Calculating Areal Ranfall Using A More Efficient Idw Interpolation Algorithm
Abstract
The estimation of areal rainfall is an important part of solving various hydrological problems utilizing rainfallrunoff and other models. Traditionally used Thiessen polygons (TP) method proved to be inaccurate mainly in mountainous areas and in catchments with insufficient number of rainfall gauges. One of the alternative to this method is an inverse distance weighting (IDW) method giving better estimates of areal rainfall even on places with rugged orography. However, this and similar methods are far less efficient than the simple TP method restraining its use to tasks where computational efficiency is not important. In this study a new algorithm accelerating the traditional IDW method is proposed and applied to three mountainous catchments situated in the central part of Slovakia. The method is compared with traditional IDW and TP methods in terms of both computational efficiency and estimated values. The results showed that while the new method gives the same results as the traditional IDW method it is far more efficient when the computational time was in all three catchments reduced by more than 96%.
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Introduction
In hydrology and other water related disciplines rainfall is one of the most important variables determining the amount of water entering the system. Rainfall measurements are essential for solving a wide range of tasks including those in meteorology, hydrology, agriculture, climate research or hydropower [7]. Even though the recent years and decades gave rise to very sophisticated measurement techniques such as meteorological radars or satellites most of the data are still collected with elevated can-type rainfall gauges. These enable to collect precipitation amounts only at single sites. However, rainfall data have very strong spatial variability and thus it is not reasonable to presume that a larger area around the rainfall gauge will share the same precipitation amounts. In order to account for the spatial variability a network comprising of several rain gauges needs to be deployed to correctly represent the rainfall amounts over a certain area. Generally, the denser the network is the more accurate is the information about the distribution of rainfall amounts over a certain area. Over the decades a large number of models of various complexity have been developed to utilize this information to solve various tasks in hydrology and other disciplines. Some of these tasks are of high importance such as predicting floods, managing water resources, evaluating crop yields and development or assessing the impact of land use or climate changes on the runoff from a catchment. Most of these models are lumped dealing with the catchment as a single unit and thus require estimates of mean areal rainfall as model input. The estimation of areal rainfall from a number of rainfall gauges could be performed using a number of various techniques. The overview of these techniques and their comparison is given in [1], [5], [8] or [9].
One of the most conventional and traditional one is the Thiessen polygon technique [10]. This method is favoured due to its simplicity, ease of use and minimum requirements on computational power. However, several authors (see e.g. [6], [8], [2] and [3]) comparing the Thiessen polygon technique with other techniques concluded that due to its fundamental principles it may produce inaccurate results especially on places with high topographical variation and the limited number of available rainfall gauges [9]. One of the possible issues with the Thiessen polygon method is displayed in (Fig. 1a), where the rainfall gauge situated in the upper right corner has no influence on the mean areal rainfall even though it is in the very near vicinity of the catchment.
Another approach in calculating areal rainfall uses inverse distance weighting (IDW) technique belonging to a family of distance weighting techniques ([8] and [4]). This technique eliminates the drawback of the Thiessen polygon method by dividing the catchment into a grid and interpolating rainfall amounts into each cell of the grid. The technique utilizes data from all rainfall gauges while assigning higher weights to stations closer to the interpolated cells. Fig. 2b shows that using the IDW technique the upper right station effects the mean areal rainfall over the catchment of interest. One of the disadvantages of the method used in calculating areal rainfall is that the precipitation must be interpolated into each cell of the grid lying inside the catchment boundaries.
Conclusion
The process of estimation areal precipitation is of high importance and is an integral part of solving various tasks in hydrology and other disciplines utilizing mainly conceptual rainfall-runoff models which are not distributed but work as lumped models. The number of methods for estimation of areal rainfall differ mainly in their accuracy but also in computational requirements. One of the methods which proved to be more accurate than the simple method utilizing the Thiessen polygons is an inverse distance interpolation method [8]. However, this method is computationally far less effective than the TP method. The aim of this study was to propose an algorithm which would significantly reduce the time needed for estimation of areal rainfall using the traditional IDW-t method. The new algorithm was tested not only in the terms of computational efficiency but also in the values areal rainfall it estimated. The algorithm was compared with the TP and IDW-t methods on data from three mountainous catchments situated in the central part of Slovakia. Two tests on two different datasets were performed for each catchment: 1) the whole dataset with missing data and 2) one year with no missing data.
The results showed that when comparing the values of estimated areal rainfall the IDW-a and IDW-t methods give practically identical results with mean absolute error in all three catchments lower than 0.001 (Table 2). As expected the estimates of the TP method when compared to the IDW-t method were slightly different with the maximum absolute error ranging from 3.8 mm (Hron@Brezno) to 7.3 mm (Vah@LM). When comparing the computational efficiency of the methods the new IDW-a method has significantly reduced the time needed for the estimation of areal rainfall using the traditional IDW-t method. In all three catchments the calculation was reduced by more than 96% in the case of analyzing the whole dataset (Table 3) and by more than 92% in the case of analyzing only 1 year (Table 4). In the case of the Hron@Brezno catchment the computational time was reduced by more than 99% when with the traditional IDW-t method it took 4h 26m 56s and with the new IDW-a method only 2m 20s. Similar results were obtained in the remaining catchments.
The significant reduction of computational time in the new IDW-a method could be used in solving many practical tasks. One of them is preparing areal rainfall data for zonal conceptual rainfall-runoff models where for each zone (most often based on altitude) a separate dataset of rainfall depths is prepared. The computational efficiency of the new method enables to divide the catchment into various number of zones and select the best division based on the selected criteria (e.g. comparison of observed and simulated flows, simulation of accumulation and melting of snow cover).
The process of accelerating the IDW method described in this study could be also applied to other linear estimators such as Kriging.