temporal disaggregation of monthly rainfall data for water...

The Influence of Climate Change and Climatic Variability on the Hydrologie Regime and Water Resources (Proceedings of the Vancouver Symposium, August 1987). IAHSPubl. no. 168. 1987.

Temporal disaggregation of monthly rainfall data for water balance modelling

Thomas W, Giambelluca Department of Geography University of Hawaii at Manoa Honolulu, Hawaii 96822, USA Delwyn S. Oki Water Resources Research Center University Of Hawaii at Manoa Honolulu, Hawaii 96822, USA

ABSTRACT Water balance computations are often done using a monthly time step because of data or computation time constraints. However, because of the variable intensity of rainfall during a month, évapotranspiration is generally overestimated and groundwater recharge underestimated when balancing is done using a monthly interval. Water balance calculations using hourly, daily, and monthly intervals for a site in Hawaii result in recharge estimates of 40%, 37%, and 19% of precipitation, respectively. For locations where daily rainfall data are unavailable, but for which daily rainfall characteristics may be estimated, a method is developed for simulating sequences of daily rainfall which equate with a known monthly total. Using this method for the Hawaii sample site, soil moisture, évapotranspiration, and recharge estimates were significantly improved over monthly based estimates. However, using the simulated data, recharge estimates were consistently about 23% lower than estimates based on hourly data.

Desaggregation temporelle des données sur la pluie mensuelle afin de modeler le bilan de l'eau

RESUME En calculant le bilan de l'eau, on emploie généralement des intervalles d'un mois à cause des données ou des contraintes du calcul. Mais, à cause de l'intensité variable de la pluie pendant un mois, ordairenment l'évapotranspiration est surestimée et la recharge de l'eau phréatique est sous-estimée quand on calcule le bilan sur une base mensuelle. Pour les besoins de cet article, des calculs du bilan de l'eau, effectués aux intervalles d'une heure, d'un jour et d'un mois dans un site à Hawai ont donné des estimations de recharge de 40%, 37%, et 19% de la précipitation, respectivement. Pour les endroits où les données quotidiennes ne sont pas disponibles, mais où l'on peut estimer caractéristiques de la précipitation journalière, une méthode est développée pour simuler des séquences de précipitation journalière,

255

256 T.W. Giambelluca & D.S. Oki

lesquelles sont égales à un total mensuel connu. En employant cette méthode pour le site d'Hawai, il apparaîl que les estimations du contenu de l'eau dans le sol, de l'évapotranpiration, et de la recharge sont meilleures que les estimations fondées sur les intervalles d'un mois. Cependant, en utilisant les données simulées, les estimations de recharge sont restées à un niveau environ 23% inférieur aux estimations fondées sur un intervalle horaire.

Introduction

Water balances are applied not only to practical water resource problems such as streamflow prediction, groundwater recharge estimation, evaluation of land use effects on the hydrologie cycle, determination of irrigation requirements and crop yield modeling, but also to problems related to climatic variation such as drought assessment (Palmer 1965).

Because of data or computation time constraints, water balances are often carried out using a monthly time step. Several authors (Rushton & Ward 1979; Howard & Lloyd 1979; Alley 1984) have noted that use of a monthly interval tends to result in overestimation of évapotranspiration and underestimation of groundwater recharge. However, the monthly water balance remains popular, perhaps because alternatives are lacking. In this paper, the effect of different time intervals on the results obtained from a simple water balance model are compared for a sample site in Hawaii. Subsequently, a method is suggested for disaggregating monthly rainfall into daily sequences for water balance computations.

Water balance model

The water balance model used in this study is a variant of the Thornthwaite and Mather (1955) bookkeeping procedure. Water fluxes through the plant-soil system are determined for each time interval in the following manner. A state variable X^ is first computed as,

xi = Si-1+ pi - Ri - Ei O )

where S^_^ = ending available soil moisture (difference between total soil moisture content and the wilting point) for previous time interval, P^ = precipitation during time interval i, R^ = surface runoff during time interval i, and E^= évapotranspiration during time interval i. (All variables are expressed as equivalent water depths.) On the basis of X-, groundwater recharge (drainage flux) and end-of-interval soil moisture are determined according to the following drainage rules,

S. = 0 l

Qi = 0 for Xj <. 0 (2)

Ei = Si-1 + Pi - Ri

Water balance modeling 257

Si = Xi

Qi = 0 for 0 < Xi£ $ (3)

Si = $

Qi = Xi - * for Xi> $ (4)

where Si= available soil moisture content at the end of time interval i> Qi= groundwater recharge during time interval i, and $ = available soil moisture capacity.

Direct runoff, under the assumptions of this model occurs as an instantaneous response to rainfall. Runoff may be estimated from streamflow measurements or computed using a rainfall-runoff model such as the Soil Conservation Service (1972) runoff curve number method. Runoff for this study was estimated using a regression-type rainfall-runoff model calibrated with field data measured locally by the U.S. Agricultural Research Service (El-Swaify & Cooley, 1980).

Evapotranspiration is determined as a function of environmental demand, potential évapotranspiration (PEi ), and soil moisture availability during the interval. Unlike the usual Thornthwaite procedure, depression of Ei below PEi is strictly determined by soil moisture availability and no differentiation is made according to whether Pi exceeds PEi. Soil moisture availability at the beginning of an interval (Z^) is determined as,

zi " si-l+ pi - Ri (5)

The instantaneous rate of évapotranspiration (E) is assumed to vary as a function of instantaneous soil moisture (S) according to the following rules:

E = PEi for S K j (6)

E = SC^1 ¥Ei for S < Ci (7)

The quantity Ci, sometimes called the root constant (Penman 1949), may be interpreted as the available soil moisture content below which E is depressed below the potential rate. A model was developed to estimate Ci (Giambelluca 1983) having the form,

C± = min[a + b(R00T) + c(PEi), 1] $ (8)

where a,b,c=calibration coefficients, ROOT = root depth (mm), and where PEi is in units of mm d~ . Data from lysimeter studies by Ekern (1966) were used to calibrate the model for conditions in Hawaii: a= 1.25, b=-1.87xl0~3, and c=5.20xl0~2 for PE < 6 mm d-1; a= 1.41, b= -1.87xl0~3, and c= 2.20xl0~2 for PE > 6 mm d~r.

Based on this model E- is determined as,

Ei = PEiTi + CiU-expI-a-Cl-T.)]} for Z- > C £ (9)

E i = Z.[l - exp(-a.)] for Z . < C . (10)


where T̂ is the fraction of the current time interval during which soil moisture is above C^,

Ti = min[(Zi - Ci)PEi"1, 1] (11)

and where,

<x± = PEiCi - 1 (12)

PE is estimated from pan evaporation measurements (Ekern & Chang 1985) adjusted by a crop factor to represent the surface of interest»

Effect of time interval

The assumptions of the above model differ somewhat from the many Thornthwaite-type water balance models. However, as with all such models, there is a tendency to overestimate Ej_ and underestimate Qj_. This bias results from the failure of the model to account for within-interval variability in soil moisture input. For longer time intervals, the importance of the rainfall variability and the resulting bias increases,

To examine the effect of time interval on soil moisture, évapotranspiration, and recharge estimation, the water balance model was applied to 3 sequences of input data derived from a single set of measurements and differing only in degree of aggregation. Specifically, an 8-year record of measured hourly data from an autographic gage, U.S. Weather Bureau (USWB) Station 0300 (Camp 84, maintained by Del Monte Corporation) in Central Oahu, Hawaii was aggregated into daily and monthly sequences. Daily runoff was computed using the previously mentioned regression model. Hourly and monthly runoff were derived from the daily sequence. Monthly pan evaporation records from a nearby station were used to estimate potential évapotranspiration. Daily potential évapotranspiration was assumed to be uniformly distributed, while the hourly sequence was computed as a function of the seasonally-adjusted diurnal cycle of incident solar radiation.

The intention here is to focus on the effect of the temporal resolution of rainfall. Non-periodic variability in potential évapotranspiration is low and direct runoff ( a small portion of the total water balance in this region) variability is substantially accounted for by the variability of rainfall.

The water balance model was run with each set of input data and with the following parameter settings: $ = 33.0 mm, ROOT = 300 mm, SQ (initial soil moisture content) = 24.75 mm, crop factor =1.00 (PE = 100% pan evaporation).

In Table 1, the 8-year water balance totals are presented. The degree of bias in the monthly-based estimates of évapotranspiration and recharge are evident. Recharge as a percentage of precipitation is 19% for the monthly run, 37% for the daily run, and 40% for the hourly run. Figure 1 shows the monthly recharge time series for each run. Hourly and daily run values are aggregated into 96 monthly totals for comparison. Using the hourly run as a standard for comparison, it is evident that the daily interval accurately


TABLE 1 Eight-year (1966-1973) water balance totals using hourly, daily, and monthly intervals for USWB Station 0300, Oahu, Hawaii

Interval Hourly Daily Monthly

P(mm) 6585 6585 6585

R(mm) 354 354 354

ET(mm) 3582 3807 4968

Q(mm) 2641 2420 1269

Note: P=precipitation, R=runoff, ET=evapotranspiration, Q=groundwater recharge.

Note: Because beginning and ending soil moistures differ, the output terms do not sum to the precipitation.

RECHARGE

300

200

I100

If ° ^ 200 o g 100

5 o S 300

200

1 00

I j a L l JpL^ lipi

JL_L

l l H * P h

H • [ j j _

HOURLY RUN

< 1 H DAILY RUN

H h MONTHLY RUN

24 48

TIME (month)

Figure 1 Monthly recharge time series for hourly, daily, and monthly-interval water balance runs for USWB station 0300, 1966-1973.

estimates the occurrence of recharge events and only slightly underestimates their magnitude. On the other hand, use of the monthly interval results in a failure to recognize many recharge events as well as a consistent and substantial underestimation of the magnitude of the recharge. Daily and monthly (versus hourly) results are shown in Figure 2 in the form of scattergrams. The corresponding coefficients of determination and standard errors of the estimate are listed in Table 2. For soil moisture, end-of-month values are used. Average soil moisture is obtained for each month by averaging end-of-day values from hourly and daily runs. This, of course, was not possible for the monthly-interval run.

The foregoing comparison indicates that use of the monthly interval in water balance computations, at least for the central Oahu sample site, results in unacceptably large error variance and bias when compared with hourly-interval computations. The daily interval, however, produces results which are very close to the hourly calculation, though still slightly biased.

260 T.W. G i a m b e l l u c a & D .S . Oki

DAILY VS HOURLY

End-of-Uonth Soil Moisture (mm)

H 1 H

10 20 30 HOURLY

Average Soil Moisture (mm)

- t 1 h

10 20 30 HOURLY

MONTHLY VS HOURLY

End-of-Month Soi! Moisture (mm)

-H— —f-7

10 20 30 HOURLY'

No within-month averages for monthly run.

100 -

>-—I < o 50 -

0 _]

ET (mm/mo)

/ .

y

1 1 —

50 100 HOURLY

Recharge (mm/mo) - I 1 h-

100 200 300 HOURLY

^ 100 -

X

z: § 50 -

0 _

ET

J

(mm/mo)

• ' , ? • ; / .

J? ! 1

50 100 HOURLY

Recharge (mm/mo) 1 H

0 100 200 300 HOURLY

Figure 2 Scattergrams of monthly water balance results for daily and monthly vs. hourly runs.

Table 2 Coefficients of determination (R2) and standard errors of the estimate (SEE) for daily and monthly vs hourly intervals( N=96 months)

Daily vs Hourly R2 SEE (mm)

Monthly vs Hourly R2 SEE (mm)

Soil Moisture End-of-Month Average

Evapotranspiration Recharge

0.989 0.995 0.993 0.998

1.01 0.52 2.06 2.75

0.340

0.789 0.853

9.26

17.62 17.78


Rainfall disaggregation model

Water balance models based on a daily time step require daily rainfall as input. Accurate and continuous daily rainfall records, however, are scarce relative to monthly records . A first-order Markov chain model can be employed to disaggregate monthly rainfall at a particular location into daily rainfall. To do so, som* daily rainfall data must be available to facilitate the estimation of rainfall distribution functions and transitional probability matrices, the basic elements of this model. The presumption here is that although the particular location and period of interest may lack continuous daily rainfall records, measurements at the site during another period or measurements at a nearby site may be available.

In order to simulate daily rainfall and at the same time utilize the information provided by the known monthly rainfall totals at the location of interest, values of the non-dimensionalized parameter Y, expressed as,

Y = PdV1 (13)

where P(j= daily rainfall (mm d-1) and Pm= monthly rainfall(mm d

_1 ), are simulated. The chain of dimensionless Y values can subsequently be multiplied by the known monthly rainfall (mm d ) to obtain a simulated daily rainfall chain.

Using historical data transformed according to equation 13, a probability histogram of Y can be constructed. The range of non-dimensionalized daily rainfall can be divided into a finite number(n) of discrete classes or states. A rainfall distribution (probability density function, PDF) must be fit over each of the non-zero rainfall states of the histogram.

Transitional probability matrices are also based on the available historical data. The matrix elements, p^ i, represent the probabilities of transition of Y from state i to state j on successive days (i=l,n; j=l,n). With n states defined, an nxn transitional probability matrix PM can be expressed as

PM = [p. .] for i,j=l,2,...,n. (14) 1 J J

The year can be divided into a finite number of seasons assuming that the transition probabilities for each season remain constant throughout that season. Using this criterion, it would certainly be acceptable to form 12 seasons corresponding to the months of the year. However, this may require the estimation of parameters beyond the capacity of the data ( Allen & Haan, 1975). By grouping the months to form a fewer number of seasons, the number of transition probabilities which must be estimated can be reduced. Each season has its associated state definitions, probability density functions, and PM.

Daily rainfall simulation

In order to apply a Markov chain model, there must exist a dependence between successive daily values of Y. Such a dependence is a


distinguishing characteristic of a Markov process» In addition, the historical time series of Y must be stationary within each season» Thus, a Markov property and stationarity test must be performed in order to validate statistically the use of a Markov chain model.

A Markov chain model was employed to disaggregate monthly rainfall at Station 0300. A 34-year record of daily rainfall from a nearby rain gage at USWB Station 8945 (Wahiawa Dam, Oahu, Hawaii) was used to define the Markov transitional probabilities. Markov property and stationarity tests were successfully passed. The key steps in the model application are briefly described below:

(a) Based on median monthly rainfall values at Station 8945, two seasons were defined for the area of interest. A dry season was found to occur during the months of April to October while a wet season was found to occur during the months of November to March.

(b) Seasonal probability histograms of Y were formed based on the historical data at Station 8945.

(c) For each of the seasons, the range of Y values was found to be adequately described by 10 discrete states or classes.

(d) In order to describe the nine non-zero rainfall states, a Weibull distribution function was fitted to each of the seasonal histograms (Figure 3).

0.20 -+- -+- - H -+-HISTOGRAM OF NON-DIMENSIONALIZEO RAINFALL STATION 8945 - WET SEASON

HISTORICAL DATA WEIBULL DISTRIBUTION

- t - -+- -+- h-5 10 15 20 25 30

DAILY RAINFALL (mm/d)/MONTHLY RAINFALL (mm/d )

a 0.06 -

-+- -+- -+-HISTOGRAM OF N0N-DIMENSI0NALI2ED RAINFALL STATION 8945 - DRY SEASON

HISTORICAL DATA WEIBULL DISTRIBUTION

5 10 15 20 25 30 DAILY RAINFALL (mm/d)/MONTHLY RAINFALL (mm/d )

Figure 3 Histograms of non-dimensionalized wet season (Nov.-Mar.) and dry season (Apr.-Oct.) rainfall for USWB station 8945 and the fitted Weibull distributions.


(e) Two 10 x 10 transitional probability matrices (one for each season) were formed based on the historical Y sequence.

(f) A random number was generated to determine the next day's rainfall state (k) given the current state (i).

(g) A second random number was generated to obtain a simulated value of Y with state k.

(h) The daily rainfall value was obtained by multiplying the simulated Y value by the monthly rainfall.

The cumulative distribution of simulated daily rainfall at Station 0300 compares favorably with the historical distribution at that station(Figure 4). It should be noted, however, that the Markov model based on the transition probabilities and distribution functions of Station 8945 seems to underestimate the number of extreme daily rainfall events at Station 0300. It is likely that the distribution

CUMULATIVE DISTRIBUTION OF DAILY RAINFALL

HISTORICAL DATA - STATION 0 3 0 0

SIMULATION !

DAILY RAINFALL ( m m )


HISTORICAL DATA - STATION 0 3 0 0 SIMULATION 2

DAILY RAINFALL (m

s 0 , 9

E


HISTORICAL DATA - STATION 0300 SIMULATION 3

DAILY RAINFALL (rr

Figure 4 Cumulative daily rainfall distributions for simulations 1, 2 and 3 in comparison with historical data for USWB Station 0300.


functions based on Station 8945 data (Figure 3) fail to account fully for extreme rainfall events at Station 0300»

Water balance with simulated rainfall data

Using the Markov-Weibull disaggregation technique described above, three daily rainfall sequences were generated from monthly data for Station 0300 during 1966-1973 » Using these data, daily runoff sequences were generated and adjusted to be equivalent to the runoff sequence computed for the actual daily data» Monthly pan evaporation was assumed to be uniformly distributed over each monttu Water balance calculations were done for each daily rainfall simulation, using the same model parameters as in the real data runs previously describedo

TABLE 3 Eight-year (1966-1973) water balance totals using 3 daily rainfall simulations and using monthly and hourly data for USWB Station 0300, Oahu, Hawaii

Run P(mm) R(mm) ET(mm) Q(mm)

Simulation 1 Simulation 2 Simulation 3 Simul» Ave o Monthly Data Hourly Data

6585 6585 6585 6585 6585 6585

354 354 354 354 354 354

4234 4205 4195 4211 4968 3582

2020 2021 2038 2026 1269 2641

Note: P = precipitation, R= runoff, ET = évapotranspiration, Q = groundwater recharge «

Note: Because beginning and ending soil moistures differ, the output terms do not sum to the precipitation »

Eight-year water balance totals for each simulation are given in Table 3* Also shown for comparison are the results using monthly and hourly data* Estimates based on the daily rainfall simulations consistently overestimate évapotranspiration and underestimate recharge, but to a much lesser degree than the monthly-interval run., Scattergrams of soil moisture, évapotranspiration, and recharge for each daily rainfall simulation run vs the hourly data run are given in Figure 5<> Table 4 lists corresponding coefficients of determination and standard errors of the estimateo

As the scattergrams and R^ values indicate, water balances using simulated daily rainfall result in évapotranspiration and recharge estimates which are in reasonable agreement with results of the hourly data calculations0 While significant bias remains, the use of simulated data greatly improves the model accuracy over monthly data»

Discussions and conclusions

The consistent underestimation of the recharge estimate based on


SIMULATION NO 2

30 •

P20^

?:- 10 -

Soi: U0!«Ur* (mm)

. " / •

';:̂ /.'

10 20 30

30 •

F 20 '

S 10

0

Soi! Uo

; • - ,

é_i

-Month

' • / ^

, / / ' •

.̂ •. ' '-

0 10 20 30 DATA

300

200 •

100

n

Rechnrg

/ • " ' ,

, [ . . : . . ]

0 100 ?00 300 DATA

0 10 20 30 DATA

30 •

§ 2 0 5

5 1 0 -

0

So? U.i.tur. (mm)

• ' ' .X

^ :

S\

0 100 200 300 DATA

Figure 5 Scattergrams of monthly water balance results for simulated daily rainfall runs vs hourly rainfall data runs.

Table 4 Coefficients of determination (R ) and standard errors of the estimate(SEE) for water balance results based on simulated daily rainfall vs hourly rainfall data (N=96months)

SI vs Hourly S2 vs Hourly S3 vs Hourly R2 SEE(mm) R2 SEE(mm) R2 SEE(mm)

Soil Moisture End-of-Month 0.319 7.66 Average O088I 2*82

Evapotranspiration 0o819 Recharge 0.957

12*38 11.08

0.485 7.10 0.895 2.58 0.842 11.79 0.956 11.26

0.405 7.98 0.882 2.74 0.838 11.91 0.951 11.61

Note: Sl=Simulation No.l, S2=Simulation No.2, S3= Simulation No.3

simulated daily rainfall, averaging about 23% compared with the hourly data results, is higher than expected. Using real data, the daily recharge estimate was biased by only 8%. The discrepancy here seems to be the result of the imperfect fit of the rainfall simulation model. While comparison with actual rainfall data (Figure 4) showed the fit to be reasonably good, presumably, the water balance is sensitive to even minor differences in the frequency

266 T.W. Giambelluca & D.S Oki

distribution or persistence of daily rainfall* The results of this study confirm that the use of a monthly

interval in water balance models results in a consistent overestima-tion of évapotranspiration and underestimation of recharge. The bias is much less pronounced for the daily interval. If daily rainfall data are available at the site for some period, or if daily data from a nearby site exist, a Markov-Welbull model calibrated to non-dimensionalized daily rainfall can be used to disaggregate monthly data. Use of simulated sequences of daily rainfall improves estimates of évapotranspiration and recharge but still results in a significant bias.

It is likely that the assumptions of the water balance model can be modified in such a way as to reduce the interval biasing effect. Specifically, the bias stems from the assumption that all water input is available at the beginning of an interval. It is possible that by shifting the time of input to some point within the interval (to the mid-interval point for example), or by distributing the input within the interval according to a fixed pattern, the interval bias can be alleviated. It is doubtful that such a solution will have much effect on the error variance produced by long interval data, however, it may offer a simple method of reducing the bias error.

The results presented here are of a preliminary nature, having been based on one sample site. Subsequent testing will be done using data from contrasting climates. Additionally, various alternative assumptions regarding the within-interval timing of water input will be tested.

References

Allen, D. M. & Haan, C.T. (1975) Stochastic simulation of daily rainfall. Res. Rep, no. 82, Water Resources Research Institute, University of Kentucky, Lexington, Kentucky, USA.

Alley, W.M. (1984) On the treatment of évapotranspiration, soil moisture accounting and aqulfier recharge in monthly water balance models. Water Resour. Res. 20(8), 1137-1149.

Ekern, P.C. (1966) Evaporation by Bermuda Grass sod, Cynodon dactyon L. Pers., in Hawaii. Agron. J. 58(4), 387-390.

Ekern, P.C. & Chang, J.H. (1985) Pan evaporation: State of Hawaii, 1894-1983. Rep. R74, Department of Land and Natural Resources, Division of Water and Land Development, State of Hawaii, Honolulu, Hawaii, USA.

El-Swalfy, S.A. & Cooley, K.R. (1980) Sediment losses from small agricultural watersheds in Hawaii (1972-1977), A.R.M.-W.-17 Agricultural Research, Science & Education Administration, U.S. Department of Agriculture, Oakland, CA, USA.

Giambelluca, T.W. (1983) Water balance of the Pearl Harbor-Honolulu basin, Hawai'i, 1946-1975. Tech. Rep. no.151, Water Resources Research Center, Univ, of Hawaii at Mamoa, Honolulu, Hawaii, USA.

Howard, K.W.F. & Lloyd, J.W. (1979) The sensitivity of parameters in the Penman evaporation equations and direct recharge balance. J. Hydrol. 41, 329-344.

Palmer, W.C. (1965) Météorologie drought. Res. Pap. 45, U.S. Weather Bureau. U.S. Government Printing Office.


Penman, H.L. (1949) The dependence of transpiration on weather and soil conditions. J. Soil Sci. 1(1), 74-89.

Rushton, K.R. & Ward, C. (1979) The estimation of groundwater recharge. J. Hydrol. 41, 345-361.

Thornthwaite, C.W. & Mather, J.R. (1955) THe water balance. Publ. Climatology. 8(1), 1-104.

U.S. Department of Agriculture (1972) Hydrology. National Engineering Handbook. Soil Conservation Service, Government Printing Office.

temporal disaggregation of monthly rainfall data for water...

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