- Unrestricted Report Combination of radar and raingauge data for precipitation estimation Report for the NFR Energy Norway funded project "Utilisation of weather radar data in atmospheric and hydrological models" Author(s) Jean-Marie Lepioufle SINTEF Energy Research SINTEF Energy Research 2012-01-26
Document history DATE DESCRIPTION 1 2012-01-26 1 2 of 33
Table of contents 1 Introduction... 4 2 Radar combined with conditioned rainfall space-time simulation... 4 2.1 Conditioned rainfall simulation scheme...4 2.1.1 Free simulation of rainfall fields in the meta-gaussian framework...4 2.1.2 Simulation of non-gaussian random field conditioned by point values...5 2.2 Combining radar and rainfall simulation using a Kalman filter; a light theory, from Todini (2001)...6 3 Application to the Rissa radar area covered with hourly raingauges... 8 3.1 Study region...8 3.2 Modeling methodology... 13 3.3 Results... 20 4 Conclusion and discussion... 24 A Raingauge description... 25 B Variogram for gaussian random fields... 27 1 3 of 33
1 Introduction Combination of different precipitations measurement has already been established for the last 10 years. A base of our work can be found in Todini (2001). This work has been done in order to improve some points cited hereafter: - Presence of intermittency - Space-time rainfall process - Non-Gaussian fields - Kriging is an interpolation that only provide smooth field 2 Radar combined with conditioned rainfall space-time simulation 2.1 Conditioned rainfall simulation scheme 2.1.1 Free simulation of rainfall fields in the meta-gaussian framework The simulated variable is assumed to be a random variable entirely defined by its space-time structure (space-time variogram) and its marginal cumulative distribution function (cdf, describing the frequency distribution of the values at point locations). The simulation of random fields characterized by any spacetime structure function is relatively simple as long as the distribution of the simulated variable is Gaussian (Guillot, 1999). The Turning Bands Method (Montoglou and Wilson, 1982) is the method used here to generate random fields. The generation of non-gaussian random fileds is less straightforward but a method was proposed by Journel and Huijbregts (1978) consisting of the two following steps: 1. The simulation of a Gaussian random field, defined by its variables,, that have a Gaussian cdf and a space-time variogram. 2. The transformation of the Gaussian field into a non-gaussian filed by using an anamorphosis function Φ, defined so that the non-gaussian fields can be written as, Φ, (1) An illustration of the use of the anamorphosis function is given in Figure 1. According to Guillot (1999), an anamorphosis funtion Φ can be derived from any cdf. However as Φ may also transform the spatial structure, and more generally, the space-time structure of the Gaussian field, one must properly define in order to obtain the expected space-time structure after the anamorphosis. In equation (1), the anamorphosis function Φ transforming Gaussian to non-gaussian random fields may also transform the space-time variogram γ. It is thus necessary to define a suitable space-time variogram function γ that will be transformed by the anamorphosis into space-time variogram function γ expected for the non-gaussian fields. If the cdf of the variables Z x, t is continuous, γ can be analytically deduced from γ. If the cdf of the variables Z x, t is discontinuous (i.e presenting accumulations of values or atoms), 1 4 of 33
γ has to be adjusted by a trial and error method in order to retrieve γ, after transformation by the anamorphosis function (Guillot, 1999). More detailed are presented in Appendix 1 2.1.2 Simulation of non-gaussian random field conditioned by point values The conditioning of random fields by point values is relatively easy to implement if the simulated variables are Gaussian (Lantuejoul, 1994). A few complex method exist to simulate non-gaussian random fields conditioned by point values (Emery, 2002, ). A simple anamorphosis procedure is presented hereafter: (i) transforming the conditioning non-gaussian values into Gaussian values, (ii) using the transformed values to conditioned the Gaussian simulated fields, (iii) transforming the conditioned Gaussian simulated fields into non-gaussian random fields, based on equation (1). The first two steps are described hereafter. Transformation of non-gaussian conditioning values into Gaussian values. In the case of simulated random fields described with continuous non-gaussian cpd, the transformation of the non-gaussian values conditioning can be achieve by using the inverse of the anamorphosis function:, Φ, (2) where represent the vector coordinates of the raingauges. In the case of simulated random fields with discontinuous non-gaussian cdf, the anamorphosis function is no more injective. Equation (2) is only valid on the continuous interval of cdf. For the transformation of the atom values, several Gaussian values can be potentially assigned. An illustration of this problem is given in Figure 2 for the case of a distribution with an atom at zero, which is the case for the application to rainfall over the Rissa radar area, Norway, presented in this report. The vector of conditioning points, is separated into, a vector of non-zero values and the complementary vector, of zero values. Every elements of, is in the continuous part of the cdf function thus the corresponding Gaussian conditioning values can be deduced from Equation (2), as followed:, Φ, (3) We recall here two methods for estimating the vector, : (1) The first method consists of assigning to all elements of the vector, the unique value, where represents the Gaussian distribution function and represents the frequency of positive values 0. (2) The second method consists of randomly simulating the elements of the vector, inside the interval ;. A third more complex and rigorous method exists. It is based on the Gibbs sampling algorithm. The vector, is estimated by iteratively kriging the complementary vector. This method should be implemented in a near future in order to get more robust simulation output. 1 5 of 33
Conditioning of Gaussian random fields by several points values. The usual method to generate Gaussian random fields conditioned by point values is described in several papers (Delhomme, 1979; Lamtuéjoul, 1994). It is based on the kriging of a field to which a simulated kriging error field is added. This is a processed in three steps: (1) Kriging the conditioning point values over the space-time domain D to obtain the field, (2) Simulating a kriging error field Σ,. (3) Adding Σ, to, to obtain a Gaussian random field, conditioned by the point values. The simulation of the kriging error field Σ, is achieved by: (1) Free simulating a random field, (2) Extracting from, the values at the conditioning point locations and kriging the extracted point values to obtain the field,. (3) Computing the difference between, and its kriging estimate,. Finally the Gaussian field conditioned by the point values can be written as:,,,, (4) Delhomme (1979) shows that the field, is a conditional simulation since (i), is characterized by the same variability as the simulated variable, (same cdf and spatial structure), (ii)it is merged with the conditioning points since the kriging is an exact interpolator, meaning that the error at the conditioning station is equal to zero. 2.2 Combining radar and rainfall simulation using a Kalman filter; a light theory, from Todini (2001) Bias and variance between precipitation estimated from weather radar and precipitation estimated using block-kriging Let us assume a random field describing the spatial radar field with a pixel resolution (i.e 1km²) at time. The information from the raingauges network is provided by the conditioned rainfall simulation. Pereira Filho and Crawford (1997) and then Todini (2001) point out that it is reasonable to assume that the two types of measurement, radar and raingauges, provide independent measures of the same unknown quantity. Todini (2001) added an assumption that precipitation estimated from block-kriging has the same variability as the precipitation estimated from weather radar. We assume here that the rainfall simulation has the same variability as the radar data. In order to avoid the problem of intermittency, we decide to work in the Gaussian world, therefore to transform all the values into a Gaussian distribution usig the anamorphose. Thus we worked with Gaussian radar data ), and Gaussian conditioned rainfall simulation. The difference of the two estimated rainfall fields is: (6) 1 6 of 33
Incorporating the true but unknown random field, it becomes: (7) The expression (3) is then used to assess the stochastic proprieties of the errors from the radar rainfall fields: - the expectation: - - the variance - We assume that conditioned simulation is a non-biased method ( 0), which enables to provide the covariance of the error prediction, (4) and (5) become: (10) (8) (9) (11) Assuming that both precipitation estimated from weather radar and conditioned simulation respect the second order spatial assumption for a given weather type, therefore and can be estimated from the equation (2) and the times series of radar data and conditioned simulation values. Combining the two sets From a Bayesian point of view and using the Kalman filter equation, Todini (2001) shows that if one considers the radar image as the a priori estimate information, conditioned simulation values can be used to estimate the a posteriori rainfall values at each time step, by: (12) And the a posteriori error prediction variance: (13) Where is the Kalman gain: (14) 1 7 of 33
3 Application to the Rissa radar area covered with hourly raingauges 3.1 Study region The radar data, the kriged values, the combination using the method developed by Todini (2001) and the combination using the Kalman filter on radar data and conditional algorithm are used to simulate rain fields observed between 2006 and 2010 (5years) over the Rissa radar area by a raingauge network from different data providers. The network consists of 28 raingauges providing data at hourly time-step. The main characteristics of the gauges are presented in Appendix 1, Table1. The simulation is carried out on a 210 220 km² subarea where the density of the network is the highest (Figure 1) Nineteen raingauges have been selected inside this simulated area to be used to compute the statistical characteristics required as input parameters of the rainfall stochastic model. The nine unselected raingauges have either too short time series to provide a robust statistics (raingauges #1 to #6) or represent a nonhomogeneous signal to the one measured by the other gauges (#9, #20, #24). Statistical characteristic maps of the non-zero rainfall variance, the percentage of intermittency and the non-zero mean are respectively represented in the Figures 2, 3, 4. In those figures, rainfall characteristics don t look very homogeneous. But due to a too few amount of raingauges it has been decided to keep all those raingauges. In future work the use of weather types classification would help for a better respect of the homogeneity within the signal. Twenty six raingauges have been selected to condition the simulations by point observations. Only gauges with no values have been evinced. Using raingauges measuring a rainfall signal different from the major part of the raingauges will enable to bring back a part of heterogeneity within a simulation based on stationarity. 1 8 of 33
Figure 1:location of the different raingauges used in the study. Statkraft (filled triangle), Orkla (filled circle), met.no (filled square), Rissa radar location (blank circle) 1 9 of 33
Figure 4: Figure 2: Variance of the non-zero rainfall 1 10 of 33
Figure 3: Expectation of the rainfall indicator 1 11 of 33
Figure 4: Mean of the non-zero rainfall 1 12 of 33
3.2 Modeling methodology Different products of rain fields have been used. First, it has been produced time series over the whole period 2006-2010: (i) radar data, (ii) spatial kriged rain fields, (iii) spatial rain field provided by a combination of radar data and raingauges values using the method shown in Todini (2001). Secondly, others product have been produced over one chose event from the 19 th July 2009 at 05:00 to the 21 st of July 2009 at 05:00: (iv) space-time rain fields provided by a combination using the Kalman filter on the radar data and a punctual conditioned rain field, (v) temporal expectation of a space-time rain fields provided by a conditioned simulation using the climatological rain distribution parameters, (vi) temporal expectation of (iv). For points (iv), (v), (vi) a rainfall distribution model and a space-time structure model are required. For points (ii), (iii) a spatial structure is required which is only a restriction of the space-time model to two dimensions. Point rainfall distribution The determination of the distribution of the simulated variable is required to implement the rainfall model. From previous studies (on daily rainfall), one can assume the rain depth cpd can be represented by the mixture of an atom at zero and a distribution function representing the nzr cpd. Figure 5 represents quantilequantile plots of different tested distribution functions. Functions such as Gamma, Pearson III, Generalized Pareto Log-Normal and Inverse Gaussian (Folks and Chhikara, 1978) have been tested. All the estimated parameters are presented in the Table 3. No result has been satisfactory enough to a take a robust decision about the choice of the distribution function. So we chose the distribution according to two constraints: i- Distribution functions with a number of parameters greater than two have been evinced. Indeed, the rainfall generator is based on a two-parameter model (mean and variance). In order to use a three-parameter distribution function, we should know the evolution of the variability in space and time according to a spacetime variogram taking into three parameters. ii- As expressed in part 2.1.2, the rainfall simulator generates first Gaussian values that have to be transformed into an asymmetric distribution using anamorphose. In appendix 1, it has been shown that all combination of theoretical distribution functions and theoretical variogram model are not possible in the correction presented in Guillot (1999). Matheron (1989), Armstrong (1992) and Leblois et al (2012 under review) highlighted as well this problem. According to i- and ii-, the log-normal distribution has been selected. Looking at the figure 7, we are aware that the simulation will provide slightly higher values than those observed. The distribution function can be written as: 1 1 0 0 (15) The two parameters of the Inverse Gaussian distribution are: - the mean: (16) 1 13 of 33
- the standard deviation: (17) where et are the mean and the variance of the non-zero values within the entire dataset. These two parameters are given in table 3. The frequency of positive value has been also calculated using the entire data, and is equal to 0.344: the ratio of rain over the entire dataset is 34.4 %. Due to non-homogeneity of the parameters in space (figures2, 3 and 4), an alternative would be to make parameters varying in space according to the climatological map build using kriging. We decide to keep a set of parameters homogeneous in space in order to not force simulation by parameters got from interpolation and thus avoiding getting parameters indirectly from observation tools. Figure 5: quantile-quantile plot of non-zero rainfall values with different theoretical punctual distributions 1 14 of 33
Table 1: parameters of the different punctual theoretical distributions Distribution p1 p2 p3 Gamma Shape=0.261 Scale=1.450 - Inverse Gaussian Mean=0.379 Shape=0.099 - Pearson III Mean=0.379 Variance=0.550 Skewness=6.261 Generalized Pareto Xi=-0.243 Alpha=0.331 k=0.325 Log-Normal Mean=0.379 Standard deviation=0.741 - Space-time variogram, and its transformed into the Gaussian world The space-time variogram used to represent the hourly rainfall are based on the exponential function. The parameters of this variogram have been estimated using the experimental variogram computed using both the space variogram cloud and the temporal variogram cloud based on the hourly raingauge values. Results for the spatial variogram are presented in figure 6, and structures are defined as: 0.229 exp 35.30 (18) Variance parameter in (9) is different from the one computed from the data set (0.224 mm²/hour²). Indeed, the estimated variance parameter value is influenced by higher variographic values as a consequence of the spatial heterogeneity. Despite a presence of anisotropy viewable on the variographic map (figure 7), no coefficient has been injected in the spatial variogram. This climatologic anisotropy represents a climatologic feature of the rainfall over Norway. However anisotropy is highly related to the weather and then vary from one day to another. Further works on the variography should be done to determine the different anisotropy coefficient according to the different weather types. Results for the temporal variogram are presented in figure 8, and structures are defined as: 0.218 exp 3.896 (19) Due to the presence of a nugget and a slightly trend reaching the variance at the infinity, fitting the temporal cloud variogram has been more awkward. The nugget hasn t been taken into account. Due to the seasonality, varying atmospherical conditons and other climate variation, the variance is lower than the one from the entire dataset. Estimated parameters values are resumed in table 2. Given that the space-time variogram is one function characterized by a single variance parameter, we chose this value equal to the variance value within the entire dataset: 0.224 mm²/hour². The space-time variogram reads: 1 15 of 33
0.224 exp 35.30 3.896 (20) After having established correction of the variogram using the Hermite polynomial expansion (Appendix 2), the space-time variogram of the Gaussian random fields becomes:, exp 28.812 2.608 (21) As information, spatial and temporal experimental variograms are shown in figures 9 and 10 with the respective variograms resulting from the Hermite polynomial expansion. Figure 6: Spatial cloud variogram and its fitted model 1 16 of 33
Figure 7: spatial cloud variogram 1 17 of 33
Figure 8: temporal experimental variogram, and its fitted models (dashed line: exponential, dot: spherical) Table 2: Estimated parameters of the spatial and temporal variogram before and after using the HErmite polynome expansion Spatial variogram Temporal variogram σ² (mm²/hour²)range (km) σ² (mm²/hour²)range (hour) Natural world 0.229 35.309 0.218 3.896 MGRF 1.036 28.812 0.945 2.608 1 18 of 33
Figure 9: Experimental spatial variogram (circle) and its fitted model (line), its transformation after using the Hermite polynom expansion (dashed line), and as information the normalized fitted variogram with the variance (blue line) and the cloud values normalized with the variance (triangle) 1 19 of 33
Figure 10: Experimental temporal variogram (circle) and its fitted model (line), its transformation after using the Hermite polynom expansion (dashed line), and as information the normalized fitted variogram with the variance (blue line) and the cloud values normalized with the variance (triangle) Other modeling features The rain field is simulated on a regular grid with a 1 km² resolution. In order to establish a coherent comparison between kriging, conditioned simulation and the combination using the conditioned simulation with the Todini method, we produced 100 runs of the event occurring between the 19 th July 2009 at 05:00 to the 21 st of July 2009 at 05:00 for the conditioned simulation and the combination using the conditioned simulation with the Todini method. All of our scripts have been run on R (x32) using a laptop with 4 processors Intel Core i5.2540m cpu 2.60Ghz. 3.3 Results It has needed 9.5 hours to produce two daily time series between 2006 and 2011: one set of kriged rainfall fields and one set of Todini rainfall fields. It has needed 9 minutes to produce two hourly time series for the chosen event: one set of kriged rainfall fields and one set of Todini rainfall fields. 1 20 of 33
It has needed 10 minutes to produce two hourly time series for the chosen event: one set of conditioned simulation rainfall fields and one set of the combination conditioned simulation with Todini rainfall fields. As an illustration, a picture of each of the output at the date 20 th July 2009 at 05:00 is presented in Figure 11. Though radar picture (figure 11a) is not the truth, it is fair to choose it as a reference for a qualitative comparison with others products. All the pictures have a common scale of value. First of all, one can see that the kriged product (figure 11b) provides smoother information with an improvement with Todini method (figure 11c). The two conditioned simulation products (figure 11 d and e) provide realistic variability. They respect all the values from the raingauges network, but provide very high values in comparison with the radar. This last effect is not a mistake but results from the characteristic of a conditioned stochastic generator: it provides a probable rainfall scenario according to the statistics extracted from the raingauges data, the conditioning data, and the ungauged areas. The combinations between the conditioned simulation and Todini method (figure f and g) provide realistic variability. Values close to the raingauges are more trusted and decreasing very high values. However, these product represent two probable scenarios. Kriging is an interpolation method which provides expectation values in between each raingauge. In figure 12, the expectations of 100 runs of respectively the conditioned simulation (figure 12 b) and the couple Conditioned simulation and Todini (figure 12c) have been computed. The field resulted from the expectation of the conditioned simulation (figure 12 b) gives a similar shape to the kriging but with higher precision. The expectation of the couple condition simulation and Todini (figure 12 c) can be used to have an idea of the possibility of the Todini combination: the radar information is too alterated. a 1 21 of 33
b c d e f g Figure 11: Different outputs at the date 20 th July 2009 at 05:00, a- radar, b- blocked-kriged, c-todini (2001), d- Conditioned simulation run 21, e- Conditioned simulation run 72, f- Combination conditioned simulation with Todini (2001), run 21, g- Combination conditioned simulation with Todini (2001), run 72, 1 22 of 33
a b c Figure 12; Comparison between the kriged fields and different time expectation of 100 runs at the date 20 th July 2009 at 05:00, a- radar, b- block-kriged, c- expectation of the conditioned simulation, d- expectation of the combination of the conditioned simulation and Todini (2001) 1 23 of 33
4 Conclusion and discussion This report synthesizes the different rainfall products we can provide: radar, kriged data, combination of radar and raingauges data using the Todini method, stochastic simulation conditioned with raingauges, combination of radar and conditioned stochastic simulation. All these product can be produce (except raw product as radar data) using open source R scripts. Cpu speed can be increased by rewriting the scripts in C. Some of these products look similar and some other are very different. Hereafter is explained which product to use according to the utilization of the user. Radar data provides excellent spatial information with a realistic dynamic. It can be used as input into a distributed hydrologic model to optimize a hydropower network. Kriging is an optimal interpolation which respects the amount of rain over the studied area. The areal mean of this product can be used as input into lump hydrological model. The expectation of the conditioned simulation can be used as well in this case. Furthermore, this last product respects a temporal correction of the rainfall phenomena. The combination using the Todini method is a fast method to combine both radar and raingauges. It enables to provide quickly long time series to use as input into hydrological model. The conditioned simulation provides several probable scenarios coherent in space and in time. This product enables to take into account some unobserved but probable scenarios. This product is useful for testing extreme non-observed event. The combination of the conditioned simulation and Todini method highlights the too high weight the combination method put on the rain gauge values. The weights assigned to the radar and the interpolated rain gauge values are proportional to the uncertainty of each of them. The weights are calculated for each pixel. However, the uncertainty within the radar data is unknown. It is indirectly estimated from the raingauge data and chosen uniform in space. In order to improve the combination of several rainfall measurements, it is highly recommended to provide independent information about the uncertainty in the radar measurements. I.e. a simple conceptual uncertainty feature related to the distance to the radar location would be a beginning. 1 24 of 33
A Raingauge description Table 1: name, belonging, coordinates and elevation of each hourly raingauge of the network # name dataset x(m utm32v) y(m utm32v) z (m) 1 9580 eklima 590 6910 482 2 61630 eklima 458 6900 579 3 63420 eklima 477 6950 6 4 67280 eklima 560 6980 299 5 67560 eklima 579 6980 127 6 68860 eklima 573 7030 127 7 69150 eklima 594 7040 40 8 BERKÅK orkla 552 6970 425 9 KVIKNE orkla 566 6940 550 10 LUSO orkla 575 7000 415 11 NERSKOGEN orkla 536 6970 665 12 SVARTELVA orkla 559 7060 124 13 SYRSTAD orkla 537 6990 125 14 YA orkla 580 6940 127 15 ØVREDØLVAD statkraft 560 6930 848 16 ÅMOTE statkraft 541 7000 227 17 HERSJØEN statkraft 610 7010 420 18 LØDØLJA statkraft 635 7000 540 19 NESJØEN statkraft 643 6990 725 20 SAKRISTIAN statkraft 673 6970 860 21 SELLISJØEN statkraft 636 6990 510 22 STUGUSJØEN statkraft 643 6980 630 23 SYLSJØEN statkraft 660 6980 840 24 AURA_AURSJØEN statkraft 476 6920 873 25 AURA_EIKESDAL statkraft 459 6930 61 26 AURA_HÅKODALSELV statkraft 480 6920 913 27 SVOR_SOLÅSVATN statkraft 494 6990 341 28 TROL_GRÅSJØ statkraft 507 6980 476 1 25 of 33
Table 2: # _ global 0.703 0.311 0.383 6.261 0.208 1.837 0.079 824773 1 NA NA NA NA NA NA NA NA 2 0.150 0.238 0.293 4.446 0.572 0.511 0.167 501 3 NA NA NA NA NA NA NA NA 4 0.385 0.107 0.495 3.139 0.638 0.776 0.316 2273 5 0.597 0.146 0.474 11.19 0.376 1.258 0.178 3175 6 2.340 0.073 0.890 4.476 0.339 2.624 0.302 2739 7 0.877 0.358 0.439 16.94 0.220 1.996 0.096 13792 8 0.374 0.287 0.302 8.632 0.244 1.237 0.0741 10070 9 20.21 0.251 0.465 67.93 0.01 43.39 0.005 8827 10 0.731 0.245 0.501 9.883 0.343 1.459 0.172 8606 11 0.443 0.323 0.298 10.85 0.201 1.486 0.060 11340 12 0.754 0.384 0.522 4.140 0.361 1.445 0.188 13492 13 0.553 0.255 0.413 5.691 0.308 1.338 0.127 8960 14 0.381 0.343 0.227 15.27 0.135 1.680 0.030 12028 15 0.246 0.305 0.201 12.83 0.165 1.221 0.033 10699 16 0.453 0.309 0.356 6.409 0.280 1.272 0.099 10852 17 0.441 0.412 0.381 10.11 0.329 1.157 0.125 13984 18 0.467 0.293 0.452 5.615 0.437 1.032 0.198 10182 19 0.467 0.339 0.376 5.699 0.303 1.241 0.114 11678 20 4.997 0.263 0.311 75.16 0.019 16.057 0.006 8996 21 0.469 0.291 0.402 6.234 0.345 1.165 0.1393 10039 22 0.396 0.286 0.371 5.648 0.347 1.067 0.1292 9892 23 1.834 0.413 0.213 103.04 0.02 8.607 0.005 14079 24 6.426 0.292 0.340 83.75 0.018 18.88 0.006 8945 25 0.837 0.283 0.510 6.182 0.311 1.638 0.159 9736 26 0.743 0.665 0.209 21.61 0.058 3.552 0.012 11215 27 0.863 0.398 0.639 5.259 0.472 1.351 0.302 13478 28 0.579 0.538 0.395 6.261 0.270 1.464 0.107 17417 1 26 of 33
B Variogram for gaussian random fields Relation between the two variograms and Let be a grf such that. Let denote the Gaussian univariate cdf, (resp ) the variogram of (resp ), then is related to by the following relation (Kolmogorov and Fomine, 1974; Rivoirard, 1994; Guillot, 1999):! (A1) where the are the coefficient of the Hermite polynomial expansion of. Hermite polynomials From Abramowitz and Stegun (1972), we get the relation between the physicist and probabilistic Hermite polynomials : 2 1 2 (A2) where the probabilistic Hermite polynomial is: 1 (A3) where is the Gaussian density and its n-th order derivative. Hermite polynomials expansion If Φ is square itegrable for the Gaussian density, then the following represenatation holds: Φ Φ H! (A4) where Φ is the inner product of Φ with H : Φ Φ H (A5) From Rivoirard (1994), we have: 1 27 of 33
Cov Φ,Ψ Φ ψ C! (A5) Figure A1: Hermite polynomials, n=1 to 5 1 28 of 33
Figure A2: Evolution of the variogram with values respecting the Inverse Gaussian distribution with increasing coefficent of variation (1 to 4) against the variogram with Gaussian values 1 29 of 33
Figure A3: Variogram of a Inverse Gaussian field (dashed line) computed from the Hermite polynomial expansion from a variogram of a Gaussian field whose parameters are equal to one. 1 30 of 33
Figure A4; Evolution of the variogram with values respecting a mixture of a Inverse Gaussian distribution with a mode at zero with increasing coefficients of variation (1 to 4) against the variogram with Gaussian values 1 31 of 33
Figure A5: Variogram of field described by a mixture distribution (mode at zero and an Inverse Gaussian distribution) (dashed line) computed from the Hermite polynomial expansion from a variogram of a Gaussian field whose parameters are equal to one. 1 32 of 33
Figure A6: Figure A6: Non-realistic temporal variogram for a Gaussian field estimated using the Hermite polynomial expansion from a field described by a mixture distribution (mode at zero and Inverse Gaussian distribution) 1 33 of 33
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