Improving the disaggregation of daily rainfall into ...

Hydroinformatics in Hydrology, Hydrogeology and Water Resources (Proc. of Symposium JS.4 at the Joint IAHS & IAH Convention, Hyderabad, India, September 2009). IAHS Publ. 331, 2009.

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Improving the disaggregation of daily rainfall into hourly rainfall using hourly soil moisture SAT KUMAR Department of Civil Engineering, Indian Institute of Science, Bangalore, 560 012, India [email protected]

M. SEKHAR Department of Civil Engineering, Indian Institute of Science, Bangalore, 560 012, India [email protected]

D. V. REDDY National Geophysical Research Institute, Hyderabad, 500 007, India [email protected]

Abstract For various applications in hydrology the rainfall at short time steps (hourly) is needed. As direct measurements of sub-hourly rainfall are hardly possible in practice (especially in developing and underdeveloped countries), a potential alternative may be to try to derive such data from the available daily data, through a disaggregation procedure. Random multiplicative cascade (RMC) models appear as an appealing solution to disaggregate the daily rainfall into sub-hourly rainfall. The RMC model provides a number of ensembles of hourly rainfall, taking the ensemble average one can get the hourly rainfall. This model is reasonably good in providing the magnitude of hourly rainfall, but fails to give the exact time of occurrence of hourly rainfall. Improvements can be made in the time of occurrence of hourly rainfall with the help of variables, which are sensitive to hourly rainfall. Soil moisture is one such a variable, which is directly sensitive to hourly rainfall. This paper deals with the potential of using hourly soil moisture in the improvement of hourly rainfall. Using this RMC model, ensembles of time series of hourly rainfall are generated. Among these ensembles of hourly rainfall, best ensembles are identified using generalized likelihood uncertainty estimate (GLUE) approach combined with the soil moisture model. The RMC model is applied and the performance is tested on several station data in the semi-arid and humid regions of the South India. The model is able to capture the magnitude of the hourly rainfall. Further, the RMC model is combined with a soil moisture data at a station in a semi-arid zone having hourly soil moisture profile data over a monsoon season. A soil moisture model was developed based on the Richards’ equation and parameters of the model were calculated through laboratory experiments. The hourly rainfall is modelled through the calibration of soil moisture data at this station. It is shown that using the hourly measured soil moisture, improved estimates of hourly rainfall is feasible. Key words: rainfall disaggregation, soil moisture, multiplicative random cascade, GLUE;

INTRODUCTION In most of the hydrological application the rainfall data at short time step is required. However rainfall data at short time scale is limited worldwide, mainly because of the high cost and low reliability of monitoring such data. Since rainfall data at daily time step is available worldwide because of the low cost and ease of collection, therefore a possible alternative to provide the short time rainfall data is to use the daily data to retrieve short time scale data. There are various models available for disaggregation of daily rainfall into Copyright © 2009 IAHS Press

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short time scale, and reproduces the statistics such as mean, variance, autocorrelations and the probability of an interval being dry (referred to as dry probability). Such disaggregated rainfall can be used only in the hydrological application where only statistics of the data are used, rather than the data itself. For a wide variety of hydrological application, where directly short time scale rainfall is required e.g. in case of distributed hydrological model, model should reproduce the truth short time scale rainfall. There have been various successful attempts to disaggregate the rainfall e.g. direct approach (e.g. Arnaud 2002), Poisson cluster model (e.g. Deidda 2000), artificial neural network (e.g. Burian and Durrans 2002), as well as models based on the principle of simulated annealing (e.g. Bardossy 1998). Among these models, the multiplicative cascade disaggregation model appears to be promising and therefore has received a growing attention during the last decade (Menabde et. al. 1997; Olsson 1998, Deidda 2000). Under certain conditions, these models produce rainfall series exhibiting scaling (multifractal) properties. The invariance scaling behaviour of rainfall has been found over a large range of space (Schertzer and Lovejoy 1987; Tessier et. al. 1993; Olsson and Niemczynowicz 1996; Svensson et. al. 1996; Over and Gupta 1996) and time steps (Olsson et. al.1993; Olsson 1996; Tessier et. al. 1996; Burlando and Rosso 1996). The scaling nature of rainfall is still discussed (Marani 2003, 2005). Nevertheless, multiplicative cascade models appear as appealing rainfall simulation tools because of their link with the multifractal theory (Gaume 2007). They are moreover equally adapted for the simulation of rainfall in space and time, are parameter parsimonious and theoretically easy to calibrate and use. Nevertheless, their calibration and validation give rise to some difficulties and questions which have seldom been commented. The main objective of this study is to illustrate the calibration and validation of a random cascade rainfall model and to elaborate their capability to produce the truth. It has been shown that with the help of the variables which are sensitive to hourly rainfall e.g. hourly soil moisture, improvement can be made to produce the disaggregated rainfall for the simulation of point-rainfall time series to be used in distributed hydrological model. This paper is organised in three parts. The first part presents the general principles of random multiplicative cascade models, their multifractal properties and the choice of a random cascade model. The issue involved in the calibration and validation i.e. to produce the truth is discussed in the second part of the paper. The third part deals with the improvement in the disaggregated rainfall with the help of hourly soil moisture.

THEORY Firstly, the concept of Random Multiplicative Cascade (RMC) model to disaggregate the rainfall has been discussed. Then, the statistical moment scaling function method of scale identification to fit the RMC model has been illustrated. Various types of distribution, which can be used to generate the RMCs, have been given. Further, the Generalized Likelihood Uncertainty Estimate (GLUE) approach has been discussed, which will be used to incorporate the hourly soil moisture into rainfall disaggregation scheme to improve the disaggregated hourly rainfall. Finally, to use the soil moisture data, soil moisture model based on the Richards’ equation has been discussed.

Improving the disaggregation of daily rainfall into sub-hourly rainfall using hourly soil moisture

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Random multiplicative cascade model Random multiplicative cascade models distribute mass of an interval on successive regular subdivisions in a multiplicative manner. This multiplicative random cascade models of rainfall disaggregation were successfully applied to rainfall modelling (e.g., Schertzer and Lovejoy, 1987; Over and Gupta, 1994, 1996; Olsson, 1998). There are two types of RMC model available namely, canonical model which preserves mass on the average in disaggregation and a microcanonical model which preserves mass exactly in disaggregation. This section summarizes the basic theory behind random cascade models. The basic cascading structure of the multiplicative random cascade model distributes rainfall on successive regular subdivisions with b as the branching number. The ith interval after n levels of subdivision is denoted as Δin (there are i=1,. . .,bn intervals at level n). The dimensionless spatial scale is defined as

λn = b − n ,

i.e., k0=1 at the 0th level of

subdivision. The distribution of mass occurs via a multiplicative process through all levels n of the cascade, so that the mass in subcube

μ n (Δ ) = R0 λn ∏ W j (i ) , n

i n

i =1

for

Δin

is:

i = 1,2, K, b n ; n > 0 .

(1)

Here R0 is an initial rainfall depth at n =0 and W is the so-called cascade generator. The basic structure of the discrete multiplicative random cascade model as used in this study is illustrated in Fig. 1, with b =2 as the branching number and n =0, 1, and 2.

Fig. 1 Graphical representation of a random cascade process with b=2.

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Scaling Identification: Statistical Moment Scaling Function Method Properties of the

cascade generator W can be estimated from the moment scaling behaviour across scales. Sample moments are defined as: bn

M n (q ) = ∑ μ nq (Δin ),

(2)

i =1

here q is the moment order ( q>0). For large n, the sample moments should converge to the ensemble moments. However, because the ensemble moments diverge to infinity or converge to zero as n → ∞ , instead the (rate of) convergence/divergence of the moments with scale should be considered. In a random cascade, the ensemble moments are shown to be a log–log linear function of the scale of resolution kn. The slope of this scaling relationship is known as the Mandelbrot–Kahane–Peyriere (MKP) function (Mandelbrot, 1974; Kahane and Peyriere, 1976) expressed as:

χ b (q ) = 1 − q + log b E (W q ).

(3) The MKP function contains important information about the distribution of the cascade generator W, and thus determines the scaling properties of rainfall. The slope of the sample moment scaling relationship can be found as:

log M n (q ) . λn →0 − log λ n large n (as n → ∞ ) and

τ (q ) = lim

(4)

for a specific range of q, slopes of the moment scaling For relationships for sample and ensemble moments converge, i.e., χb q = τ q . (5)

( )

( )

( )

In data analysis, the scaling exponent function τ q will be used to identify the best suited distribution for the random generator W of the cascade and to calibrate the parameters of this distribution. Some examples of PDF candidates for W and the corresponding expression of the τ q for the IID multiplicative cascade are presented in Table 1. The least square method is used to calibrate the parameters of each tested W’s PDF on the empirical scaling exponent functions. Once the parameters of the PDF have been found, the cascade approach can be preceded in two ways. If the quantity R0 is preserved on average i.e. the expectancies of the W random numbers are equal to 1, then on average, the cascade process is then called canonical. If moreover, the quantity R0 is preserved on each of the b sub time steps, the cascade process is called micro-canonical.

( )

The Generalized Likelihood Uncertainty Estimate (GLUE) The Generalised Likelihood Uncertainty Estimation (GLUE) technique was introduced partly to allow for the possible equifinality (non-uniqueness, ambiguity or nonidentifiability) of parameter sets during the estimation of model parameters (inverse problem) in the overparameterised models. These problems may arise in the disaggregation of rainfall coupled with soil moisure. Hence in this work GLUE has been used to identify the optimal rainfall. The basic idea of this approach follows from recognizing the equifinality of models and parameter sets. It is possible to give different degrees of belief to different models or parameter sets, and among them it may be possible to reject many of them because they clearly do not give the right sort of response for an


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application. The ‘optimum’, given a data set for calibration, will have the highest degree of belief associated with it. The implementation of GLUE requires the likelihood measure of different ensemble. Once such available likelihood measure is the modelling efficiency E of Nash and Sutcliffe (1970) expressed as,

σ r2 E = 1− 2 , (6) σo 2 2 where, σ r is the variance of the observation, and σ o is the variance of the errors. Table 1 Expression of Probability density function of W and expression forτ (q ) .

Model

β α

Probability density function

P(W = b ) = b , P(η = 0) = 1 − b c

−c

−c

P(W = b γ + ) = b −c , P(W = b γ − ) = 1 − b −c

LogPoisson

c m exp(−c) W = Aβ , P(Y = m ) = m!

Uniform

1 P(W ) = , b ∈ [0, b] b

LogNormal

W = β m+σY , P(Y ) =

Y

1 exp(− Y 2 / 2) 2π

τ (q ) τ (q ) = q(1 − c ) + c ⎡ qγ +− c (1 − bγ + , − c )q ⎤ + ln ⎢b q −1 ⎥ ( 1 − b −c ) ⎦⎥ ⎢ ⎣ τ (q ) = q − ln(b ) q q (1 − β ) + β − 1 q = −c

ln (b )

⎛ bq ⎞ ⎟ ln⎜⎜ q + 1 ⎟⎠ ⎝ q− ln (b )

( σ ln β )2 (q 2 − q ) q− 2 ln(b )

Soil Moisture Modeling Flow in the the unsaturated porus media can be described by the Darcy’s law. Insertion of Darcy’s law into continuity equation leads to the well known Richards’ equation.

∂θ ∂ ⎛ ⎛ ∂ψ ⎞⎞ = ⎜ K⎜ − 1⎟ ⎟ ∂t ∂z ⎝ ⎝ ∂z ⎠⎠ ,

(7)

where θ is volumetric soil moisture content, z is the elevation relative to a plane (positive downward), K is hydraulic conductivity, ψ is pressure head and t is time. Analytical solution of the Richards’ equation for the field boundary conditions is not available. So one has to discretize the equation into space and time using either finite difference or finite element methods. A soil moisture model based on the finite element method has been developed in the MATLAB using the pde toolbox. The model has been tested with the standard numerical solution available for infiltration and exfiltration dynamics in the presence of stratification of stratified soil layers.

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STUDY AREA For testing the model, datasets of four stations located in South India, with extremely different climate condition are used: Wailapally, Gundelpet, Maddur and Ambalvayal. Wailapally is situated in the western part of the Nalgonda district, Andhra Pradesh. Hourly rainfall and hourly soil moisture at three depths 50 cm, 110 cm and 200 cm from ground surface, are measured at this station. Apart from these the daily rainfall is also measured separately. Due to battery failure and some other problem, sometime the hourly rainfall data could not be measured, so only daily rainfall data is available. Since application of distributed hydrological model requires the fine rainfall, there is a need to get the hourly data from measured daily data. Gundelpet and Maddur come under the Chamrajanagar district of Karnataka. At Gundelpet station, 15 min rainfall data is measured but length of data is very short. Maddur is an experimental watershed and have a automatic rain-gauge which can measure the hourly rainfall. Ambalvayal is located in the Wayanad district of Kerala and have the hourly rainfall measurements. Table 2 shows the location and approximate annual rainfall in all the four stations. It can be seen from the table the approximate annual rainfall of stations varies from 600 mm to 2000 mm which correspond to semi-arid to humid zone. Table 2 Stations names, their latitude and longitude and approximate annual rainfall.

S.N. Station Name 1 Wailapally 2 Gundelpet 3 Maddur Ambalavayal 4

Latitude (N) 17.095° 11.808° 11.776°

Longitude (E) 78.892° 76.691° 76.558°

11.557°

76.301°

Approximate average annual rainfall (mm) 650 600 950 2000

PROCEDURE AND RESULT S Procedure of the disaggregation of daily rainfall data into hourly data can be described as: (1) Firstly, the observed rainfall at available minimum time step has been taken. (2) Then this rainfall is aggregate with b=2 up to the higher time of available data e.g. for station Wailapally the hourly rainfall is aggregated to two hourly and two hourly to four hourly and so on. (3) Then the sample moments from equation (2) has been calculated with q varying from 0 to 5 at the interval of 0.5. (4) Then the value of the τ q from equation (4), e.g. slope of the line of λ vs M for constant q’s on a log-log plot has been calculated. (5) Then appropriate distribution of W has been found by fitting the curve of τ q as given in Table 2. (6) Once the distribution of W has been found out, W can be generated using the PDF of the distribution. (7) W can be used in equation (1) to get the ensemble of hourly rainfall from the daily rainfall. (8) Using these ensembles of disaggregated hourly rainfall, the ensemble of soil moisture has been generated. GLUE has been used to identify the best ensemble of soil moisture and corresponding rainfall.

( )

( )


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Scaling behaviour of the q-moments: Fig. 2 shows the empirical q-moments of the rainfall amount series with the time step on log–log plots for all the four stations. Moment of all orders shows linear relation with time step on log-log scale. Although some of the previous studies on rainfall series shows the different linear relationship with different range of scale, the number and the limits of the identified ranges of time scales vary from one study to another and both are sensitive to the rainfall patterns and hence to the climate. Calibration of a rainfall disaggregation model

( )

The scaling exponent function τ q has been used to identify the best suited distribution for the random generator W of the cascade and to calibrate the parameters of this distribution. The least square method is used to calibrate the parameters of each tested PDF of W on the empirical scaling exponent functions. Fig. 3 shows the observed scaling exponent function along with the fitted scaling exponent function. In all the four stations the log-Poisson leads to best fits to the empirical scaling exponent functions. However, the parameters of log Poisson cascade model differ for all the stations.

(a) Wailapally

(c) Maddur

(b) Gundelpet

(d) Ambalavayal

Fig. 2 Log–log plots of the empirical q-moments versus time step for all the stations.

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(a) Wailapally

(b) Gundelpet

(c) Maddur (d) Ambalavayal Fig. 3 Calibration of a cascade model on the basis of the scaling exponent function τ (q ) for all the stations.

Disaggregation of rainfall from RMC model: Disaggregation of rainfall has been done using the RMC model with parameter fitted on the scaling exponent function. In all the stations only log Poisson model for the PDF of W has been selected for generation of W. With the help of W the daily rainfall has been disaggregated into 6 hourly rainfall (b=2). Then 6 hourly rainfall has been disaggregated into 3 hourly rainfall (b=2), again this 3 hourly rainfall has been disaggregated into hourly rainfall (b=3). Fig. 4 shows the CDF of the disaggregated hourly rainfall along with the measured hourly rainfall. It is clear from the figure 4 that CDF of disaggregated hourly rainfall match well with the CDF of observed hourly rainfall except for rainfall of lower magnitude. Hence the RMC model is able to simulate the magnitude and frequency of hourly rainfall. Fig. 4 shows the comparison of disaggregated hourly rainfall with the measured hourly rainfall for the Wailapally station for one of the monsoon season. It is clear from the figure 4 that disaggregated hourly rainfall is not matching with the measured one. Hence the time of occurrence of rainfall is not exactly matching with the time of occurrence of hourly rainfall. Here, RMC model is able to predict the statistics of the hourly rainfall (CDF) but fails to give the exact time of occurrence of hourly rainfall. The time of


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occurrence of hourly rainfall can be made through hourly soil moisture because of its sensitivity to hourly rainfall.

(a) Wailapally

(b) Gundelpet

(d) Ambalavayal (c) Maddur Fig. 4 CDF of measured vs. simulated hourly rainfall as simulated by RMC for all the four stations.

Fig. 6 shows the disaggregated hourly rainfall as retrieved by combining the hourly root zone soil moisture with RMC. The combination of root zone soil moisture with the RMC model retrieved hourly rainfall better than the RMC alone. However, the rainfall of lower magnitude is not retrieved well. This may be due to the fact that, either RMC model itself is not giving the hourly rainfall of lower magnitude correctly as shown in figure 4, or due to the less sensitivity of root zone soil moisture with the rainfall. The rainfall of lower magnitude goes to atmosphere through evaporation and does not reach upto root zone. Hence root zone have no information about the rainfall. Availability of surface soil moisture can be useful in retrieving the hourly rainfall because of its sensitivity to rainfall of lower magnitude. Nevertheless this simulated hourly rainfall from hourly soil moisture can be used to predict the root zone soil moisture and deeper percolation more accurately.

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25

20

20

15

15

10

10

5

5

0 0

5

10 15 20 Retrieved hourly rainfall

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Fig. 5 Relation between measured and simulated hourly rainfall as simulated by RMC. the hourly soil moisture.

0 0

5

10 15 20 Retrieved hourly rainfall

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Fig. 6 Relation between measured and simUlated hourly

rainfall retrieved after combining

CONCLUSION This study illustrates the usefulness of RMC approach which is based on moment scaling method, for disaggregation of daily rainfall into hourly rainfall. This has been done by disaggregation of rainfall at four stations located in extremely different climate condition. All the four station used in this study shows the scaling behaviour irrespective of the climate variability. The RMC model is able to predict the magnitude and frequency of hourly rainfall but fails to give the exact time of occurrence of hourly rainfall. Further, hourly measured root zone soil moisture is used to get the exact time of occurrence of hourly rainfall. Use of root zone soil moisture is able to improve the time of occurrence of hourly rainfall. But the root zone soil moisture fails to predict the time of occurrence of hourly rainfall of lower magnitude because of its less sensitivity to rainfall of lower magnitude. Since the rainfall of lower magnitude is not very sensitive to root zone soil moisture, this approach can be used to simulate the root zone soil moisture and deep percolation.

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