) and Kingston (1976). Such probability information is a logical aid to decision making, allowing estimates to be made of the range of financial outcomes possible in a given situation, the likelihood of each, and the expected profitability of adopting various management alternatives. Farmers vague subjective probabilities of rainfall amounts are replaced by more accurate objective information; this may be especially useful for new entrant farmers or when contemplating introduction of enterprises new to an area. Probabilities of various rainfall amounts during periods of one month, two months and so on have been computed by the Bureau of Meteorology for a large number of recording stations throughout Australia. Information on rainfall persistence could be provided as conditional probabilities of various rainfall t Substantialland developmenttakes place during the first 8 or 9 yearsand essentialincomeis maintained by forced sales of livestock. Thus incomeduring the developmentphase is maintained at the expense of earning capacity in the immediate post-development years.

300

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amounts given the rainfall recorded in the previous time period. Rainfall levels could be expressed in absolute amounts, or in relative terms as in Tables 4 and 5. These conditional probabilities may be viewed simply as an extension or refinement of the unconditional figures, and could be used by farmers for the same purposes. As an example in a Queensland grazing setting, suppose January and February had been unseasonally dry and a grazier was contemplating whether to sell cattle or hold onto them (with the possibility of forced sale later and at a lower price). If the (unconditional) probability of substantial rain in March is high he may decide to hold on. But it may be that the probability of adequate March rain conditional on a dry January and February is somewhat lower, and the optimal policy is to sell now. This example illustrates the usefulness of information on rainfall persistence. One problem which arises is that conditional probability information could become very bulky, a complete distribution being required for each current state. However, it is suggested that these data need relate only to critical rainfall amounts at key decision-making points in the farm calendar, e.g. during prolonged dry periods on grazing properties; at planting or harvest time for grain crops; at irrigation periods; or during crop drying (of hay, grain, raisins, etc.). Since the information is situation specific it would best be compiled by extension officers with a close knowledge of the farming methods. However, there is a clear need for meteorological bureaux to provide forward forecasts of rainfall in the form of probability statements, as has been done in the USA for more than a decade (Waite, 1967). (Persistence effects would be built into these forecasts.) There is also a need to educate farmers in the use of rainfall probability information, whether as tables of past relative frequencies or as forward-looking forecasts. A quantification of the value of meteorological information on rainfall persistence to farmers would be most difficult. In fact, no assessment of the value of unconditional rainfall probability information is known to the author. Since the use made of such information would vary with the location, farming system, decision situation, information-using behaviour and goals of the farmer and so on, it is likely that any evaluation would have to be on a case-by-case basis. Aggregation of benefits throughout a country, even for just one rural industry, would be a daunting task. The simulation technique could again be used as an evaluative device, together with a modern (Bayesian) decision theory approach. That is, for a given decision situation and given pay-off rules the value of a perfect predictor (perfect information) could be used to set an upper limit on the returns from additional information. The value of probability information expressing various degrees of uncertainty could likewise be assessed.+ In this regard conditional probabilities would presumably have greater value to the decision maker than unconditional probabilities. A measure of the usefulness of conditional probabilities could also be + An interesting study along these lines by Byerlee (1968) suggests that substantial value could accrue from a long-range annual rainfall predictor when planning nitrogen fertiliser policies for wheat and fodder conservation decisions.

RAINFALL PERSISTENCE

301

obtained using the definition for information content of a probabilistic forecast as put forward by Theil (1967), which in this case would become n

Ik = ~'Pij[logpij

-- l o g p i ]

i=1

where p~ are the unconditional or prior probabilities over n rainfall states and p~j are the conditional or posterior probabilities of transitions between states. However, this statistical measure of information content would need to be translated into economic terms. 7.

CONCLUSIONS

Statistical analyses provide strong evidence that for some locations rainfall levels in succeeding intervals of up to one month duration are persistent; that is, they possess serial correlation in excess of seasonality. For modelling purposes, use of historical rainfall data for complete years intrinsically captures this short-term persistence. However, sometimes one will wish to introduce more generality into simulated rainfall sequences by sampling from probability distributions for each time period. If. say, weekly or monthly rainfall were sampled at random from the historical distributions (in either empirical or functional form) there would be a danger of both underestimating the variability and overestimating the level of financial performance. This is overcome by building into the sampling procedure a mechanism for relating rainfall amounts in successive periods. The mechanism may take the form of a simple Markov model as described here or more complex models such as are used for streamflow generation.

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COUGHLAN, M. J. (1975). The evidence for climatic change in some Australian meteorological data, Australian Conference on Climate and Climatic Change, Monash University. DUMSDAV,R. G. ( 1971 ). Evaluation of soil conservation policies by systems analysis, in Systems Analysis in Agricultural Management, eds. Dent, J. B. & Anderson, J. R., Wiley, Sydney. FILAN, S. J. (1972). Tests for autocorrelation: applications in agricultural research, University of New South Wales. FREUND, J. E. & WILLIAMS,R. J. (1959). Modern Business Statistics, Pitman, London. GOODSPEED, M. J. & PIERREnUMaERT,C. L. (1975). Synthetic Input Data Time Series for Catchment Model Testing, Prediction in Catchment Hydrology, Australian Academy of Science. GREVE,R. W., PLAXICO,J. S. & LAGRONE,W. F. (1960). Production and income variability of alternative farm enterprises in northwest Oklahoma, Okla. Agr. Expt. Sta. Bul. B-563. HAAN, C. T., ALLEN, D. M. & STREET, J. O. (1976). A Markov chain model of daily rainfall, Water Resources Research, 12(3), 443- 9. HARRISON, S. R. (1976). Optimal growth strategies Jor pastoral ,firms in the Fitzroy Basin land development scheme, unpublished Ph.D. thesis, University of Queensland. HARRISON, S. R. & FXCK,G. W. (1978). Computer Package for Validation of Agricultural Systems Models, Lincoln College. HARRISON, S. R. & LONGWORTH, J. W. (1977). Optimal growth strategies for pastoral firms in the Queensland Brigalow scheme, Australian J. Agric. Economics, 21(2), 80-96. HILDRETH, R. J. (1959). Bunchiness of precipitation data and certain analytical implications, Methodology Workshop, GP-2 Technical Committee, Lincoln, Nebraska. KINGSTON, G. (1976). Rainfall probabilities. agricultural applications, Forty-Third Conference, Queensland Society of Sugar Cane Technologists, Cairns, pp. 45- 56. KLEMES, V. (1974). The Hurst phenomenon: A puzzle? Water Resource,; Research, 10(4), 675 88. LAMB, H. H. (1973). Climatic change and foresight in agriculture: the possibilities of long-term weather advice, Outlook on Agriculture (ICI), 7(5), 203 10. LELP~R, G. W. (1953). A note on the alleged run of dry and wet years. J. Australian hist. Agric. Sci., 19( 1). 48- 50. LONGWORTH, J. W. (1969). The central tablelands/arm management game, unpublished Ph.D. thesis, University of Sydney. MAHER, J. V. (1967). Drought assessment by statistical analysis q] rainJall, Report to the ANZAAS Symposium on Drought, Melbourne. M CMAHON. T. A. (1977). Some statistical characteristics of annual streamflows of northern Australia, Hydrology symposium 1977; The l~vdrology qf Northern Australia, pp. 131 135, Institute of Engineers, Brisbane. O'MAHONEY, G. (1961). Time series anaO'sis q/ some Australian rainJa/l data, Meteorological Study No. 14, Bureau of Meteorology, Melbourne. PATIISt)N, A. (1964). Synthesis ~?/ rain[all data, Civil Engineering Technical Report No. 40, Stanford University. PHILIIPS. J. B. (1969). RainJall distributions and simulation models, ANZAAS Congress, Section 8, Adelaide. PHILt,IPS, J. B. (1971). Statistical methods in systems analysis. In Systems analysis in agricultural management, Wiley, Sydney. ROBINSON, 1. B. and MAWSON,W. F. 11975). Rainfall probabilities, Queensland Agric. J., Mar. Apr., pp. 3 22. RUTHERFORD,J. (1950). The measurement of climatic risk in the western division of New South Wales. Review q/" Marketing and Agricultural Economics, 18(1), 246- 265. SRIKANTHAN, R. & MCMAnoN, T. A. (1978a). A review of lag-one Markov models for generation of annual flows, J. Hydrology, 37, 1 2. SRIKANTHAN,R. & MCMAHON,T. A. (1978h). Comparison of fast fractional gaussian noise and broken line models for generating annual flows, J. 14vdrology, 38, 81 92. THHt., H. (1967). Economics and Information Theory, North-Holland, Amsterdam. TRE/3ECK, D. B. (1971). Spatial diversi[ication of beef producers in the Clarence river valley, unpublished M.Ec. thesis, University of New England. TROUP, A. T. (1965). Southern Oscillation, Quarter O' J. Royal Meteorological Sot., 91,490~506. VERHAGEN,A. M. W. & HIRST. F. (1961). Waiting timcs Jbr drought relieJin Queensland, Division of Mathematical Statistics Technical Paper No. 9, CSIRO, Melbourne. WAITE, P. J. (1967). Rainfall probability forecasts: what do they mean'?, Iowa Farm Science, 22(4), 68 9. WRIGHT. A. (1970). Systems anah'sis and gra'-ing systems: management-oriented simulation, Farm Management Bulletin No. 4. ],Jniversity of New England.