RAINGUAGE NETWORK DESIGN Pankaj Mani Scientist D, National Institute of Hydrology, Patna mailofpmani@yahoo.com

Definition A raingauge network is an organized system for adequately sample the rainfall information, in space and time, to avoid chances of over/ under design of WR projects. The objective of a raingauge network, in addition to adequately sample the rainfall, includes the estimation of its variability within the area of concern.

Definition: WMO Guidelines As per WMO guides to hydrological practices (WMO, 1976), the network design covers following three main aspects: Number of data acquisition points required, Location of these points and Duration of data acquisition from a network.

Categories of Station Measurement stations are divided into three main categories, namely: Primary stations: Long term reliable stations expected to give good and reliable records. Secondary or Auxiliary stations: Placed to define the variability over an area. The data observed at these stations are correlated with the primary stations, and if and when consistent correlations are obtained secondary stations can be discontinued or removed. Special stations: These are established for specific studies and do not form a part of minimum network or standard network.

Stages of Network Design Data Collection Catchment and existing raingauge information; their location, duration, etc. Collection of data & estimation of precipitation characteristics, Determination of accuracy of existing network, Determination of number and location of new stations required, if any.

Methods of Network Design Cv Method Key Station Network Method Spatial correlation Method Entropy Method WMO Guidelines

Cv Method The coefficient of spatial variation of rainfall from the existing stations is used to determining the optimum number of raingauges. N = optimal number of stations, P = allowable degree of error in the estimate of mean rainfall and Cv = coefficient of variation of rainfall values at the existing m stations. P1, P2, ........Pm is the recorded rainfall at a known time at 1, 2, .......m station If m > 30, σm can also be used

Cv Method: Computational Steps Use the average annual rainfall data for individual station. (period accumulated; monthly, 10 days average etc. can also be used) Determine the statistics (mean, standard deviation and coefficient of variation) Obtain the value of P Compute N (number of station) for given permissible error in mean rainfall estimates using formula .

Key Station Network Method The station showing highest correlation with the catchment average is assumed to be most representative station. The correlation coefficient between the average of storm rainfall and the individual stations are computed and the stations are arranged in decreasing order of correlation. The station exhibiting highest correlation coefficient is called the first key station and its data is removed for determination of next key station .

Key Station Network Method The second and successive key stations are determined after removing the data of already selected key stations. Now the RMSE of estimated rainfall from combination of key stations (1,2, 3,….all) is determined and a graph is plotted between RMSE and corresponding number of stations in combinations. It will be seen that a stage comes when the improvement in the rainfall estimates is insignificant with the addition of more stations.

Key Station Network Method Sl No. No. of stations in combination Combination of stations Key station 1 9 1,2,3,4,5,6,7,8,9 2 8 2,3,4,5,6,7,8,9 4 3 7 2,3,5,6,7,8,9 6 2,3,5,6,7,9 5 2,3,6,7,9 3,6,7,9 3,6,7 3,6

Key Station Network Method No of stations 1 2 3 4 5 6 7 8 RMSE 21.45 11.65 6.957 6.893 5.197 3.812 2.047 1.148

Spatial Correlation Method The basis of this method is the correlation function ρ(d) exists between two stations. The ρ(d) is a function of the distance between the stations, and the form of which depends on the characteristics of the area under consideration and on the type of precipitation. The function ρ(d) can frequently be described by the following exponential form: where ρ(0) is the correlation corresponding to zero distance and d0 is the correlation radius or distance at which the correlation is ρ(0)/e.

Spatial Correlation Method For an area ‘s’ with the center station, and assuming ρ(d) exists and described as above, the variance of the error in the average precipitation over ‘s’ is given by Kagan (1966) as: The relative RMSE for an area ‘S’ with ‘N’ stations evenly distributed so that S = N × s, is defined as: where, the first term is attributed to random error and second term with spatial variation in the precipitation field.

Spatial Correlation Method S=N X s L L=1.07√s s S=N X s L L=√s

SCM:Computational Steps Theoretically, ρ(0) should equal to unity but is rarely found so in the practice due to random errors in precipitation measurement and micro climatic irregularities over an area. The correlation between two rainfall stations with concurrent data are computed and plotted against its distance. For ‘n’ number of existing stations the possible combination would be nC2. The inter-stations distance is plotted on X-axis while the correlation is plotted on Y-axis (on log scale).

SCM:Computational Steps Station Name Distance Correlation 6 1 6.635933 0.281523 4 6.720109 0.487851 2 3 7.234885 0.448008 8.698046 0.280573 9.633291 0.480566 5 10.21089 0.17029 10.71486 0.346644 10 12.02968 0.254124 13.37846 0.355068 7 8 13.61854 0.368347 14.43904 0.296527 14.46311 0.645556 14.61809 0.24556 15.72948 0.476863 16.2681 0.427164 16.85225 0.382571 17.21371 0.451507 18.79007 0.25819 20.59654 0.468624 9 20.63936 0.396006 20.87076 0.281303 23.24876 0.343275 Station Name Distance Correlation 3 4 23.42803 0.415388 5 23.76972 0.24444 8 25.3396 0.281992 7 31.36899 0.214145 6 31.99795 0.299989 34.21938 0.907909 34.91605 0.257129 2 36.98174 0.282765 37.68367 0.422015 38.2408 0.369154 1 38.24285 0.399103 38.67579 0.39844 40.85275 0.298737 10 9 42.0461 0.248119 43.41789 0.335313 45.93341 0.193902 51.81685 0.265757 55.24055 0.177415 56.30718 0.430246 58.28824 0.36881 59.21265 0.335097

SCM:Computational Steps The slope of this line will 1/d0 and the Y intercept will give log[ρ(0)], hence ρ(0) and d0 is computed.

SCM:Computational Steps S= total catchment area, n = no. of station evenly distributed representing area ‘s’ by each station, so that S=n x s. Compute the sigma and mean for each station from time series rainfall data, estimate Cv = sigma/mean. Compute s=S/n Compute RMSE, Z1 (for estimation average rainfall over area S) for n=1, 2, 3, ….n

SCM:Computational Steps No. of Stations RMSE 1 0.776178 2 0.388089 3 0.258726 4 0.194045 5 0.155236 6 0.129363 7 0.110883 8 0.097022 9 0.086242 10 0.077618 11 0.070562 12 0.064682 13 0.059706 14 0.055441 15 0.051745

Entropy Method The hydrological information and regional uncertainty associated with a set of precipitation station are estimated using Shannon’s entropy concept. Since time series of rainfall data can be represented by gamma distribution due to the presence of skewness, the entropy term is derived using single and bivariate gamma distribution. .

Entropy Method A quantitative measure of the uncertainty associated with a probability distribution or the information content of the distributions termed Shannon entropy can be mathematically expressed as: Marginal entropy for the discrete random variable X is defined as: where H(X) is the entropy corresponding to the random variable X; k is a constant that has value equal to one, when natural logarithm is taken; and i p represents the probability of ith event of random variable X.

Entropy Method Marginal entropy for the discrete random variable X is defined as: The marginal entropy H(X) indicates the amount of information or uncertainty that X has. If the variables X and Y are considered as independent, then the joint entropy [H(X, Y)] is equal to the sum of their marginal entropies defined by: where p(Xi ) is the probability of occurrence of Xi, computed by N2 and LN2 distributions, and N is the number of observations.

Entropy Method Similarly, the joint entropy in a region with m station and precipitation variables (X1, X2, ……Xm) can be extended to where, X1, X2, ……. Xm represents precipitation variables measured at m station and p(xj1, xj2, ….xjm) is the joint probability of occurrence of jth event at mth station.

Entropy Method It means, the entropy H(Xi) at a station, which is also the transmitted information about the variable Xi at station I can be decomposed into: the net information H(Xi) at i, common information T(Xi, Xj) between station pair i and j which will remain constant throughout the segment i-j considering station pair (i,j), which depends on the the distance between the station pair and is maximum at station location

Entropy Method z and w are normalized variates of X and Y respectively, with a mean of zero and standard deviation of unity. If ρzw is the correlation coefficient between z and w, then the information transmitted by variable Y about X, T(X,Y) or by variable X about Y, T(Y,X) is given by Shannon and Weaver

EM: Computational Steps Using the rainfall time series data, the mean and variance for each station are estimated. Using these values compute the scale and shape parameters (σ and λ) for gamma distribution; mean = σ λ and variance = σ2 λ. The entropy of rainfall variable measured at each station, H(X) were computed using formula;

EM: Computational Steps Normalize the rainfall distribution at a station using its mean and sigma value (use x-sigma/mean). Make the possible pair of inter-station by making triangular grids from existing rainfall stations such that: correlation between all three stations at the vertices exist, and none of triangles has any obtuse angle. Now compute the correlation coefficient (ρzw ) between all pair of stations using their normalized series. Compute T(X,Y)= T(Y,X) by formula:

EM: Computational Steps Considering a triangular element shown below formed by joining three precipitation station (i,j,k) measuring precipitation Xi, Xj, Xk respectively. Interpolate H(XiXj) at intermediate points (points at 10 equal interval spacing between two stations) considering the figures and assuming linear transformation of information between two stations.

EM: Computational Steps Now plot the information contours using computed H(XiXj). Identify the pockets of inadequate information to propose the new stations.

Network Design: WMO Guidelines These recommendations are based on the 1991 review of Members’ responses regarding the WMO basic network assessment project

Network Design: WMO Guidelines