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Optimal selection of number and location of rain gauge stations for areal estimation of annual rainfall using a procedure based on inverse distance weighting estimator

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Abstract

One basic demand toward advancement of economic growth is the need for reliable data on quantity and quality of water. Optimum design of rain gauge network in space leads to reliable data on water input. A conventional paradigm in rain gauge network design is to cast the optimization problem in a stochastic framework using geostatistical tools, which calls for an extensive matrix inversion to compute measure of accuracy. Deterministic schemes rely solely on network topology for interpolation and do not require matrix inversion and they are quite easy to use and understand. This feature might be a good reason to invest on network design based on deterministic methods. Changing the support size and assigning a measure of accuracy to the block-wise estimate are two basic challenges associated with working on a deterministic scheme. A new areal variance-based estimator using stochastic inverse distance weighting (Stc-IDW) is developed to design a rain gauge network. A new criterion is defined to move from point to block and cast the measure of accuracy for the entire study area. To evaluate the effectiveness of the proposed methodology, the coupled algorithm is applied to a case study with 25,000 km2 and 34 rain gauge stations in Iran. Development of measure of accuracy versus number of stations is achieved via both Stc-IDW and block kriging estimators, and the results are compared and contrasted to one another. Surprisingly, the optimum network configuration for various combinations of rain gauges shares almost identical goodness of fit criteria. Based on the results, the minimum of eleven stations are found to reach the maximum accuracy for both methods.

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Abbreviations

a :

Range (distance whereby the correlation value tend to a small ignorable value)

A Areal :

Percentage of area with acceptable accuracy

A point(s 0):

“Acceptable probability” at an un-sampled point s0

C(h):

Covariance function

d(s i, s j):

The Euclidian distance between two points si and sj

h ij :

Separation vector between two spatial locations

k :

Frequency factor defined for a specific distribution

M :

The number of points inside a typical block

N :

Total number of rain gauge stations

N(h ij):

Number of data pairs whose separation vector is hij

n :

Number of holding rain gauge stations

P(s 0):

True value of annual rainfall at s0

P o(s 0):

Observed rainfall at spatial locations s0

\(\hat{P}\left( {\varvec{s}_{0} } \right)\) :

Estimated value of annual rainfall at s0

\(\hat{P}_{\text{V}} \left( {\varvec{s}_{0} } \right)\) :

Estimated value of mean annual rainfall over block V index at s0

R(s 0):

Residuals of point-wise estimation at point s0

R V(s 0):

Residuals of mean annual rainfall over block V index at s0

\(R^{ *} \left( {\varvec{s}_{0} } \right)\) :

Standardized estimation error

s i, s j :

Corresponds to spatial location i, j

α :

The percent of acceptable probability

\(\lambda_{i} \left( {\varvec{s}_{0} } \right)\) :

Weighting coefficient corresponding to observed value of rainfall depth at s0

\(\hat{\gamma }\left( {\varvec{s}_{i} ,\varvec{s}_{j} } \right)\) :

Experimental variogram at two points whose separation vector is hij

\(\gamma \left( {\varvec{h}_{ij} } \right)\) :

Theoretical variogram at two points whose separation vector is hij

\(\mu \left( {\varvec{s}_{0} } \right)\) :

Lagrange multiplier

β :

Distance decay parameter (power)

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, School of Engineering, Shiraz University, P.O. Box 71348-51156, Shiraz, Iran

    M. Shahidi & M. J. Abedini

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  1. M. Shahidi
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  2. M. J. Abedini
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Correspondence to M. J. Abedini.

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Shahidi, M., Abedini, M.J. Optimal selection of number and location of rain gauge stations for areal estimation of annual rainfall using a procedure based on inverse distance weighting estimator. Paddy Water Environ 16, 617–629 (2018). https://doi.org/10.1007/s10333-018-0654-y

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