A Multiple-Point Statistics Approach to Generate Rainfall Maps at High Spatial-Temporal Resolution Merging Radar and Rain-Gauge Measurements

Transcription

1 A Multiple-Point Statistics Approach to Generate Rainfall Maps at High Spatial-Temporal Resolution Merging Radar and Rain-Gauge Measurements G. Ghiggi 1, G. Mariéthoz 2, A. Berne 1 1 Environmental Remote Sensing Laboratory, EPFL, Switzerland 2 Institute of Earth Surface Dynamics, UNIL, Switzerland Abstract Precipitation is a complex process, strongly varying over a large range of spatial and temporal scales. Rain gauges provide direct measurements of rainfall intensity but unfortunately with limited spatial representativity. On the other hand, weather radar systems cover extended areas with high spatial and temporal resolutions suitable to capture the high dynamics of rainfall, but the indirect estimates of rain rate that these systems provide are affected by significant uncertainties. The approach we suggest allows to partially correct errors and biases present in radar images, which arise for example from non-uniform vertical profiles of reflectivity and conversion of radar reflectivity into rain rate (Z-R relationship). The precipitation pattern is extracted directly from the radar image and then used to interpolate rain gauge data over a simulation grid. Cumulated rainfall maps can be generated with customized temporal resolutions from 1 day up to 10 minutes, namely the integration time of the raingauges for the study case. The proposed technique is an alternative to classical geostatistical approaches and does not require the definition of a spatial variability model (correlogram or variogram) of precipitation. The method has been tested for two rainy events covering different topographic Swiss regions; a cross-validation and a cross-comparison scheme have been applied to assess the skills of this statistical approach in comparison with a geostatistical approach operationally implemented by MeteoSwiss [45-47]. Validation analysis shows that MPS estimates perform similarly to geostatistical methods: the mean error is smaller than the resolution of the rain-gauge and the bias in the radar estimate is reduced up to 50 %. Additionally, during the convective event, the representation of the precipitation fields seems to be better characterized by the MPS approach, with a lower smoothing of the highest rainfall intensity, compared to the geostatistical method. 1. INTRODUCTION Many environmental and hydrological processes such as runoff, subsurface and groundwater flow derive from precipitation. Natural hazards such as floods, landslides and avalanches that generate severe economic (and sometimes unfortunately human) consequences are also influenced by (excessive) precipitation. The distribution of the precipitated atmospheric water can also have significant consequences on the social organization. Actually rain and snow heavily affect the economic growth of our society: besides the touristic impact on beach resorts or mountain regions, in particular ski resorts, agricultural productivity directly depend on distribution, intensity and variability of precipitation. It is not a simple coincidence if economic growth has been related to water availability since the Greek and Roman civilizations. Therefore, from both the hydrological and the economic points of view, precipitation distribution and accurate quantitative estimation are essential for our society. Focusing on the estimation of hourly precipitation, in the last decade several multivariate geostatistical methods were proposed, based on Kriging with External Drift (KED) and Co-Kriging formulations. The published literature on the subject prove that geostatistical methods were generally found to outperform deterministic methods and simpler merging techniques [17, 28]. Haberlandt et al. (2007) [20] used radar, daily precipitation and elevation as secondary variables in KED. Velasco- Forero et al. (2009) [56] proposed a non-parametric technique to automatically compute correlograms, incorporating radar data as auxiliary variable using KED. Schiemann et al. (2011) [38] used a geostatistical radar-rain gauge merging technique based on KED with nonparametric correlograms and parametric semivariograms. Verworn et al. (2011) [57] also investigated the introduction of topography data as an auxiliary variable combined to radar data in KED formulation. Sideris et al. (2014) [45-47] proposed a co-kriging technique with external drift that involves the fitting of three variograms (two direct and one cross-variogram) using also temporal information. This approach is actually applied in real-time procedures by MeteoSwiss to produce hourly rainfall map estimates (CombiPrecip operational product). These maps are used as benchmark to the proposed technique. The proposed merging technique was tested over the whole Swiss territory, characterized by a varied 1

2 topography mostly mountainous with the Alps in south and the Jura massif range in the northwest (see Figure 1). The Alps acts as a climatic barrier between northern and southern Switzerland. The South presents almost a Mediterranean climate with warm dry summers and mild wet winters, while the northern side is strongly influenced by westerly winds coming from the Atlantic Ocean. These are responsible for precipitation, often characterized by the socalled orographic effect, which refers to the rise in precipitation rates due to uplift, adiabatic cooling and resulting condensation of humid air masses on windward mountainsides [3]. The inner-alpine valleys are characterized by a distinct climate, because of the rain-shadow effect due to the fact that they are sheltered against precipitation from both the north and south weather system. This is the reason why dry conditions prevail in these Swiss regions. Switzerland relies entirely on radar and rain gauges for precipitation estimation. The spatial resolution of these two sources of data is several order of magnitude different. Rain gauges provide direct (almost) point measurements of cumulated precipitation over a specific period of time, but do not allow to capture the large variability in space and time of the precipitation fields, even in presence of dense gaugenetworks. Weather radars provide indirect but detailed and continuous spatial information (reflectivity), also in areas where lack of rain gauges does not allow the survey of rainfall depths. On the other side, considering that radar signals in some valleys can be blocked by the surrounding mountains, a carefully designed rain-gauge network turns out to be really important. It is therefore particularly attractive to combine both sources of information taking advantage of each. In alpine environments such as the Swiss territory, characterized by thousands of catchments, some of which highly exposed to natural hazards, the merging of these two types of data sources is necessary. Although rain-gauge measurements are assumed to be fairly accurate, they are seldom free of error [43]. Strong winds for example can lead to underestimation of precipitation during strong convection events; whereas remote and high altitude locations can be underrepresented. On the other side, uncertainty linked to radar signal propagation, such as attenuation and echoes, may affect radar data leading to biased estimates. The complex shielding by the mountains and the strong ground clutter increase the complexity of the employment of this technology over the Swiss territory. For example, when retrieving the rainfall estimation at ground level, additional uncertainties arise because of the vertical profile of reflectivity (VPR) and the conversion of radar reflectivity into rain rates (Z-R relationship) [17]. Radar rain rate estimation can also be altered by the presence of hail [10] or by the bright band phenomenon at a given distance from the radar. In this study, we assume that each rain-gauge (point) measurement integrated over 10 minutes is exact and represents the precipitation over an area of 1 km 2 which is the resolution of the radar composite that we used. We also consider that radar estimated precipitation at a certain altitude corresponds to the real-time situation on the ground. Although it is known that if stratiform precipitation conditions prevail such assumption may hold, when convective cells arise in weather pattern, rain-gauge and radar measurements can be misleading because of significant microvariability [46]. Fig. 1. Topography of Switzerland 2

3 For those who may not be familiar with multiple-point geostatistics, Section 2 provides the basics of this technique and introduces the Direct Sampling algorithm. Section 3 gives details about the applied approach and Section 4 includes a brief description of the data used in the case studies. Section 5 shows how the performance of this technique has been tested, whereas Section 6 presents the results of cross-validation and the comparison with MeteoSwiss CombiPrecip hourly estimates. Finally, Section 7 concludes this study. 2. MULTIPLE-POINT STATISTICS Multiple-Point statistics (MPS) was proposed by Guardiano and Srivastava in 1993 for modelling subsurface heterogeneity [19]. Strebelle developed the first efficient implementation of the method in 2002 [43]. Other algorithms followed during the last decade [1, 48, 53, 61]. The characterization of precipitation spatial pattern has always been a challenge. Traditional geostatistical approaches use variograms to identify spatial variations of different types of phenomena. The variogram takes into account the correlation between only two points in a while, but in case of non-linear continuity (e.g. intermittency of precipitation) it cannot distinguish different pattern of heterogeneity [24]. For this reason, the variogram is an incomplete measure of uncertainty in complex and non-linear pattern structure [49]. To overcome the limitation of the variogram and its twopoint based statistics, training images have been introduced. When MPS were originally created, a training image (TI) corresponded to a conceptual model describing the expected pattern of the simulation, characterized by only categorical variables. Today, with different algorithms that have emerged and increasing computational efficiency, a TI has become an image representing continuous variables that provide information about structures and patterns (geometric features) of the simulated phenomenon (e.g. weather radar images). A set of training images allows the identification of the dominant spatial variation patterns. MPS characterizes the spatial structure by considering the configuration of several points within the TI (instead of the two with variogram approaches), extracts multiple-point statistics and hence enables the reproduction of complex patterns. The additional information used to condition the simulation and limit the range of possible patterns are called hard data (e.g., rain-gauge measurements). At the beginning of the sequential simulation, the simulation grid (SG) is filled with the available hard data (e.g. raingauge measurements). The simulation proceeds using a random path, the grid is progressively filled with simulated nodes values that become in turn conditioning data for the following simulated nodes. Classical MPS consist in computing a sequence of all possible patterns, in order to generate a sort of catalog, and then copy one of the possible pattern in the simulation grid at each simulated node, according to conditional probability distribution function imposed by hard data and previously simulated nodes. In other words, each unsampled node x of the simulation grid is considered as a single data event and the neighborhood conditioning data dn (including hard data and previously simulated nodes) are used to generate a cumulative conditional probability distribution function F from which is extracted the following random value of the target simulated variable Z(x): F (ZTI,x, dn) = Prob [ Z(x) ZTI dn] (1) Unfortunately, given the size of the radar images used as TI, this method turns out to be too expensive in terms of computation costs. In order to avoid explicitly constructing of the cumulative distribution function F, the Direct Sampling technique has been applied to generate stochastics fields representing complex statistical and spatial properties directly from the TI [30, 31, 48]. In fact, the focus does not reside in the probability of a data event to occur, but only in one representative outcome of it. Instead of building conditional joint distributions and then sampling them, the TI is directly sampled, without storing probabilities. As soon as a satisfactory configuration depending on the neighboring conditioning data (dn) is found in the TI, the value of the corresponding pixel is assigned to the simulated node (x). In order to determine if a good configuration in the TI is found and to avoid the enormous number of iterations required for the scan of the entire training image to search out the perfect matching pattern, a function quantifying the dissimilarity between conditioning data and TI data has been defined. If the resulting value exceeds the acceptance threshold, the algorithm seeks for other configurations until a possible correct pattern is found. If an excessive number of iterations (defined by the user) is reached, the best of the combinations previously identified is retained. For continuous variable, the distance dissimilarity function is defined as follows: D(d n, Z TI ) = 1 n d n j j Z TI,j (3) where n is the number of conditioning data and Z TI,j is the pixel value in the training image situated at the same distance as d j from the simulated node in the simulation grid. 3

4 The steps of the algorithm can be summarized as follows: 1. Pattern extraction from conditioning data in SG, composed by rain-gauge measurements and previously simulated nodes. 2. Pattern identification in the TI using the distance dissimilarity function until one combination does not exceed the acceptance threshold or the maximal number of iterations is reached and the smallest distance is selected. 3. Pattern reproduction in the SG, consisting in copying the value of the selected pixel of the TI in the simulated node in the SG, which becomes a conditioning data for the following simulations steps. At the beginning of the simulation, founding a corresponding pattern in TI may require more time because of the lack of conditioning data (only hard data are present in the SG). As long as the simulation evolves, an increasing number of conditioning data arises in the simulation grid, and an adaptive neighborhood search radius reduces the distance from the simulated node from where conditioning data are considered in the training image. During the final steps of the simulation, faster combinations are found and less time is required to simulate the final nodes in the SG. 3. DESCRIPTION OF THE METHOD The training image (TI) for each event simulation is composed by the original cumulated radar composite and a series of bias maps ; the latter representing the same spatial structure of precipitation extracted from the original cumulated radar composite with the addition of a uniform bias over the entire image (Figure 3). This essentially allows the algorithm to choose values in agreement with the gauge data that maybe are not present in the original cumulated radar composite and therefore increase the variability of the simulation. The number of bias maps depends mainly on the aggregation time of the required estimation: 10 minutescumulated radar estimates produce lower bias of rainfall amounts than hourly estimates because of the different amount of precipitation. The same concept applies to stratiform and convective events since the latter involve higher quantity of rainfall and hence the under/overestimation can be greater (in absolute terms). Moreover, a 4-dimension layer, composed of two 2D-grid containing north and east coordinates, has been added to each map layer of the TI (Figure 4), in order to ensure that the accepted value of the distance-dissimilarity function (see Section 2) comes from the region near the simulated node and hence reflects the regional spatial structure of precipitation. Fig. 2. Illustration of the direct sampling (DS) method. (a) Defines the data event in the simulation grid. The question mark represents the node to be simulated. The two white and the black pixels represent the conditioning data (b) Defines a search window in the TI grid by using the dimensions a, b, c, d of the data event. (c) Scans of the search window starting from a random location till (d) the simulation data event is satisfactorily matched. (e) Assigns the value of the central node of the first matching data event to the simulated node. (From Mariéthoz et al., 2010 [31]). 4

5 In fact, if the difference between the coordinates of the neighboring conditioning data d and those of the TI-vector Z is elevated, the distance-dissimilarity function does not accept the value from the training image and goes on searching for another value closer to the simulated node. The construction of this multi-dimensional TI (Figure 3) allows the algorithm to take advantage of the precipitation spatial pattern, considering at the same time different possible local biases in radar data. In this way, the algorithm is able to find the best combination in order to optimize the estimation, whereas misleading values in radar data are neither selected nor "imported in the simulation grid because they exceed the dissimilarity threshold implemented in the algorithm. The spatial flexibility seeking for the correct node is very useful because it allows the algorithm to reckon spatial errors affecting the radar estimation, such as the wind advection of precipitation that can displace the effective ground precipitation from the area estimated by radar at a given altitude. Since radar data are considered a reliable source to indicate dry regions, in order to improve the computational efficiency and reduce the simulation time, the sequential simulations are restricted to areas up to 3 km from the edge of rainy zones indicated by the radar; the rest of the simulation grid is set to zero rainfall. The tolerance of 3 km allows to avoid possible error in radar estimation such as beam attenuation during strong storm events and wind drift at the surface. The output of the simulation at each pixel is an ensemble of plausible rain rate values. If a single value at each pixel is requested, the user can compute a representative one like the mean, the median or the mode for instance.. Fig. 4. Illustrative representation of the 4-dimensional space of the TI with the coordinates matrix Est and North 4. DESCRIPTION OF AVAILABLE DATA The Swiss Federal Office of Meteorology and Climatology, MeteoSwiss, operates four C-band radars and a network of over 200 automatic rain-gauges (SwissMetNet network). In this study only 104 automatic rain-gauges, submitted to strict high quality control by MeteoSwiss, have been employed as conditioning data for the merging technique and then used for cross-validation. The remaining gauges have been used only for the cross-comparison scheme (see Section 5). The average distance between two rain-gauges is about 25 km. The radars are situated at Albis (925 m a.s.l.), La Dole (1675 m a.s.l.), Monte Lema (1625 m a.s.l.), Plaine Morte (2942 m a.s.l.) and perform a scan every 2.5 minutes with reflectivity measurements up to 240 km. The location of the radars and the rain gauges used for this study can be seen on Figure 5. The radar composite has a temporal resolution of 2.5 min, while the rain-gauge network has a temporal resolution of 10 minutes. Radar data have been aggregated according to the time of interest of the cumulated estimation, assuming stationarity between the sampling periods. Similarly, rain-gauge data have been accumulated when the desired cumulated precipitation estimates exceeded 10 minutes. 5. VALIDATION METHODOLOGY Fig. 3. Bias maps in the TI The performance of the technique has been tested with two different evaluation schemes: cross-validation and crosscomparison. The cross-validation scheme (CV) is based on removing (in turn) any rain-gauge location from the conditioning data, in order to compare the estimate of several sequential simulations of the algorithm (Ẑ) to the value registered by the removed gauge (Z). 5

6 For this scheme, we have used 104 rain-gauges (colored in red in Figure 5) uniformly distributed over the Swiss territory and submitted to high quality control by MeteoSwiss. The cross-comparison scheme (CC) compares MPS estimates (Ẑ) that utilize all the previous 104 MeteoSwiss gauges as conditioning data to generate the simulations, with the remaining SwissMetNet gauges (see Figure 5), highly clustered in the Wallis region, that undergo a less strict quality control and are therefore more prone to errors. Both verification approaches however imply the assumption that rain-gauge values are error-free. Several quality parameters have been employed for each test case to evaluate the estimation of cumulated precipitation of the rain gauge-radar merging technique: 1. The mean bias (BIAS) and the mean error (ME) allow to evaluate the presence of systematic errors in the method. A positive (negative) value of the parameters indicates a general overestimation (underestimation). BIAS = 10 log 10 N i Ẑ i N i Z i for Z i > 0 [db] (4) ME = 1 N (Ẑ N i=1 i Z i ) (5) 2. The mean absolute error (MAE) and the root-meansquare error (RMSE) are used to assess the general quality of the merging technique. MAE = 1 N Ẑ N i=1 i Z i (6) RMSE = 1 N (Ẑ N i Z i ) 2 i=1 (7) 3. The median absolute deviation (MAD) gives a measure of the dispersion being less influenced by outliers [38]. MAD = median( Ẑ Z ) (8) 4. Scatter, which is defined as half the distance between the 16 % and 84 % quantiles of the error distribution, is a robust measure of the spread of the multiplicative error of the estimate and is also insensitive to outliers [47]. Since the algorithm estimates precipitation only where it is assumed to be rainy (see section 3), cross-validation and cross-comparison have not been applied at rain-gauge location where both radar and gauge data indicated no precipitation. Fig. 5. MeteoSwiss C-band radars and rain gauge networks. 6

7 6. RESULTS The validation schemes have been applied to two different rainfall events: the first one characterized by stratiform precipitation over the whole Swiss territory (test case 1) and the second one representing a typical summer convective storm that battered both south and north of Alps during the evening of June 7th Test case 1 The first rain event occurred between April 30th and May 1 st 2015 and was characterized by a stratiform rainfall concerning almost all the Swiss territory. In 24 hours, more than 100 L/m 2 of water have fallen in various Swiss regions, with the Arve River in Geneva recording a volumetric peak flow rate that exceeded 900 m 3 /s, the highest since measurements started in In St. Gingolph, at the border between Switzerland and France, the River Morge flooded the village transporting mud and rubbles in the houses, causing damages for several hundred thousands of francs. During the 12h-events analyzed, the radar underestimated precipitation constantly in comparison to rain-gauges, both in flat and mountainous regions. In several districts, especially in central Switzerland and in the Geneva area, the radar underestimation reached more than 20 mm precipitation during 12 h. This can be observed in Figure 6, where rain-gauge measurements have been superimposed with circles on rainfall maps, as well as in Figure 7 where the OLS fitting slope of the scatterplot between rain-gauges and radar (the left-one) shows that the radar underestimation increases by growing precipitation intensity. In Figure 6, we can also observe that the MPS approach tends to overestimate compared to the radar and hence tries to correct the supposed underestimation of the radar measurements. The same behavior can be observed in CombiPrecip estimations, which results are highly correlated to MPS approach. The right-scatterplot of Figure 7 shows a correlation of 90% between the two methods: a similarity that is evident in the image comparison of Figure 6. Fig. 6. Comparison between radar and MPS estimate of cumulated precipitation over 12 hours. For illustrative purposes, cumulated rainfall less than 5 mm are not represented. Fig. 7. Scatterplots between radar,mps and MeteoSwiss estimates with the addition of a kernel density smoother and an arbitrary color scale to enhance features due to the different value frequencies. Red-dashed lines indicate OLS fitting of the data. 7

8 Fig. 8. Comparison between MPS and CombiPrecip estimates of cumulated hourly precipitation with superimposed raingauge measurements. Remarkable differences between the two methods appear only outside the Swiss territory where no gauge measurements are present to inform about possible bias in the radar composite. As a consequence, both methods rely on rain-gauges situated along the Swiss border to estimate bias outside Switzerland. It should be mentioned that CombiPrecip uses all the 200 gauges indicated in Figure 5 to construct the spatial variability model and that this leads to a bigger amount of information mainly in Wallis (due to the dense gauge-network) and in the Geneva region, where the radar composite shows some blocking due to interference of the sky radar of the Geneva airport. In Figure 8, it can be noticed the presence of a mis-calibrated gauge (indicated by an arrow) that has been used as conditioning data for both methods. From the cumulated rainfall maps, it seems that MPS approach may be less sensitive to an erroneous rain-gauge measurement than Combiprecip. However, the overall validation analysis should not be influenced too much by these gauges. In Table 1 are reported the comparison criteria of the two validation schemes as well as the original radar estimation for aggregation time scales of 10 and 60 minutes. The mean errors (ME) of CV and CC schemes for both aggregation time scales show that the systematic underestimation of the radar during the event is well eliminated and that the resulting ME is lower than the resolution of the rain-gauges (0.1mm/10min). It is also immediately noticeable that biases are halved compared to the radar estimate for both aggregation scales, with 60-minutes cumulated estimations showing a better performance in reducing bias than 10-minutes simulations. Table 1. Quality parameters for the test case 1 (stratiform event) over 7h validation analysis ( am, 1 May 2015) Agg. time ME [mm] Bias [db] MAE [mm] RMSE [mm] MAD [mm] Scatter[mm] Correlation [%] Radar 10 min MPS CV 10 min MPS CC 10 min Radar 60 min MPS CV 60 min MPS CC 60 min

9 Fig. 9. Evolution in time of the comparison criteria for aggregation times of 10 minutes and 60 minutes (1 May 2015) This could be explained by the fact that possible errors affecting gauges or radar measurements are compensated by cumulating data over 1 hour. This behavior is also visible in the correlation between gauges and MPS estimations, with an increase of 8 % and 16 % for 10-minutes and hourly estimates respectively. The greater robustness of the 60-minutes estimates seems to appear also in Figure 9 which illustrates the evolution in time of some quality parameters. As a matter of fact, it can be observed a greater variability of the validation scores for the 10-minutes estimates than the hourly ones. Analyzing in detail the scores evolution in time of the two validation schemes (Figure 9), the correlation time series between MPS estimates and gauge data also show that this variability affect more CV than CC scheme. Nevertheless, it should be considered that the removal of a gauge from the conditioning data in CV may lead the algorithm to rely only on one gauge information to correct the possible radar bias that can vary spatially, leading to a kind of extrapolation that not always produces satisfactory results. Concerning this issue, an extended analysis of the gauge density s influence on the estimation accuracy should be conducted. Anyways, the MPS merging approach is shown to improve significantly the stratiform precipitation estimation, although for the 10-minutes aggregation time simulations the improvement is not markedly visible because of the low rainfall quantity involved in this type of hydrometeorological event. 6.2 Test case 2 The second event happened in the evening of June 7th 2015 and was composed by two typical summer convective storms that affected the Swiss Plateau, the western part of Austria, and the southern part of the Alps. Due to the heavy and persistent rainfall, several landslides and floods occurred in the central part of Switzerland, one of which killed a mother and her 5-year old daughter in the region of Dierikon, near Lucerne. In contrast to test case 1, the radar seems to overestimate precipitation if compared to the amount recorded at raingauge stations (Fig.11 right). However, it must be considered that the convective event was characterized by storm cells with localized high rainfall intensity (between 60 and 120 mm/h), as we can observe in Fig. 10. As a consequence, microvariability and wind advection can lead to an underestimation from rain gauges. Comparing CombiPrecip estimates with the MPS approach, in case of large rainfall the MPS method estimates higher precipitation rate (almost 5 mm of precipitation each hour). This could be related to a lower smoothing of the convective cells that inevitably arise with the construction of the spatial variability model of the kriging/cokriging formulations. However, the precipitation spatial patterns represented by both methods are highly comparable, as proven by a global correlation of 96% over the three analyzed hours. This similarity is clearly visible also in Figure 12, where an hourly MPS estimate is compared to the corresponding CombiPrecip evaluation. 9

10 Fig. 10. Illustration of an MPS estimation with 10 minutes temporal resolution compared to the corresponding aggregated radar composite. Rain-gauge measurements are superimposed as circles over the rainfall maps. The validation methodology applied to this convective events case did not show representative results due to the low quantity of available wet gauges for validation and the temporal evolution analyzed. Besides, the wide difference in rainfall quantity over the Swiss territory and the missing of some gauges to register high precipitation, due to microvariability of the cells storms over short aggregation Fig. 12. Comparison between MPS hourly estimate and CombiPrecip. Rain-gauge measurements are superimposed as circles over the rainfall maps. time scales, give a distorted vision of the performance of the MPS approach. However, it must be considered that the flexibility of the MPS approach to move a precipitation pattern extracted from the radar composite to a nearing gauge recording a rainfall-depth in agreement to the radar estimation at a given altitude, should avoid the creation of misleading conditioning data that would alter the estimation. Fig. 11. Scatterplots between MPS and MeteoSwiss estimates using a kernel density smoother with a zoom for hourly cumulated precipitation higher than 40 mm/h. Red dashed-lines indicates OLS fitting of the data. (7 June 2015) 10

11 7. CONCLUSION The approach presented in this report introduces a new and essentially non-parametric statistical method which is able to generate rainfall fields in agreement with rain-gauge measurements and spatial precipitation patterns detected by radar. Our approach is conceptually simple and avoid the fitting of spatial variability model; an operation that can be complicated when one must deal with short aggregation time scales. Moreover, it shows great flexibility in the choice of the temporal resolution of the estimate and that can be interesting for a variety of hydrological applications (ie. flash flood forecasting). According to the analysis of the two test cases presented in this study, the algorithm is very efficient under stratiform conditions and increases remarkably the quality of precipitation estimates compared to the radar alone. It appears also to be less influenced by erroneous gauge measurements. Besides, during convective events, the comparison with CombiPrecip estimates suggests that the proposed approach may be useful to characterize the spatial distribution of precipitation in presence of convective cells with highly-localized rainfall intensity, because it avoids the characteristic smoothing of extreme values that usually appear using classical geostatistical methods. Further work is however required to increase the computational efficiency of the algorithm for possible realtime application. The removal of the 4th dimension of the TI could be achieved by restricting the freedom of moving of the search window in the TI to a given neighborhood from the simulated node. At the same time, the 3rd dimension (and the corresponding bias maps) can be substituted by the use of an invariant-mean distance dissimilarity function. Additional analysis are also needed to assess the sensitivity of the algorithm to the number of gauges used as conditioning data and to verify if the use of additional rain gauges with minor precision and less quality control as secondary conditioning data would improve the accuracy of rainfall estimates. Finally, an evaluation over a prolonged period of time should be conducted to assess the accuracy and robustness of the proposed technique with different types of precipitation. Anyway, the proposed method shows particularly promising results to obviate to the problem of merging radar and gauge estimates, both during stratiform and convective events. REFERENCES [1] Allard, D., Froidevaux, R., Biver, P., Conditional Simulation of Multi-Type Non Stationary Markov Object Models Respecting Specified Proportions Math. Geosci., 38(8), [2] Berne, A., Krajewski, W. F., Radar for hydrology: Unfulfilled promise or unrecognized potential? Journal of Hydrology 51, [3] Chow,V.T., Maidment, D.R., Mays L.W., Applied hydrology. International ed. Singapore: McGraw-Hill, Inc. [4] Cole, S. J., Moore, R.J., Hydrological modelling using rain gauge and radarbased estimators of areal rainfall. Journal of Hydrology 358, [5] Comunian, A., Renard, P., Straubhaar, J., D multiple-point statistics simulation using 2D training images. Computers & Geosciences 40, [6] Creutin, J.D., Delrieu, G., Lebel, T., Rain measurement by raingauge-radar combination: a geostatistical approach. Journal of Atmospheric and Oceanic Technology 5, [7] Creutin, J. D., Borga, M., Radar hydrology modifies the monitoring of flash-flood hazard. Hydrological Processes 17, [8] Chugunova, T., Hu, L. Y., Multiple-point simulations constrained by continuous auxiliary data. Mathematical Geoscience 40, [9] Dell Arciprete, D., Bersezio, R., Felletti, F., Giudici, M., Comunian, A., Renard, P Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments. Hydrogeology Journal 20, [10] Doviak, R. J., and D. S. Zrnic, Doppler Radar and Weather Observations. Dover [11] Erdin, R., Frei, C., Kunsch, HR., Data transforation and uncertainty in geostatistical combination of radar and rain gauges. Journal of Hydrometeorology 13, [12] Frei, C., Schar, C., A precipitation climatology of the Alps from highresolution rain-gauge observations. International Journal of Climatology 18, [13] Gabella, M., Notarpietro, R., Improving operational measurement of precipitation using radar in mountainous terrain. IEEE Geosciences & Remote Sensing Letters 1, [14] Germann, U., Joss, J., Operational measurement of precipitation in mountainous terrain. In Weather Radar: Principles and Advanced Applications. Series Physics of Earth and Space Environment,

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