Comparison of rainfall distribution method

Transcription

1 Team 6 Comparison of rainfall distribution method In this section different methods of rainfall distribution are compared. METEO-France is the French meteorological agency, a public administrative institution in charge of forecasting and the study of meteorological phenomena. Their website allows us to access to different data about the instruments of hydrological measurements. We were therefore able to reap the Lambert II extended coordinates for each rain gauge, shown in Table 1. Rain gauge Lambert II extend X Y Z Carros Levens St Martin Vésubie Guillaumes Roquesteron Puget Theniers Table 1 : Rain gauge location. Lumped and quasi distributed methods Thiessen Polygons Method Thiessen proposed a method to evaluate spatial averages over a domain from anecdotal. One way underlying the "method of Thiessen" is an interpolation technique based on the "law of the nearest neighbour. Clearly, the main advantage of this method is its simplicity. There is no objective information on the representativeness of interpolations. As illustrated in the diagrams below, where the problem is presented in one dimension, if the density of sampling points is "very strong", representativeness may be about right (left hand side graph), in other cases Instead the results can be very disappointing (right hand side graph). Page 1

2 Figure 1 : Comparison of Thiessen method efficiency, depending on sampling density Correlation coefficient 1,0 Gandin 0,9 0,8 0,7 Krigeage Spline 0,6 0,5 0,4 Ariytmétic mean Of 5neighbour Thiessen 0,3 0% y 50% 100% Pourcentage reconstructed rainfall periode Figure 2 : Comparison between Thiessen and BLUE s method Thiessen polygons assign a weight to the gauges and then distribute the values homogeneously to the sub-basins. However, we can consider this traditional method of interpolation as quasi Distributed. Indeed we allocate different weights to rain gauges and we do not consider an equal distribution of rainfall across our main basin. Fully lumped models consider a uniform distribution over the entire Var basin. By going a little deeper, we can say that a lumped model, for each sub catchment, would consider that no rainfall sub-basins would not undergo rainfall. Distributed methods Here, three methods are compared to find a more accurate way to distribute rainfall. We will detail three methods of interpolation; BLUE (Best Linear Estimators Unbiaised). We will begin with the Spline method (minimum curvature), then we will focus on the method of inverse distance weight (IDW) and we will then finish with the Kriging method. Spline The basic form of the minimum curvature Spline interpolation imposes the following two conditions on the interpolant: The surface must pass exactly through the data points. The surface must have minimum curvature. Page 2

3 The basic minimum curvature technique is also referred to as thin plate interpolation. It ensures a smooth (continuous and differentiable) surface, together with continuous first-derivative surfaces. Rapid changes in gradient or slope (the first derivative) can occur in the vicinity of the data points; hence, this model is not suitable for estimating second derivative (curvature). Inverse Distance Weights (IDW) This is a deterministic interpolation technique that creates surfaces from measured points, based on either the extent of similarity. The inverse distance weight technique provides an interpolation using a linearly combination of the rain gauge locations. To predict a value for any unmeasured location, IDW will use the measured values surrounding the prediction location. Those measured values closest to the prediction location will have more influence on the predicted value than those farther away. Thus, IDW assumes that each measured point has a local influence that diminishes with distance. Kriging This is a geostatistical interpolation technique that utilizes the statistical properties of the measured points. Geostatistical techniques quantify the spatial autocorrelation among measured points and account for the spatial configuration of the sample points around the prediction location. Kriging generates an estimated surface from the points introduced (rain gauge stations). Kriging is a spatial interpolation method, sometimes considered the most accurate from a statistical point of view, which allows a linear estimate based on the expectation and also on the variance of the spatial data. As such, Kriging is based on the calculation, interpretation and modelling of the variogram, which is a measure of the variance depending on the distance between data. This interpolation method differs from other methods (inverse distance, Thiessen polygons) because it has advantages. Kriging is the best linear unbiased, the means are identical and the variance is minimal. Page 3

4 Three rainfall distributions were built using SURFER 8 software: Figure 3 : Rainfall distribution with Spline method. Figure 4 : Rainfall distribution with IDW method Figure 5 : Rainfall distribution with Kriging method A variogram was built with a spherical method (scale = 375, length = 15100), also reflect uncertainty measures by imposing a Nugget effect (N = 10) Problems were met when building the variograms. Indeed it is difficult to establish a regression with so few data, (whether in a linear exponential or spherical for the variograms components). Finally ArcGIS software was used, that allows us to self calibrate the correlation and so the variogram could be more accurate. Page 4

5 The distribution of rainfall was therefore rebuilt in ArcGIS. The Spline method was rejected because this method does not give a smooth result. It also gives a massive distribution of rainfall spread over Tinée while the sub basins do not have a rain gauge. Figure 6 : Rainfall distribution with IDW and Kriging method, made in ArcGIS In ArcGIS the interpolation was used to raster from spatial analysis. Then IDW and Kriging methods are used, with a spherical variogram for Krigging, and we choose an inverse distance squared weighted interpolation. A zonal statistic was applied to work out the rainfall distribution in all the sub-basins, shown if Figure7. Figure 7 : ArcGIS print screen of zonal statistic, of IDW rainfall distribution in all the sub-basins Page 5

6 Initially the plan was to compare the sums of hourly rainfall (from , 12:00 to 11/06/1994 1:00 am) on each sub-basins. Comparisons of the Thiessen polygon results with IDW method and Kriging method could then be carried out, which are shown in Table 2 below: Comparison: Kriging (Spherical) vs. IDW vs. Thiessen Polygons Kriging IDW (Power = 2) Thiessen Polygons Catchments mean (mm) Vesubie Esteron Lower Var Tinee Upper Var VAR Table 2 : Comparison between Kriging, IDW and Thiessen polygons method. Then the hourly rainfall for the method of Thiessen polygons and krigging were compared. See below the summary table giving rainfall in mm for each sub-basin: Thiessen Kriging Thiessen Kriging Thiessen Kriging Thiessen Kriging Thiessen Kriging Date Time Up Var Low Var Tinée Vésubie Esteron 3/11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ Total (mm) : Tableau 3 : Comparison between kriging and Thiessen rainfall distribution, with hourly rainfall. Page 6

7 We can now compare the rainfall on each sub-basin by editing graphics. We note the similarity of rain on the various sub-basins. See below the rain by the two methods considered in the basin Up Var: Rainfall (mm) UP VAR : comparison of rainfall distribution (Thiessen Vs. Kriging) Time (hours) Up Var (Thiessen) Up Var (Kriging) Figure 8 : Graph representing the rainfall distribution with Kriging and Thiessen method. We now need in order to confirm our observations, compare the peaks of flood generated by these different methods. For this, we used our model calibrated for HEC-HMS. We have not changed any other settings, on the purpose to only observe the influence of the rainfall distribution method. Discharge (m 3 /s) Comparison of the Hydrograph produced by Thiessen and Krigging distribution method Observed Hydrograph SCS Unit Hydrograph with Thiessen Polygons SCS Unit Hydrograph with Kriging Method date / hours Figure 9 : Influence of the rainfall distribution method on the flood peak. Page 7

8 Conclusion The choice of interpolation method depends primarily on the information available. One can distinguish between Climate methods (where there are several observations of the field, or using purely spatial (this is our case). In our case the number of rain gauges is too small for the numerical values to be reliable. Indeed we note that there are not enough points to be able to assimilate the information for the variogram. Consequently when the variogram was set up to perform a method of Best Linear Estimators Unbiaised (IDW, spline, Krigging) the gridding was not accurate and we will lose here all the interest of the correlation. It would be very interesting to have more data and to make a correlation between rainfall and the distance to the sea and altitude. It would require us to apply the exponential law for the distance to the sea and make a log normal regression to the altitude. This work could be done with SURFER software. By observing the three different methods we can notice that the results are almost similar. The impact of the distribution of rainfall over the flood peak is negligible. We can therefore conclude that in our case study the rainfall distribution method should be Thiessen polygons. This method is faster to implement and gives us more extreme rainfall, which may be interesting to take into account to model the event having the greatest impact. (To see the implementation of coefficient Thiessen, please refer to the hydrological analysis report.) Page 8