Spatial disaggregation of rainfall datahydrologie.org/hsj/430/hysj_43_01_0091.pdf · 2015. 1....

Hydrological Sciences—Journal—des Sciences Hydrologiques, 43(1) February 1998 91

Spatial disaggregation of rainfall data

R. MEHROTRA & R. D. SINGH National Institute of Hydrology, Jalvigyan Bhavan, Roorkee 247667, India

Abstract Use of output from global Circulation Models (GCMs) by regional or small scale rainfall-runoff models necessitates the disaggregation of the hydrological information available from GCMs to smaller scales. The hydrological processes of interest commonly occur at much smaller scales than those being modelled by GCMs. The present work examines the disaggregation of areally averaged monthly rainfall values of a basin or a region into point rainfall values. It uses some statistical methods based on a frequency analysis approach, a correlation approach and a disaggregation approach. A total of ten different methods have been tried and their relative performances compared based on some error criteria evaluated from observed and disaggregated point rainfall and mean areal rainfall values and their statistics. The results show the superiority of methods based on disaggregation techniques over other methods. The methods presented and discussed in the paper may very well be applied to disaggregate mean areal rainfall values into point rainfall values and also for infilling missing rainfall records.

Désagrégation spatiale de données de précipitations Résumé L'utilisation à l'échelle locale ou régionale, dans des modèles pluies débits, des résultats de modèles de circulation générale, nécessite la désagrégation de l'information hydrologique de ces modèles à une échelle plus fine. Les phénomènes hydrologiques les plus intéressants se déroulent en effet à une échelle beaucoup plus petite que celle des phénomènes modélisés par les modèles de circulation générale. Ce papier s'intéresse à la désagrégation de valeurs mensuelles de précipitations moyennées sur un bassin ou une région en précipitations ponctuelles. Il utilise certaines méthodes statistiques fondées sur l'analyse fréquentielle, l'analyse des corrélations et la désagrégation. Un total de dix méthodes différentes ont été expérimentées et leurs performances respectives ont été comparées sur la base de critères d'erreur calculés à partir des données observées et des données obtenues par désagrégation ainsi que de leurs statistiques. Les résultats montrent la supériorité des méthodes fondées sur la désagrégation. Les méthodes présentées et discutées ici peuvent parfaitement être appliquées pour désagréger des valeurs surfaciques moyennes en valeurs ponctuelles comme pour reconstituer des valeurs manquantes dans des enregistrements pluviométriques.

INTRODUCTION

The implications of global climate change on water resources have been studied by many workers in the past two decades. Whether these changes can be explained by natural variation or are a result of greenhouse warming is still debated. Currently, most GCMs operate globally with a resolution varying from 2° x 2° to 10° x 10° latitude/longitude. Thus the resulting climatic prediction from GCMs cannot be directly used for developing localized responses to climate change impacts at basin/catchment scale using hydrological models. The hydrological processes of interest commonly occur at scales in the order of tens of thousands of square

Open for discussion until 1 August 1998

92 R. Mehrotra & R. D. Singh

kilometres. Also, for a better understanding of the physics of various related processes involved, the variation in hydrological parameters at local or regional scale needs to be examined (Moss & Tasker, 1987; Riebsame, 1988; Hay et al, 1992; Barros & Lettenmaier, 1993; Bardôssy & Plate, 1992). Some of those authors discussed scale issues concerning the use of GCM output for climate change impact studies at local and/or regional scale. Hay et al. (1992) proposed the use of a stochastic disaggregation model for such studies. The coupling of GCM output to mesoscale rainfall models and deterministic rainfall-runoff models was studied by Ayers & Leavesley (1988). As has been also pointed out by many researchers (Gleick, 1987; Wagonner, 1990; Epstein & Ramirez, 1994), studies related to climate change, based on the results of GCMs, do require atmospheric response at local scale. The present work aims at building this link between the large scale change and regional and/or local change.

METHODOLOGY

All the approaches used herein need point rainfall values located in and around a basin under study and mean areal rainfall values for the basin. Several methods are available in the literature to estimate the areal mean from point rainfall values but no method can provide more than an estimate of the true value. Studies done by many workers on comparisons of some of these methods (Singh & Birsoy, 1975a,b; Singh & Chowdhury, 1986; Mandeville & Rodda, 1970; Allerup et al, 1982) indicate that for larger time intervals i.e. monthly and yearly, the estimates by different methods are within 10% of one another. For the present study, the Thiessen polygon method, being the most common and extensively used method for the estimation of mean areal rainfall was adopted. Various approaches used for the disaggregation of mean areal rainfall to point rainfall values are described hereunder.

Frequency analysis approach

Frequency analysis is the most common and extensively used statistical approach to deal with the frequency of occurrence of a hydrological variable. The present study considers three distributions, i.e. normal, extreme value type 1 (EV1) and Pearson type III (PT3). The Box-Cox power transformation technique for normalising a series was also used in the analysis.

For each distribution, reduced variâtes corresponding to values of rainfall at a point (station) were computed using the recommended plotting position formula for that distribution. A simple linear regression equation, Yt = M, Z, + C, was fitted between the variâtes (monthly rainfall) (Yt) and the corresponding reduced variâtes (Zt) using the least square approach and values of the parameters M, and C, were calculated for station t. The same procedure was repeated for all stations and also for mean areal values. Thus if NS stations are considered, then NS + 1 relationships

Spatial disaggregation of rainfall data 93

had to be developed and NS + 1 sets of values of the parameters M, and C, computed. The relationships could then be used to disaggregate the mean areal rainfall values (YNS+l) into point values, as follows.

From the established relationship between mean areal values and their reduced variâtes, YNS+l = Mm+lZNS+l + Cm+l, the value of the reduced variate (ZNS+i) corresponding to a mean areal value (YNS+l) is obtained by substituting the values of parameters Mm+l and CNS+l in the above equation. Corresponding to this ZNS+1, the rainfall value (Y) at this station is calculated from the developed relationship of station t by using the equation Y, = M,ZNS+l + Ct. Here, ZNS+l is the value of the reduced variate which will not change with station and M, and C, are the parameters for station t, already computed.

In the case of a power transformation, the Box-Cox transformation technique was used. Mathematically, the Box-Cox transformation can be represented as:

Yt = ( 7 / - l) /aifa + 0 (1)

Y, = ln(F,)ifa = 0 (2)

Here, Y, is the monthly rainfall value at a station t and value of a is selected in a manner such that the transformed series is normalized and that means that the coefficient of skewness of the transformed series is close to zero.

Disaggregation approach

Here the disaggregation technique used was as primarily proposed by Valencia & Schaake (1973) for the modelling of hydrological time series in temporal as well as in spatial domain.

The general form of the disaggregation model was:

Y = AX + Be (3)

For the present analysis, X was the mean areal rainfall and Y a column matrix (NS x 1) containing corresponding (NS) point rainfall values at the NS stations considered in the analysis. The X and F values were transformed values obtained by subtracting the observed mean values for each series from the series values. A(NS x 1) and B(NS x NS) are the parameter matrices and e is a (NS x 1) matrix of independent identically distributed normal deviates. Parameter estimation of equation (3) was done by using the method of moments as follows (Salas et al., 1980):

A = Syx- Sxxl (4)

BBT = Syy - SyxS^Sxy (5)

where SYY is the matrix of co-variances among the station values, 5 K is the matrix of co-variances between the station values and the mean areal rainfall value and SH is the matrix of co-variances among the mean areal rainfall values. Lane (1979) gives a solution technique to solve BBT for obtaining B.

From the above parameter estimation, it is clear that this approach considers


rainfall series at all the stations and mean areal rainfall series simultaneously, and thus preserves the correlations among all the station values and also between each station value and mean areal value. In order to preserve these correlations, the parameters required to be preserved are also large in number which make the matrix notation necessary. However, these matrices can be solved easily with the help of computer routines.

In this approach also, similar to the previous approach, the first parameter matrices A and B were computed using the observed rainfall series at different stations and the mean areal values series. Now a given mean areal value can be disaggregated using equation (3) to get corresponding station values.

The relationship Y = AX + Be contains a deterministic component AX and a stochastic component Be. In the analysis, as a possible alternative, the equation Y = AX, considering only the deterministic component was also considered. Thus under this approach, four possible alternatives were tried: (a) considering a single relationship for all twelve months; (b) twelve different relationships for each month; (c) matrix A only i.e. the deterministic component and a single relationship for all twelve months; and (d) matrix A only and twelve different relationships for all the months.

Correlation approach

In this approach, for each station the relationship between the station values and mean areal values was established. Thus if there were NS stations under consideration then NS relationships needed to be developed. A linear relationship was assumed to hold good for the analysis. Using these relationships, a given mean areal rainfall could easily be disaggregated to get the station values.

In this analysis two different alternatives were tried: considering (a) data that followed normal distribution; and (b) the data were normalized after power transformation. In all, ten methods were used in the study. They are listed and described in brief in Table 1.

Table 1 Description of methods used in the study.

Method no. Description

1 Disaggregation approach, considering one matrix A and different relationships for all twelve months.

2 Disaggregation approach, considering one matrix A and one relationship equally good for all twelve months.

3 Disaggregation approach, considering both matrices A and B and different relationships for all twelve months.

4 Disaggregation approach, considering both matrices A and B and one relationship for all twelve months.

5 Normal distribution frequency analysis approach. 6 Power transformation frequency analysis approach. 7 PT3 distribution frequency analysis approach. 8 EV1 distribution frequency analysis approach. 9 Normal distribution regression approach. 10 Power transformation regression approach.


APPLICATION

The study area

The catchments studied were the Damanganga and Sher basins (Table 2). The required input data for these basins was readily available. One of the constraints may be the availability of a sufficient length of data, as the complete data required is available for a limited period only. The index maps of these basins showing the locations of gauge discharge site and rain gauge stations are given in Fig. 1.

Pvj "V f . ^ W JLVAPIWEAR

,s

f s S x V • • • ' * • . GUJARAT

it.- SO DUDtjNK^ Y ':_ JKHANVEtv'^

20° V

JAWHARQ*.

/ 0 " , VADOLI

.*

k

^MAHARASHTRA

H A R S U U /

MOK^EDA J TRWBAK

Damanganga basin

Sher basin

LEGEND © RAIN GAUGE STATSON

= ROADS —<£ RIVER

DISTT. BOUNDARY A GAUGING SITE

BASIN BOUNDARY - - - - - STATE BOUNDARY

Fig. 1 Index maps of Sher and Damanganga basins.


Table 2 General description of study area.

Basin name Damanganga basin Sher basin

Latitude Longitude Elevation Normal annual rainfall Normal annual runoff Data availability Area Region Location

19°45'-20°20' 72°40'-73°40' 950-0 m 2212 mm 1427 mm 1974-4983 2261 km2

Humid Central India

22°25'-23°55' 79°15'-79°40' 1110-450 m 1255 mm 502 mm 1978-1987 1500 km2

Dry sub-humid Central India

For these basins, more than 90% of the total annual rainfall falls during the southwest monsoon season (June-September).

Data availability

Stier basin The monthly rainfall data at the recording stations at Lakhandon, Harai and Mungwani were used. The Thiessen weights for these stations were 0.35, 0.25 and 0.40 respectively.

Damanganga basin Monthly rainfall data for 12 stations located in and around the basin were used. The individual station names along with their Thiessen weights were as in Table 3.

Table 3 Station name and Thiessen weights—Damanganga basin.

Station

Peint Mokhada Silvassa Harsul Rakholi Vadoli

Thiessen

0.1083 0.1216 0.0337 0.1829 0.0438 0.1022

weight Station

Vapi Khanval Dudhani Raima! Jawahar Trimbak

Thiessen weight

0.0135 0.1070 0.1670 0.0362 0.0640 0.0135

Application of methodology to Sher and Damanganga basins

Monthly rainfall data at various rain gauge stations for both basins were available for ten years. In order to evaluate the performances of the various methods two different approaches were adopted. In the first, data for the first seven years were used for the calibration of the methodologies and, based upon the calibrated parameters, the remaining three years of data were used to test the performances of the methodologies. In the second approach, all ten years of data were used for the


development and testing of the methodologies. It may be noted here that, if a station was used for the calculation of mean areal

value, then the disaggregation for that station would imply an a priori relationship between the point rainfall and the mean areal rainfall value. To avoid this, in the analysis first a given station was left out and the mean areal value computed based on rainfall values at only the other stations. As this station was now excluded, the Thiessen weights were also revised. All the different methods and the two approaches were applied using this revised mean areal value and the disaggregated rainfall values at the given station were obtained. A similar procedure was repeated for the other stations and corresponding disaggregated station rainfall values were calculated.

The following calculations were performed: 1. Statistical properties, viz. mean (MEAN), standard deviation (SDV), coefficient

of variation (CV) and coefficient of skewness (SKEW) were calculated for each station along with mean areal values, for observed records for the calibration and validation periods and for the complete periods.

2. The calculation of all these statistics for disaggregated station values obtained by excluding the stations one by one, and for mean areal values computed using true Thiessen weights (i.e. weights obtained by considering all the stations) and these disaggregated station values. These mean areal values were compared with the mean areal values computed using the observed records at all the stations and the true Thiessen weights. The observed statistical properties of all ten years of record for both basins are

given in Table 4. All the ten methods as explained in the previous sections were applied to both

the Sher and Damanganga basins using the approaches.

Table 4 Observed statistical properties of Sher and Damanganga basins on monthly scale.

Statistical

MEAN SDV CV SKEW

properties Damanganga basin

169.50 286.80

1.70 2.10

Sher basin

94.90 112.80

1.20 1.70

Performance evaluation criteria

The performances of all the methods was evaluated on the basis of reproduction of the statistics at the stations and for mean areal values. It is to be noted that, owing to the large number of stations and methods involved in the analysis, it would be difficult to arrive at a conclusion by comparing all the statistical parameters or rainfall values for all the stations. For a method to be consistent, it should perform well simultaneously at all the stations and also for mean areal values. To account for this, two criteria were used: total absolute error (TAE) (which accounted for the

98 R. Mehrotra & R, D. Singh

performances at all the stations and also for mean areal rainfall for statistics) and average absolute error (AAE) (which take into consideration the deviation of mean rainfall values, computed from disaggregated station values from observed mean values).

Total absolute error (TAE) At each station, the absolute difference between each of the statistical parameters of the observed records and those from the disaggregated records were computed. The weighted average of the absolute difference at each station was computed by multiplying it by the weight of that station. The absolute difference between the parameters of the observed and disaggregated mean areal values was added to get the TAE value.

Mathematically, the criterion may be expressed as:

7M£ = 2 ABS^STH) - STC^-W, + ABS(SAH- SAC) (6)

where STHj and STCj represent the values of a statistical parameter at the yth station computed from the observed and disaggregated records respectively, SAH and SAC are the values of the parameter computed from mean areal values of the observed and disaggregated records respectively, Wj is the weight associated with the jth station, and NS is the number of stations considered.

Average absolute error In the mean areal rainfall values (AAE) The sum of the absolute differences between the average rainfall values computed from the observed records and the disaggregated records gives AAE.

Mathematically, the criterion may be expressed as: N ,

:Y,ABSy ;=1

AAE = X ABS PHi - PC, (7)

where PHt and PCt represent mean areal values for the ith month computed from the observed and disaggregated records respectively, and N is the total number of observations.

ANALYSIS AND RESULTS

The TAE values computed for the calibration and validation periods and the complete periods of records, using the different methods, is given in Table 5 for the Sher basin and in Table 6 for the Damanganga basin. The AAE values for these periods are given in Table 7 for both basins. Further details of the statistics at all the stations obtained with all the methods are given elsewhere (Mehrotra, 1995).

Table 5 indicates that on the basis of TAE, for the Sher basin method 7 performed well for both calibration and complete periods and method 1 performed well for the validation period. However, for the Damanganga basin, method 1 performed well for calibration, validation and complete periods (Table 6). These


Table 5 Total absolute error (TAE) in statistics for Sher basin.

Statistics Method no. 1

Calibration period: MEAN SDV

cv SKEW

Validation MEAN SDV CV SKEW

2.40 4.54 0.00 0.13

period: 3.64

15.96 0.13 0.39

Complete period: MEAN SDV CV SKEW

1.18 2.62 0.06 0.03

2

2.32 3.48 0.02 0.12

4.43 19.08 0.12 0.41

2.69 1.61 0.07 0.03

3

6.81 7.32 0.02 0.13

7.56 16.57 0.14 0.36

10.74 8.83 0.09 0.10

4

8.14 4.03 0.09 0.18

11.93 20.95 0.15 0.37

10.20 4.31 0.16 0.09

5

0.18 0.75 0.01 0.14

4.79 18.83 0.12 0.41

0.28 1.40 0.01 0.01

6

0.10 0.58 0.00 0.14

4.77 18.83 0.12 0.41

0.29 1.49 0.01 0.01

7

0.07 0.33 0.00 0.11

4.67 19.43 0.16 0.42

0.25 0.51 0.00 0.00

8

102.78 934.47

4.08 7.34

4.67 20.25 0.16 0.42

4.11 1.90 0.08 0.47

9

1.50 6.51 0.03 0.24

4.42 19.09 0.12 0.41

3.63 5.43 0.04 0.14

10

110.75 1004.01

1.83 2.37

17.69 15.95 0.30 0.49

8.45 27.54 0.31 1.36

Areal average computed based on disaggregated rainfall values at a station which was excluded during analysis.

Table 6 Total absolute error (TAE) in statistics for Damanganga basin.

Statistics Method 1

Calibration period: MEAN SDV CV SKEW

1.42 12.33 0.09 0.08

Validation period: MEAN SDV CV SKEW

Complete MEAN SDV CV SKEW

102.25 177.17

0.11 0.28

period: 7.50

25.35 0.13 0.45

no. 2

5.29 15.83 0.14 0.27

102.56 176.32

0.09 0.17

8.80 27.65 0.13 0.33

3

19.41 17.90 0.13 0.26

114.30 196.48

0.15 0.47

56.66 68.11 0.14 0.52

4

39.19 10.88 0.38 0.46

118.81 186.73

0.21 0.28

76.14 30.90 0.48 0.65

5

3.58 12.42 0.11 0.21

103.86 182.16

0.15 0.23

11.21 28.03 0.09 0.26

6

1.79 12.36 0.09 0.20

103.66 182.27

0.18 0.22

8.94 28.02 0.10 0.25

7

4.57 10.59 0.11 0.20

103.90 182.82

0.15 0.25

12.44 27.52 0.09 0.27

8

5.74 11.00 0.11 0.23

103.78 184.32

0.16 0.26

14.45 29.11 0.09 0.26

9

5.30 15.85 0.14 0.27

102.59 176.31

0.09 0.17

8.81 27.65 0.13 0.33

10

269.08 466.62

0.70 2.68

276.30 495.80

0.67 2.08

310.56 435.43

1.95 5.32

Areal average computed based on disaggregated rainfall values at a station which was excluded during analysis.

tables also indicate that methods 3, 4, 8 and 10 give higher errors as compared to other methods.

On the basis of the AAE criterion (Table 7), for the Sher basin method 7 gave least error for calibration as well as for complete periods while method 2 gave less


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error for the validation period. Further, for the Damanganga basin methods 9 and 2 performed almost equally well for all the periods.

For different stations, different methods performed well for different statistics and the performance of a method varied from calibration to validation periods. In general, methods 1, 2, 5, 7 and 9 gave comparable results. However, if weight is given for the performance of a method during the validation period, then methods 1 and 2 may be preferred and recommended for disaggregation of mean areal monthly rainfall values into point (station) rainfall values.

CONCLUSIONS

Various techniques to disaggregate the mean areal monthly rainfall for a basin were examined. These techniques may prove a useful tool to deal with resolution incongruities associated with GCMs and local scale rainfall - runoff models, specifically when limited information about a catchment is available, and also for similar other problems. These techniques can also be used for infilling missing records. Based on the results of present study, a disaggregation technique which is a step forward as compared to other methods in terms of preservation of correlations may be preferred.

REFERENCES

Allerup, P., Madson, H. & Rus, J. (1982) Methods for calculating areal precipitation—applied to Susa catchment. Nordic Hydrol. 13, 263-278.

Ayers, M. A. & Leavesley, G. H. (1988) Assessment of the potential effects of climate change on water resources of the Delaware River Basin: work plan for 1986-1990. VSGS Open File Report 88-478.

Bardossy, A. & Plate, E. J. (1992) Space-time model for daily rainfall using atmospheric circulating patterns. Wat. Resour. Res. 28(5), 1247-1259.

Barros, A. P. & Lettenmaier, D. P. (1993) Multiscale aggregation and disaggregation of precipitation for regional hydroclimatological studies. In: Macroscale Modelling of the Hydrosphere (ed. by W. B. Wilkinson) (Proc. Yokohama Symp., July 1993), 183-194. IAHS Publ. no. 214.

Epstein, D. & Ramirez, J. A. (1994) Spatial disaggregation for studies of climatic hydrologie sensitivity. J. Hvdraul. EngngDiv. ASCE 120(12), 1449-1467.

Gleick, P. H. (1987) Regional hydrologie consequences of increases in atmospheric CO, and other trace gases. Climate Change 10, 137-161.

Hay, L. E., McCabe, G. J. Jr, Wolock, D. M. & Ayers, M. A. (1992) Use of weather types to disaggregate general circulation model predictions. J. Geophys. Res. 97(D3), 2781-2790.

Lane, W. L. (1979) Applied stochastic techniques (LAST computer package), user manual. Division of Planning Technical Services, Bureau of Reclamation, Denver, Colorado.

Mandeville. A. M. & Rodda, J. C. (1970) A contribution to the objective assessment of areal rainfall amounts. J. Hydrol. (NZ) 9(2), 281-291.

Mehrotra, R. (1995) Development of disaggregation techniques. NIH Publ. no. TR(BR)126, 1994-1995. Moss, M. E. & Tasker, G. D. (1987) Role of stochastic hydrology in dealing with climatic variability. In: The

Influence of Climate Change and Climatic Variability on the Hydrologie Regime and Water Resources (ed. by S. I. Solomon, M. Beran & W. Hogg) (Proc. Vancouver Symp., August 1987), , 201-207. IAHS Publ. no. 168.

Riebsame, W. E. (1988) Adjusting water resources management to climate change. Climate Change 13(1), 69-97. Salas, J. D., Delleur, J. W., Yevjevich, V. & Lane, W. L. (1980) Modeling of Hydrologie Time Series. Water

Resources Publications, Littleton, Colorado. Singh, V. P. & Birsoy, Y. K. (1975a) Studies on rainfall-runoff modelling: 1. estimation of mean areal rainfall. WRRI

Report 065, New Mexico Water Resources Research Institute, New Mexico State University, Las Cruces, NM. Singh, V. P. & Birsov, Y. K. (1975b) Comparison of the methods of estimating mean areal rainfall. Nordic Hydrol.

6(4), 222-241. Singh, V. P. & Chowdhury, P. K. (1986) Comparing some methods of estimating mean areal rainfall. Wat. Resour.


Bull. 22(2), 275-282. Valencia, D. R. & Schaake, J. C. (1973) Disaggregation processes in stochastic hydrology. Wat. Resour. Res. 9(3),

580-585. Wagonner, P. (1990) Climate Change and US Water Resources. Wiley, New York.

Received 29 May 1996; accepted 22 June 1997

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