Design information for rainfall networks in the arid region of South Australia

Transcription

1 The hydrology of areas of low precipitation L'hydrologie des regions à faibles précipitations (Proceedings of the Canberra Symposium, December 979; Actes du Colloque de Canberra, décembre 979): IAHS-AISH Publ. no. 28. Design information for rainfall networks in the arid region of South Australia KON CHIN TAI and TREVOR JACOBS Adelaide, South Australia Abstract. Background information is required for the effective design of rainfall networks in the arid region of South Australia. A line of 2 rainfall stations was selected at random along the Transcontinental Railway below the 200 mm isohyet for the purpose of sampling information. For each station, 56 years of record were used. The sampled stations reveal slight heterogeneity of hydrological behaviour. Statistical tests show the slight heterogeneity not to be significant, and the hypothesis that the region is approximately homogeneous is accepted. Furthermore, a contingency statistical test shows the spatial correlation structure to be independent of the temporal correlation structure. The effective number of stations is affected more by the spatial correlation structure than the temporal correlation structure. Estimation of the long-term regional mean rainfall is affected more by the length of the historical records than by the absolute number of rainfall stations. Informations sur l'organisation d'un réseau pluviométrique dans la région aride du Sud de l'australie Résumé. Des informations préliminaires sont nécessaires pour l'organisation efficace d'un réseau pluviométrique dans la région aride du Sud de l'australie. Une ligne de 2 stations pluviométriques a été choisie au hasard le long de la ligne de chemin.de fer transcontinentale sous l'isohyète 200 mm dans le but de recueillir un échantillon de l'information disponible. On a utilisé pour chaque station des relevés portant sur 56 ans. Les stations ainsi choisies ont mis en évidence une légère hétérogénéité de comportement. Des tests statistiques ont montré que cette légère hétérogénéité n'était pas significative et on a admis l'hypothèse que toute la région était à peu près homogène. Ultérieurement un test de contingence statistique a montré que la structure de corrélation spatiale était indépendante de la structure de la corrélation temporelle. Le nombre effectif de stations intéressées est affectée davantage par la structure de corrélation spatiale que par le structure de corrélation temporelle. L'estimation de la moyenne régionale à long terme des précipitations est affectée plus par la longueur des relevés historiques que le nombre absolu de stations pluviométriques. INTRODUCTION Arid zone hydrology may receive greater attention in Australia as the need for mineral resources and alternative forms of energy increases throughout the world. In South Australia, attention is being drawn to a need for establishing new rainfall and streamgauging stations in arid regions (Fig. ). In order to gain useful information on which future design of networks can be based, 2 rainfall stations were selected at random along the Transcontinental Railway as shown in Fig. 2. The stations stretch over a distance of approximately 960 km. The choice of the line is significant, because it is in the general direction of the 200 mm isohyet, and is also in the general direction of the frontal activity of the westerlies. Details of the stations and their annual rainfalls are given in Tables and 2. Statistical significance tests carried out on the sample data reveal the acceptance of the following hypotheses: () hydrological homogeneity across the arid region; and (2) independence of the spatial correlation function from the temporal correlation function. 205

2 206 Kon Chin Tai and Trevor Jacobs " iso FIGURE. The arid region of Australia denoted as AZ. With the acceptance of the above hypotheses, analysis of the data shows that: () the number of years the sampled station is in operation is the most significant of all the parameters in reducing the regional variance in the estimate of the long-term areal mean rainfall, (2) the spatial correlation structure is less significant than the length of historical record in reducing the regional variance; however, the effective number of stations remains sensitive to the spatial correlation function, (3) temporal correlation is the least significant of all the three parameters. BACKGROUND THEORY As an exploratory study, the random variable of annual rainfall is selected, although there is a collection of rainfall data available on a monthly basis. Some study of the rainfall process on a monthly basis for the southeast of South Australia has been done by Cornish etal. (96), and Stenhouse and Cornish (958). Hitherto, two basic assumptions have facilitated the analysis and synthesis of rainfall network design; the assumptions of homogeneity and independence of time and spatial correlation function. In particular, Rodriguez-Iturbe and Mejia (974) have shown the viability of these concepts in relation to the design of a rainfall network in Venezuela. The first assumption of homogeneity states that: () the hydrological statistical properties are the same throughout the region considered;

3 Design information for rainfall networks 207 FIGURE 2. Isohyets of average annual rainfall, South Australia. (2) there is an identical rainfall field,/(pc,-, t) for each storm event, (3) the rainfall time series are stationary in the weak sense; i.e. the series are in statistical equilibrium with the point mean and the point variance the same everywhere in the region, (4) the temporal and spatial correlation structure of the rainfall process is homogeneous, (5) if temporal correlation shows a weak memory, it could best be represented by a Markovian model with a uniform value of the first order autocorrelation coefficient, and (6) the process of decay in the spatial correlation function is assumed to be isotropic. The assumptions above are restrictive, because they apply either to a single storm event or to a small, physically homogenous area, free from any orographic effects.

4 208 Kon Chin Tai and Trevor Jacobs TABLE. General physical information of rainfall stations No. Station name Latitude Longitude Elevation [m] Hughes Cook Oldea Im marna Barton Wynbring Tarcoola Kingoonya Coondambo Wirraminna Wirrappa Hesso 30 43' S 30 37' S 30 27' S - _ 30 33' S 30 43' S - 3 4'S 3 25'S 32 8'S 29 3'E 30 25'E 3 50'E 'E 34 34'E 'E 'E 37 27'E TABLE 2. General statistical properties of annual rainfalls at stations in the arid region of South Australia No. Station name Mean [mm] Standard deviation [mm] Skewness coeff. Kurtosis First order autocorrelation coefficient Hughes Cook Oldea Immarna Barton Wynbring Tarcoola Kingoonya Coondambo Wirraminna Wirrappa Hesso The second assumption deals with stationarity in the covariance structure. Yevjevich (972) calls it stationarity in the wider sense or second-order stationarity. The important aspect of this assumption is that of independence between spatial correlation and temporal correlation structure in the covariance of the rainfall field. Rodriguez-Iturbe and Mejia (974) have assumed the following: cov[/(*,-, 0,/K?')] = â l v-r{x i -x, i).r*{t-t') () where o\ is the point variance of the rainfall field f(x iy t); r(x t x'i) is the spatial correlation structure ; and r*(t t') is the temporal correlation structure. The latter is found to be weak in terms of the years of sampled record, and can be approximated by a simple Markovian model, / *(f-f') = P lf '~" (2) where p is the first autocorrelation coefficient. In the South Australian context, the spatial correlation structure decays as a function of inter-station distance. Using an exponential decay model the best fit is obtained if -s/f is used instead of V, the distance between two points (Figs. 3 and 4). The exponential decay model is, r(vf) = exp(-/*vf) (3) where h is the exponential decay factor.

5 Design information for rainfall networks FIGURE DISTANCE ( Km) Spatial correlation versus inter-station distance. o.i FIGURE /T Kilometres Spatial correlation versus square root of distance. The aforementioned investigators have shown that in estimating the long-term areal mean rainfall as the regional variance is shown to be (4) var [X] 2AT2 n 2 N l t= l" = f(xi,t) (5)

6 20 Kon Chin Tai and Trevor Jacobs It is further shown that the regional variance is a function of: () the correlation structure of the rainfall process in both space and time (equations (2) and (3)), (2) the number of stations in the network n, (3) the sampling geometry of the network, and (4) the length of time TV the stations are in operation. It is to be noted that/(x,, t) is now regarded as the annual rainfall field, which is a composition of many storms. With the assumption of independence between the spatial and temporal correlation structures, the aforementioned investigators have deduced that the regional variance is var[a r ]=ff [/5' (JV)].[i!' 2 (»)] where a p is the point variance of the station, F^N) and F 2 (n) are the variance reduction factors due to sampling in time and in space. i-pn-l I r P - ^ F (N)=\N + 2 -p. (6) (7) and (8) It is also further shown that, for variance reduction due to random sampling of stations in space, or F 2 (n) W(o) + 2l^i^+^-*;) (9) F 2 (n) = -zln + n(n - I) e r(xi-x'i) A where e represents an expected value and e[r(x t x'i)/a] is the expected value of the correlation between two points randomly located on area A. Hxf-x-y f In R r(y/v)-f(sjv)d(^/v) where r(y/v) represents the spatial correlation which is isotropic; and/(\/f) is the frequency function of the square root of the distance -N/F between two randomly chosen points in the area A. A is equal to b\fr, where b is the average width of the region, and y/r is the square root of the largest distance in A. On this basis, the authors have deduced that r{x t - X-)' f Jo where Z = y/v; Z R = ^JR and/(z) = /Z. On integration, the integral on the right-hand side of equation (2) becomes r(x { - xl)l A J hz (0) (H) -hz âz (2) [l - e~ hz ] hyfe _ Q-H-JR (3)

7 RESULTS The results of the analysis are summarized below. Design information for rainfall networks 2 Homogeneous assumptions The long-term regional mean, X r = 88 mm of rainfall is computed from sampled data. The long-term regional variance is var [X r ] = a^-f^n) -F 2 (n), but the point variance CT P estimated from the sample is 0 4 mm 2. Based onn= 56 years and the first autocorrelation coefficient p = 0.0, the variance reduction factor due to sampling in time isf (A0_= Based on n = 2 stations, and^/z 2 = (930)( ) =.5, where A = b\fr with b the average width of the region in kilometres, a reference to Rodriguez- Iturbe and Mejia's study (974) shows that the variance reduction factor due to random sampling in space is F 2 (n) = Hence the computed long-term regional variance var [X r ] = 0 4 (0.02)(0.60) = 26 mm 2 of rainfall, or a long-term standard deviation of mm of rainfall. It is interesting to note that almost the same value of F 2 («) = 0.6 is obtained, using h = 0.04; v^= 3, and e[r(x t -x-)/a] = 0.57 using equation (3). Heterogeneous assumptions The expressions for the long-term mean and the long-term variance are still the same as the homogeneous case. Again, the values of the regional mean and regional variance are the same as values estimated for the homogeneous case. This is because only one set of sampled data is available, with statistical significance tests accepting the hypothesis that the sampled statistics belong to the same homogeneous population statistics. In this event, the variance reduction factor due to sampling in time F^N) has a value of 8 for p = 0.00; 0.02 for p = 0.0, and for p = For «=2 stations sampled, and with the data of inter-station correlation coefficients from Table 3, the variance reduction factor due to random sampling in space is F 2 (n) = The long-term regional variance for 0.00 <p <0.20 is 3 mm 2 < var X r < 75 mm 2. Hence the regional long-term standard deviation of the sample is mm < a x < 3 mm. Sensitivity level and significance level The sensitivity of F^N) to the length of historical record is very much greater than that of F 2 (n). The length of time that an historical record is kept is far more significant than the number of stations (see Fig. 5). TABLE 3. Matrix of cross-correlation coefficients of station annual rainfalls Stations The matrix is presented above in symmetric mode.

8 22 Kon Chin Tai and Trevor Jacobs F2(n),F,(N) N. ' ~ ~- _ r^tr^ "*-* 5i2L29 in _n 0.50 ^w^f, (N) against N with p* n, N FIGURE 5. Variance reduction factors: F t (N) versusn; and F^ri) versus n. The statistical Student 'f test has shown that the hypothesis that the sampled statistics belong to the same population statistics could be accepted at the five per cent level of significance. Autocorrelation analysis of the rainfall time series has shown that the first order autocorrelation coefficient is small, varying between 0 and 0.27 as shown in Table 2. The exception is Hughes station which has a value of This conclusion is reinforced by results of spectral analysis. Statistical testing using a contingency table and a x 2 test at the five per cent level of significance shows that the hypothesis for independence between spatial and temporal correlation functions could be accepted. Effective number of rainfall stations The concept of equivalence of sample size has been given by Yevjevich (972). If «stations in a region are mutually correlated in space, the pulling together of n sets of TV observations into a single size of nn can yield 'only as much information as some lesser number n e N, where n e < n. n e is the effective number of stations in the region, + r(/j - ) (4) where the regional mean of spatial dependence is f. -7i r " Z/=i Lji=j + i 'il I n(n ) (5)

9 Design information for rainfall networks 23 As n tends towards infinity, the effective number of stations tends towards the inverse of the regional mean, lim n e = =- (6)»-*<*> l/n+r(l I/n) r The importance of n e is as follows: () it is highly sensitive to r but less sensitive to n; (2) n e is the effective number of mutually uncorrelated «e series at n e points; (3) as much information about the long-term regional mean is obtained from a sequence of n e N observations as nn, where the time series at n e stations are not serially correlated in time. In the South Australian context, n e =.38 for n = 2 stations with f = 0.7 sampled from the serial correlation matrix, Table 3. The value of n e is between one and two stations in most cases studied. Ëagleson (967) has pointed out that very little information is gained in utilizing more than two properly located stations for estimating long-term mean rainfall for strictly homogeneous areas. Effective number of years of record If the n series of annual rainfall are time dependent but not periodic, then the effective length of sample record is (Yevjevich, 972): N 56 N e = ^ ^ = = 5.85 (7) l + 2(r ri) + 2(0.04) r x and r\ are the lag-one serial correlation coefficients for two rainfall stations and r x r\ is the average product (f^ri) between all stations. The total effective sample size is therefore n e N e = (.38)(5.85) = 72, instead of nn = (2)(56) = 672. The order of reduction is about 90 per cent. DISCUSSION For the particular zone of South Australia, the random variable of annual rainfall is a composition of many storms with random distribution in sizes and location in the region. But the individual storms are affected more by frontal activity rather than by convergence and convection (Cornish et al, 96). It is the persistence of the frontal activity that is relevant. It is not expected that complete homogeneity of hydrological behaviour could be attained because station locations are not in complete alignment with the principal axis of maximal correlation. The exact orientation of the axis is not known but is expected to be aligned with the general direction of the frontal activity of the westerlies. Two trade-offs are established: () a trade-off between the variance reduction factors, F t (N) and F 2 («), (2) a trade-off between the total sample size (nn) and (n e N e ), the effective sample size. In respect of the first trade-off, it is not a singular unique trade-off as is the second one. CONCLUSIONS From the time series analysis of 2 rainfall stations, it is concluded that the accuracy of measuring the long-term areal mean is affected more by the number of years the

10 24 Kon Chin Tai and Trevor Jacobs rainfall stations are in operation rather than by the absolute number of stations. The length of the historical record is more significant than the first order autocorrelation coefficient for the sampling of the annual rainfall variable in time, especially if the first order autocorrelation coefficient is very small. Statistical interpretation of autocorrelation and cross-correlation results of annual rainfall time series indicate that the assumption of independence between temporal correlation structure and spatial correlation structure is acceptable for the arid region of South Australia. The effective number of rainfall stations n e, is determined inversely by the regional mean of spatial correlation F, as the number of stations n increases to infinity. For a homogeneous hydrological region, it is found that n e does not generally exceed 2, because of the robustness of the regional mean of spatial dependence. Slight heterogeneity of hydrological behaviour has been detected in the arid region of South Australia. But the deviation from homogeneity is not statistically significant to reject the hypothesis that all the sampled values of rainfall statistics belong to the same population. The use of the square root of the inter-station distance is useful in retaining the ability to analyse the problem as an exponential decay of spatial dependence. This simplification allows for the comparison with a totally homogeneous and fully isotropic situation, which is useful as a basis for mathematical formulation. REFERENCES Cornish, E. A., Hill, G. W. and Evans, M. J. (96) Interstation correlations of rainfall in southern Australia. Division of Mathematical Statistics Technical Paper no. 0, Commonwealth Scientific and Industrial Research Organization, Australia, Melbourne. Eagleson, P. S. (967) Optimum density of rainfall network. Wat. Resour. Res. 3, no. 4, 02-, Rodriguez-Iturbe, I. and Mejia, J. M. (974) The design of rainfall networks in time and space. Wat. Resour. Res. 0, no. 4, Stenhouse, N. S. and Cornish, E. A. (958) Interstation correlation of monthly rainfall in South Australia. Division of Mathematical Statistics Technical Paper no. 5, Commonwealth Scientific and Industrial Research Organization, Australia, Melbourne. Yevjevich, V. (972) Probability and Statistics in Hydrology: Water Resources Publications, Fort Collins, Colorado, USA.