Revisited rainfall network design: evaluation of heuristic versus entropy theory methods

Abstract

Rain gauges are installed to measure pointwise precipitation and provide a comprehensive perspective of its spatiotemporal variations. Selection of an efficient and reliable rainfall monitoring network is a key role to reduce its maintenance and handling cost. The main purpose of the current paper is to compare efficiencies of various network design methods. The applied methods are entropy theory (as probabilistic multi-criteria decision-making) and genetic algorithm (as one of the heuristic methods) with three objective functions. Also, two classical (ordinary kriging; OK) and modern (Bayesian maximum entropy; BME) spatial simulation methods were undertaken to provide a comprehensive spatial simulation of precipitation. The proposed assessment was applied on spatial mean annual precipitation variability in the Namak Lake watershed located in the central part of Iran. The final efficiency of developed network design methods is evaluated in terms of three criteria known as mass estimation error, total error, and spatial bias of estimated rainfall. Based on the results, different network distributions have been proposed by the methods. Despite the reliability of the heuristic approach in nonlinear optimization due to its mathematical principle, the results indicated that the network design based on entropy theory can be used to estimate long-term mean annual precipitation more reliably and accurately. Results of the mass estimation error have shown 78 and 83% superiority of the entropy theory approach from the worst approach obtained from the OK and BME methods, respectively.

Appendix. Successive steps of network design based on entropy theory

• Step 1 Data collection and analysis. In this step and after checking for homogeneity, the annual precipitations in each rain gauge stations were averaged over 30 years of the registration period to attain long-term mean annual precipitations.

• Step 2 Measurement of transinformation and distance. The transinformation and distance were calculated for each pair of the existing rain gauges. To obtain transinformation, which requires the calculation of marginal entropy and joint entropy, a contingency table should be provided.

• Step 3 Plotting of the T-D curve. Transinformation values were drawn versus distances known as experimental T-D values (Fig. 12). To fit a theoretical T-D curve to experimental T-D values, different function structures were tested to achieve the best fit. These calculations were applied in GRAPHPAD PRISM statistical software using the least square fitting procedure (Motulsky 1999). Finally, the best curve was chosen in terms of R² and MAE. Results, which will be presented in the results section, have been shown the superiority of the exponential decay curve which has the following function structure:

$$ T\kern0.24em (d)=\left({T}_0-{T}_{\mathrm{min}}\right)\;{e}^{-k(d)}+{T}_{\mathrm{min}} $$

(11)

where T₀ is the initial value of transinformation, K is the transinformation decay rate, T_min is the minimum transinformation value, and d is the distance between rainfall stations. The plot of the exponential decay curve with its parameters is shown in Fig. 12. In this figure, the distance value L is called range in which the transinformation reaches its lowest value. Accurate estimation of range (L) plays a fundamental role in optimal positioning of rainfall stations. It can be calculated by the following equation (Mogheir et al. 2006):

$$ L=\frac{\ln \left({T}_0-{T}_{\mathrm{min}}\right)-\ln \left(\varepsilon \right)}{K} $$

(12)

where ɛ is the constant number (error term).

The percentage of the net redundant information (%NRI), which is the criterion to evaluate redundant information, is computed as:

$$ \% NRI=\frac{T\kern0.24em (d)-{T}_{\mathrm{min}}\;}{T_0-{T}_{\mathrm{min}}}\times 100 $$

(13)

• Step 4 Obtaining of square grid size. After determining the L value, the optimum grid size is calculated based on the square block with a and 0.5 L length and diagonal sizes, respectively (Sophocleous 1983; Olea 1999). Figure 13 shows the schematic of a regular square grid. The recommended grid size is obtained as:

$$ a=0.5\;(L)\;\cos {45}^{\circ }=0.354\;L $$

(14)

• Step 5 Fitting of the best grid network. Since the superimposing of the grid (with grid size obtained in a previous step) on spatial distribution of stations results in various configurations of the rain gauge network, the fitting of the best grid is of great importance. To achieve this goal, different grids were tested and the best one was selected in terms of minimum summation area of boundary cutting cells. It means that, after fitting a regular square network to the watershed, the area of boundary cut cells are calculated and the best grid network is chosen based on the minimum area. This procedure was applied in the geographical information system (GIS).

• Step 6 Selection of a rain gauge per square block. In the last step of the methodology, the stations were positioned through the optimum grid network obtained in the previous step. There was more than one station in some cells. The numbers were reduced by superimposing a square block over the study area and selecting one station per square block in a stratified way. In the following paragraph, the approach to determine the most convenient station based on the geostatistical framework is described.

The fundamental key to geostatistical estimation of mean annual precipitation is the calculation of the degree of spatial dependency which is expressed by a variogram model. Variogram modeling consists of two major parts known as experimental and theoretical variograms. Experimental variograms/cross variograms were calculated based on the observations in rain gauges. Primary fitting of theoretical and experimental variograms was carried out by GS+ software (Gamma Design Software, 2001). For this purpose, the model (e.g., exponential, Gaussian, spherical, etc.) and its parameters (nugget, sill, and range) were selected based on the maximum correlation coefficient. Then, these primary parameters of theoretical variograms were optimized using iterated nonlinear weighted least squares (INLWLS) method to guarantee the best fit to the experimental variograms (Cressie 1985). Finally, after defining the optimum structure of theoretical variograms, their covariance models were used for estimation purpose. Application of the geostatistical method reveals the best station in each cell and the optimum position and numbers of rain gauges is obtained.