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Point Versus Block Ordinary Kriging in Rain Gauge Network Design Using Artificial Bee Colony Optimization

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Iranian Journal of Science and Technology, Transactions of Civil Engineering Aims and scope Submit manuscript

Abstract

Optimal design of rain gauge stations results in making the point and/or areal estimation of rainfall more accurate. The measure of network accuracy depends on the number and spatial location of rain gauge stations. In this paper, a new areal variance-based estimator using point ordinary kriging is developed to assess the level of accuracy for prioritizing rain gauge stations in a given network with no simplification considered. To the best of authors’ knowledge, this is the first time thereby a new point-based goodness of fit criterion so-called the percentage of area with acceptable accuracy is coupled with artificial bee colony optimization (ABC) to prioritize rain gauge stations and then validate the associated measure via coupling ABC with block ordinary kriging (BOK). This measure is applied to move from point to block and obtain the measure of accuracy. The coupled algorithm is applied to a case study with 34 existing rain gauge stations. The proposed algorithm is equipped with minimum tuning parameters and mimics the spatial pattern of rainfall variability in a distributed fashion. The results of the proposed approach showed that only eight rain gauge stations are required to achieve the same level of accuracy as the original network. In addition, the computed measure of network accuracy reproduces the BOK results for all values of n. In conclusion, the proposed scheme can be considered as a benchmark in rain gauge network design to assess the correctness of other paradigms in network design for all values of \(C\left( {N,n} \right)\).

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Abbreviations

\(A_{\text{Areal}}\) :

The percentage of area with acceptable accuracy

\(A_{\text{Point}}\) :

“Acceptable probability” at an un-gauge point \({\mathbf{s}}_{0}\)

\({\mathbf{h}}_{ij }\) :

Separation vector between two spatial locations i, j

k :

A multiplier

\(M\) :

The number of points inside a typical block

\(N\) :

Total number of rain gauge stations

N (\({\mathbf{h}}_{ij}\)):

Number of data pairs whose separation vector is \({\mathbf{h}}_{ij }\)

\(n\) :

Number of holding rain gauge stations

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

The true value of annual rainfall at \({\mathbf{s}}_{0}\)

\(P_{V} \left( {{\mathbf{s}}_{0} } \right)\) :

The true value of mean annual rainfall over block V index at \({\mathbf{s}}_{0}\)

\(P\left( {{\mathbf{s}}_{i} } \right),P\left( {{\mathbf{s}}_{j} } \right)\) :

Observed rainfall at spatial locations \({\mathbf{s}}_{i} ,{\mathbf{s}}_{j}\)

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

Estimated value of annual rainfall at \({\mathbf{s}}_{0}\)

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

Estimated value of mean annual rainfall over block V index at \({\mathbf{s}}_{0}\)

\(R\left( {{\mathbf{s}}_{0} } \right)\) :

Residuals in point ordinary kriging

\(R_{V} \left( {{\mathbf{s}}_{0} } \right)\) :

Residuals in block ordinary kriging

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

Standardized estimation error

\({\mathbf{s}}_{i } ,{\mathbf{s}}_{j}\) :

Corresponds to spatial locations i, j

α :

Threshold value

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

Weighting coefficient corresponding to observed value of rainfall depth at (\({\mathbf{s}}_{0}\)) in point ordinary kriging

\(\lambda_{i}^{\text{BK}} \left( {{\mathbf{s}}_{0} } \right)\) :

Weighting coefficient corresponding to observed value of rainfall depth at (\({\mathbf{s}}_{0}\)) in block ordinary kriging

\(\hat{\gamma }\left( {{\mathbf{h}}_{ij} } \right) = \hat{\gamma }\left( {{\mathbf{s}}_{i} ,{\mathbf{s}}_{j} } \right)\) :

Experimental semi-variogram at separation distance \({\mathbf{h}}_{ij }\)

\(\gamma \left( {{\mathbf{h}}_{ij} } \right) = \gamma \left( {{\mathbf{s}}_{i} ,{\mathbf{s}}_{j} } \right)\) :

Theoretical variogram at separation distance \({\mathbf{h}}_{ij }\)

\(\mu\) :

Lagrange multiplier for point ordinary kriging

\(\mu^{\text{BK}}\) :

Lagrange multiplier for block ordinary kriging

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Acknowledgements

This research was in part funded by the Regional Water Organization of Fars Province under Contract FAW-97004. Their financial support is greatly appreciated.

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Authors and Affiliations

  1. Department of Civil Engineering, Islamic Azad University, Shiraz Branch, Shiraz, Iran

    Mahdi Attar

  2. Civil and Environmental Engineering, School of Engineering, Shiraz University, Shiraz, Iran

    Mohammad Javad Abedini

  3. Computer Engineering (Software Systems), Department of Computer Engineering and Information Technology, Shiraz University of Technology, Shiraz, Iran

    Reza Akbari

Authors
  1. Mahdi Attar
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  2. Mohammad Javad Abedini
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  3. Reza Akbari
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Correspondence to Mohammad Javad Abedini.

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Attar, M., Abedini, M.J. & Akbari, R. Point Versus Block Ordinary Kriging in Rain Gauge Network Design Using Artificial Bee Colony Optimization. Iran J Sci Technol Trans Civ Eng 45, 1805–1817 (2021). https://doi.org/10.1007/s40996-020-00484-9

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  • DOI: https://doi.org/10.1007/s40996-020-00484-9

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