selecting optimum locations of rainfall stations using...

International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 12 No: 01 36

126301-7575 IJCEE-IJENS © February 2012 IJENS I J E N S

Selecting Optimum Locations of Rainfall Stations Using Kriging and Entropy

Ayman G. Awadallah Associate Professor, Civil Engineering Department, Faculty of Engineering,

Fayoum University, Fayoum, Egypt [email protected]

Abstract—Availability of rainfall data in real time is needed for management of important water resources like that of Zamzam well in Makkah watershed. However, rainfall network, in most watersheds, is significantly poorer than the values recommended by the World Meteorological Organization. A methodology based on the sequential use of kriging and entropy principles is used to determine the spatial distribution of potential rainfall gauging stations. Kriging, which performs linear averaging to reconstruct the rainfall over the watershed on the basis of observed rainfall data, is used to compute the spatial variations of rainfall in the locations of candidate stations. The information entropy reveals the rainfall information of each of the candidate locations. By calculating the joint entropy, the candidate stations are prioritized. The methodology is applied on Makkah watershed using geostatistical analyst in ArcGIS software for kriging and R software for information entropy. The candidate locations are validated on ground and final stations locations are selected.

Index Terms—Rainfall network design, Kriging, Entropy, Makkah, Saudi Arabia

I. INTRODUCTION

Accurate rainfall data is indispensable in planning and operation of water projects. Availability of rainfall data in real time is needed for management of water resources and especially in arid zones where water scarcity is prevailing. The purpose of this research is to present the methodology undertaken to select the optimum number and locations of the proposed rainfall stations in Wadi Ibrahim, Makkah watershed, Saudi Arabia (Figure 1). Wadi Ibrahim watershed is the source of Zamzam well, which is the drinking water for millions of pilgrims coming to Makkah. The withdrawal from Zamzam well is determined based on the availability of rainfall and groundwater levels in the aquifer.

The actual density of rainfall network, in most watersheds, is significantly poorer than the values recommended by the World Meteorological Organization [1]. There has been much research done on the subject of minimum densities of rainfall networks. WMO recommends certain densities of rain gauge stations to be followed for different types of catchments. For small mountainous regions with irregular precipitation, 25 km2 per station is recommended. According to the same WMO guidelines, the density falls to 10 – 20 km2 per station in urban areas [2]. As such, for the 40 km2 Wadi Ibrahim watershed, 2 to 4 stations are required. The full WMO minimum network requirements are listed in Table 1.

Several publications have applied statistical theory for rainfall network design. Shih [2] introduced various steps based on a covariance factor among rain gauge stations to design the rainfall network. Shannon’s entropy was used to decide the addition or removal of rain gauges ([3] and [4]). Kriging method (proposed by Matheron [5] for spatial interpolation) has also been applied for network design ([6] and [7]). Chen et al. [8] proposed a method composed of kriging and entropy that can determine the spatial distribution of rain gauge stations in a catchment. The approach proposed by Chen et al. [8] was followed to select the locations of the proposed 2 to 4 rainfall stations in Wadi Ibrahim watershed.

II. AVAILABLE DATA

The nearest stations available with extent till 2009 are four stations (Azeezeah, Sharaaea, Makkah and Laith) with coordinates as shown in Table 2. The rainfall data for these stations are available for the period from 2002 to 2009. The stations are ordered in the table in descending order from the point of view of the distance away from Wadi Ibrahim. Figure 2 illustrates the locations of these rainfall stations



TABLE I. RECOMMENDED MINIMUM DENSITIES OF PRECIPITATION STATIONS

Physiographic Unit Minimum Density per Station

(Area in Km2 per station) Non-Recording Recording

Coastal 900 9000 Mountainous 250 2500 Inter plains 575 5750 Hilly / Undulating 575 5750 Small Islands 25 250 Urban Areas 10-20 Polar / arid 10000 100000

TABLE II. AVAILABLE RAINFALL STATIONS IN THE VICINITY OF THE STUDY ZONE

Station Location Coordinates Available Data Range

Lat.(N) Long.(E) Laith 21° 20.506 39° 43.682 2002-2009 Sharaaea 21° 31.451 39° 55.115 2002-2009 Makkah 21° 26.267 39° 46.133 1985-2009 Azeezeah 21° 24.411 39° 51.888 2002-2009

III. METHODOLOGY AND RESULTS

As previously stated, kriging and entropy are used sequentially for selecting station locations. Kriging, as an interpolator, performs linear averaging to reconstruct the rainfall over the catchment on the basis of the observed rainfall. It is used to compute the spatial variations of rainfall for each of the available years on a grid covering 20 by 10 km2, with a cell size of 2.5 by 2.5 km2. Thus, the rainfall data at 32 candidate locations of the rain gauge stations are reconstructed. Five intervals are adopted to construct the empirical histogram of temporal rainfall distribution at the 32 candidate locations.

Optimal bin intervals were selected using the Jenks' natural breaks classification scheme. This scheme determines the best arrangement of values into classes by iteratively comparing sums of the squared difference between observed values within each class and class means. The "optimum" classification identifies breaks in the ordered distribution of values that minimizes within-class sum of squared differences, using the total amount of data at the 32 locations. After the classes or bins are determined, the frequency of total annual rainfall in each of the classes is calculated for each of the 32 candidate locations. The locations of the candidate stations are shown by Figure 3.

Figure 1. Location of Wadi Ibrahim Watershed, Makkah, Saudi Arabia



Figure 2. Location of Exisitng Rainfall Stations

Figure 3. Location of Candidate Rainfall Stations on which kriging is applied

A1 A2

B1

B2



The information entropy, which is then applied, reveals the rainfall information of each rain gauge station in the catchment. First of all, the entropy of each candidate rain gauge station is calculated after all rainfall data are reconstructed. The most important rain gauge station is the one with the highest entropy H(x). If p denotes the probability mass function of X and I(X) is the information content or self-information of X, then the entropy can explicitly be written as

∑∑==

−==n

iibi

n

iii xpxpxIxpXH

11

)(log)()()()( (1)

where b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10.

In the case of pi = 0 for some i, the value of the corresponding summand 0

blog 0 is taken to

be 0, which is consistent with the limit

0)log(lim

0=⋅

+→pp

p (2)

Various estimators are used to estimate entropy, such as the empirical entropy, the Miller bias corrected empirical estimator [9] and the non parametric estimator of Chao and Shen [10] which is a Horvitz-Thompson [11] estimator

applied to the problem of entropy estimator. All three methods were implemented on R software [12] using the entropy package developed by Hausser and Strimmer [13]. All three estimators gave different results; however, the regions with maximum entropy were 2 regions formed by locations (2, 3, 10, 11, 17, and 18) and by locations (7 and 8). The two regions were chosen as the two first potential regions. They are shown in Figure 3 as A1 and A2 potential regions.

To choose the next two potential regions, it was assumed that the information of the 2 first potential regions was introduced. With the introduction of this information, the relocated rain gauge station with the minimum mutual information, min{H(x1) – H(x1|x2)}, is selected next. The rain gauge station providing the maximum conditional entropy H(x1|x2) is selected as the second important rain gauge station in the rain gauge network. Since 2 regions– instead of one – are selected in the first entropy round, two regions are selected in the second round based on the mutual information concept. The second set of potential regions is denoted B1 and B2 on Figure 3. Thus, the optimum spatial distribution of the 4 rain gauge stations in the network can be determined. They are shown in the centre of the previously mentioned regions and within the boundaries of the watershed, as illustrated by Figure 4.

TABLE III. CRITERIA FOR SELECTION OF RAINFALL STATIONS

Parameter Description

Collector Orientation Wet Bucket 45° of Magnetic West

Distance between dry-deposition collector and rain gauge ≤ 5 m distance < 30 m

Vertical distance between collector orifice and rain gauge orifice Distance ≤ 0.3 m

Vegetation height Height ≤ 0.6 m within 5 m of instrument base

Cultivated agricultural fields Distance > 20 m from collector

Pasture land Distance > 20 m from collector

Vertical objects (Including towers, wires, fences, etc.), angle of projection from instrumentation

Projection angle ≤ 45° from top of instrument

Trees, angle of projection from instrumentation Projection angle ≤ 45° from top of instrument

Buildings, angle of projection from instrumentation Projection angle ≤ 30° from top of instrument

Objects, > 1 m tall, > 5 cm in width or depth Distance ≥ 5 m from instrument

> 20% of annual precipitation is frozen Wind shield present on rain gauge

Wind shield, pivot axis Same height as rain gauge orifice

Rooftop installation Urban sites only Rooftop installation, equipment separation from potential emission sources (sewer vents, HVAC systems)

Maximize separation

Rooftop installation, objects, angle of projection Projection angle ≤ 30° from top of instrument



Figure 4. Locations of candidate regions within the boundary of the watershed

Figure 5. Locations of actual selected station locations

Potential ground stations



The 4 selected regions were tested on the ground with the criteria hereafter stated in Table 3. It was found that 3 sites are suited for rain gauge tipping bucket type and one site even suitable for a complete weather station, among 21 candidate sites tested for suitability. The sites are shown in Figure 5 where sites, B, C and E, are the tipping bucket sites and site I is for the weather station site. The site selection criteria are thus confirming the statistical approach based on entropy / kriging methodology and the 4 locations selected are located within the boundaries of the regions optimized by the statistical approach.

The kriging / entropy methodology proved to be well suited for selection of rainfall gauge locations in arid regions using limited information from the existing rainfall network.

IV. CONCLUSIONS

Kriging and entropy are used sequentially to determine the locations of candidate rainfall stations in the mountainous watershed of Wadi Ibrahim, Makka, Saudi Arabia; which is the source of Zamzam well, drinking water of millions of pilgrims. Using the available rainfall data, 4 station locations are selected to comply with WMO guidelines and to manage the water resource efficiently and in real time. The locations selected by geostatistical approach are validated on ground.

Further research is needed to apply the methodology on a larger scale and to confirm that the proposed rainfall gauges add information to the spatial distribution of rainfall in the region. Furthermore, a comparison between entropy methods is needed to decide on the best suited method in selection of rainfall gauge locations.

REFERENCES [1] World Meteorological Organization, Guide to Hydrological

Practices, WMO-164. WMO: Geneva, 1994.

[2] S. F. Shih, “Rainfall variation analysis and optimization of gaging systems”, Water Resources Research, 18, pp: 1269-1277, 1982

[3] P. F. Krstanovic and V. P. Singh, “Evaluation of rainfall networks using entropy: II. Application”. Water Resources Management, 6, pp. 295–314, 1992.

[4] M. Al-Zahrani and T. Husain, “An algorithm for designing a precipitation network in the south-eastern region of Saudi Arabia.” Journal of Hydrology, 205, pp. 205–216, 1998.

[5] G. Matheron, Traité de Géostatistique Appliquée, Tome II. Le krigeage, Mémoires du Bureau de Recherches Géologiques et Minières No. 24. Editions B.R.G.M.: Paris, 1962.

[6] G. Bastin, B. Lorent, C. Duque and M. Gevers, “Optimal estimation of the average rainfall and optimal selection of rain gauge locations.” Water Resources Research, 20(4), pp. 463–470, 1984.

[7] A. St-Hilaire. T.B.M.J. Ouarda, M. Lachance, B. Bobée, J. Gaudet and C. Gignac, “Assessment of the impact of meteorological network density on the estimation of basin precipitation and runoff: a case study.” Hydrological Processes, 17, pp. 651–3580, 2003.

[8] Y.-C. Chen, C. Wei and H. C. Yeh, “Rainfall network design using kriging and entropy.” Hydrological Processes, 22: pp. 340–346, 2008.

[9] G. Miller, “Note on the bias of information estimates.” In H. Quastler, ed., Information Theory in Psychology II-B, pp. 95–100, Free Press, Glencoe, IL. 1955.

[10] A. Chao and T. -J. Shen, “Nonparametric estimation of Shannon’s index of diversity when there are unseen species.” Environmental and Ecological Statistics, 10, pp. 429–443, 2003.

[11] D. G. Horvitz and D. J. Thompson, “A generalization of sampling without replacement from a finite universe.” Journal of the American Statistical Association, 47, pp. 663-685, 1952.

[12] R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2011. Available at: http://www.R-project.org.

[13] J. Hausser and K. Strimmer, “Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks.” Journal of Machine Learning Research, 10, pp. 1469-1484, 2009.

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