SINTEF Energy Research

SINTEF Energy Research

2012-01-26

TR A7187- Unrestricted

Report

Inference of a rainfall space-time

covariance model

Report for the NFR Energy Norway funded project "Utilisation of weather radar data in

atmospheric and hydrological models"

Author(s)

Jean-Marie Lepioufle

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Document history

VERSION DATE VERSION DESCRIPTION

1 2012-01-30

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Table of contents

Part 1: inference of a space-time variogram related to an elemental process using raingauge data ......................... 5

1 Introduction .................................................................................................................................................................................................................... 5

2 A space time random process representing elemental rainfall ................................................................................................. 6

2.1 Elemental rainfall as a stationary process of order 2.................................................................................................................6

2.2 Space-time variogram as a function of the Lagrangian distance only ...........................................................................7

3 Regularization of the elemental space-time variogram .................................................................................................................. 7

3.1 Notation ......................................................................................................................................................................................................................7

3.2 Without advection ...............................................................................................................................................................................................8

3.3 With advection .......................................................................................................................................................................................................8

4 Consequences of time regularization on the shape of the spatial variogram ................................................................ 9

4.1 Without advection ...............................................................................................................................................................................................9

4.2 With advection .................................................................................................................................................................................................... 10

5 Inference of the elemental space-time variogram ........................................................................................................................... 11

6 Case study .................................................................................................................................................................................................................... 12

6.1 Dataset .................................................................................................................................................................................................................... 12

6.2 Inference of spatio-temporal variogram using spatial experimental variograms ................................................ 16

6.3 Limitations due to algorithm ..................................................................................................................................................................... 17

6.4 Limitations due to statistical heterogeneities ............................................................................................................................. 18

7 Conclusion .................................................................................................................................................................................................................... 18

Part 2: inference of a space-time marginal distribution related to an elemental process, a theoretical approach ........................................................................................................................................................................................................................ 19

1 Introduction ................................................................................................................................................................................................................. 19

2 A space time random process representing elemental rainfall .............................................................................................. 19

2.1 Elemental rainfall as a stationary process of order 2.............................................................................................................. 19

3 Marginal distribution of a time-regularized random variable .................................................................................................. 20

3.1 Notation ................................................................................................................................................................................................................... 20

4 Marginal distribution encompassing several regularized variables at different time aggregation ............ 20

4.1 Notations ................................................................................................................................................................................................................ 20

5 Inference of the elemental space-time marginal distribution ................................................................................................. 21

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6 Marginal distribution encompassing several regularized variables at different spatial aggregation and the inference of the elemental space-time marginal distribution .............................................................................. 22

6.1 Notation ................................................................................................................................................................................................................... 22

6.2 Inference of the elemental space-time marginal distribution ............................................................................................ 22

7 Conclusion, limitations of the method and future adaption ...................................................................................................... 23

8 References ................................................................................................................................................................................................................... 23

A Extension variance ................................................................................................................................................................................................. 25

B Dispersion variance ............................................................................................................................................................................................... 27

C Raingauge description ......................................................................................................................................................................................... 28

APPENDICES

A Extension Variance

B Dispersion variance

C Rain gauge description

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Part 1: inference of a space-time variogram related to an elemental

process using raingauge data

1 Introduction

A good characterization of rainfall variability in space and time is necessary for many studies in meteorology, climatology or hydrology. Depending on the addressed questions and on the type of data used, the rainfall amounts are accumulated over time durations ranging from minutes or hours to years or decades. Geostatistics is among the possible mathematical theories used for interpolating or simulating rainfall fields that take into account the so-called structure of the fields modeled by a variogram or a covariance function (Journel and Huijbregts, 1978). In geostatistical jargon, the accumulation over time is called regularization and the time duration over which we accumulate is called time support. Analyzing the structure of rainfall fields regularized over increasing time supports, different studies experimentally showed that the range of the variograms, or distance of decorrelation, increases while the sill of the variograms, or free variance, decreases (see for instance Lebel et al., 1987). When respecting these two conditions, variograms taken for different durations are thus empirically considered as “realistic”. The aim of this study is to provide a theoretical framework of coherence between the field structures of rainfall accumulations over different durations. Following the initial reasoning of Journel and Huijbregts (1978), we assume there is a hypothetic underlying space-time rain process and we consider that the spatial variogram for a given duration of accumulation is a regularized form of the elemental space-time variogram of this process. Various published studies bring contributions to this issue. Kyriakidis and Journel (1999) and Ma (2008) review many different stochastic models used in environmental science to extend space statistics to time dimension. Since the pioneering work of Zawadzki (1973), different space-time structure models including rain cell advection have been proposed (see for instance Lebel et al., 1998; Mellor, 1996 or Shah et al., 1996). Gneiting (2002) underlines the link between advection and non-full symmetry of space-time variograms meaning that the cross-terms of the space-time variogram are direction dependent. Bacchi and Kottegoda, 1995, derive the spatial covariance of the time-aggregated underlying rain process and consider this theoretical formulation hard to apply in the absence of known elemental covariance function. Ali et al. (2003) established spatial variogram models evolving coherently with the accumulation duration. In their study, the underlying rain process is implicitly the rain event, which is typically less than one day in Sahel and which also corresponds to the basic available data. The correlation in time between two events is neglected and the intermittency is explicitly taken into account, being a dominant factor of anisotropy. The variogram coherence framework proposed for Sahelian conditions (western Africa) cannot be extended to daily accumulations under extra-tropical atmospheric conditions, where meteorological situations have a greater temporal continuity. Capitalizing on the above contributions, this paper provides theoretical and practical guidelines to conduct rainfall variography when a coherence of the variograms is desired for different time resolutions. The basic assumptions are i) the existence of a unique stationary process representing the rain elemental variability in space and time, and ii) a constant unidirectional advection. In Section 2, we mathematically define an elemental space-time process incorporating both a Taylor velocity (1938) describing the internal small-scale

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rainfall dynamics and an advection velocity featuring the large-scale entrainment of precipitating areas. Section 3 shows how the variogram of the elemental process can be regularized over time to derive rainfall variograms for different durations. In Section 4 we relate the properties of the regularized variograms (sill, range and shape) to the accumulation duration. In Section 5 we propose an inference method for the parameters of the elemental variogram. Section 6 gives presents our case study case of the raingauge network over the Rissa radar area in Norway. Conclusions are grouped in Section 7.

2 A space time random process representing elemental rainfall

2.1 Elemental rainfall as a stationary process of order 2

We consider the rainfall field as one realization of a random function (RF) in a 3D domain: two dimensions in space and one dimension in time. The RF is written:

, , ∈ , ∈ (1)

where is the location in (the geographical) space, and the location in time. These real-world coordinates being Eulerian, they will latter get a subscript where needed. We assume the rainfall process is homogeneous in both space and time. Formally we choose it to be member of a second order stationary RF family. For such a RF the expected value (μ) and the variance ( ) exist and are constant over the space-time domain. The structure will be described by the space-time variogram , , defined by:

,12

, , (2)

where , and , are all the couples of points separated in space by the vector and in time by the time lag . In order to describe the elemental variability of rainfall, we consider, among all the processes introduced above, a family of RF verifying the following two features:

Correlation functions in space and time having the same shape but having distinct characteristic scales (in space) and (in time) representing basically the “internal” dynamics of the precipitation system.

A uniform (in space) and constant (in time) advection velocity introducing a geographic shift that is linearly increasing with time and that represents the “external” movement of the precipitation system.

These two features introduce a transformation from Lagrangian (subscript where needed) to equivalent Eulerian coordinates as follows (here is considered parallel to the axis, with no loss of generality):

. /

(3)

It is then quite straightforward to see that, starting from an underlying 3D-isotropic 2nd-order stationary RF defined in in Lagrangian coordinates , , , we define a family of RF in Eulerian coordinates that

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satisfy the above features imposed to elemental rainfall variability. One of the three dimensions of our 3D space is converted into a time axis and a uniform constant advection is introduced. The ratio / has the dimension of a velocity, and is the so-called Taylor velocity, noted . The inverse / is called anisotropy coefficient and will be denoted α in the following. The Taylor velocity and the advection velocity are distinct parameters.

2.2 Space-time variogram as a function of the Lagrangian distance only

If we consider two real-world points of Eulerian coordinates , , and , , , separated by , where has the following components:

(4)

and , the distance of these points in the underlying Lagrangian world will be

⁄ or, in a more compact form, ⁄ . The space-time variogram of our model can thus be expressed as the following function:

, ⁄ (5)

where φ is any function suitable to be the 3D variogram of the elemental process in the underlying Lagrangian space. In other words the Eulerian space-time variogram can be expressed as an ordinary variogram applied to an equivalent Lagrangian distance.

3 Regularization of the elemental space-time variogram

The elemental space-time variogram defined in (5) can be regularized over different time supports . The regularized variograms can be directional in either space or time. In this section, we first define spatial variograms of the regularized variables. We start without advection to keep the parallel with the so-called grading of mining applications (Journel and Huijbregts, 1978). Advection is added in a second step assuming the advection velocity and the displacement vector have the same direction. Finally, we show how to develop the temporal variograms of the regularized variables.

3.1 Notation

A family of second order stationary RF , in both spatial and temporal domains is obtained by regularizing over a chosen duration the RF , representing the elemental rainfall:

,1

,∈

(6)

From the second order stationarity of , , we find identical expectation for Z x, t and , :

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,1

,∈

1

∈

(7)

Our goal is then to calculate in a first step the spatial variogram of the regularized process, namely:

, 012

, , (8)

using the elemental space-time variogram , given by expression (5) with and without advection. In a second step, the same type of derivation is used to calculate the temporal variogram of the regularized process , .

3.2 Without advection

In absence of advection, the regularized variogram , 0 is related to , using the grading expression developed in Journel and Huijbregts (1978), and to be found also in Wackernagel (1995) with an approach using the so called extension variance , (cf. Appendix 1). The -block spatial variogram (8) reads as an extension variance of the regularized function , with

the regularized function separated by a distance :

, 0 , , , (9)

The average value, , of the elemental variogram, , between -blocks separated by a distance is computed through the following double integration:

,1

,∈∈

1′ / ′

∈∈

(10)

When 0, the above expression describes the average value of the elemental variogram within a -block:

,1

′ / ′∈∈

(11)

3.3 With advection

By analogy with the expression (10), the -block spatial variogram is elaborated with exactly the same

algorithm, but the Lagrangian distance changes according to advection:

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, , 0 , , (12)

with:

,1

′ ′ / ′∈∈

(13)

and

,1

1/∈∈

(14)

In general these integral calculations are not tractable analytically. They are numerically easy but somewhat expensive in regard to the double integrals. However geometric considerations can lead to simplification: one integral being replaced by appropriate weighting.

4 Consequences of time regularization on the shape of the spatial variogram

The time regularization of the space-time elemental variogram modifies the sill, the range and the shape at the origin of the regularized variograms in space. In order to better distinguish the sources of these changes we successively examine the cases without and with advection.

4.1 Without advection

Although the below results are independent of the model used for the elemental variogram, the choice of a spherical model helps reading the range evolution better than models reaching their range asymptotically, like the exponential variogram. Let’s use, without loss of generality, a spherical model with unit spatial and temporal ranges, and unit variance as our elemental space-time variogram. Figure 1 represents different regularized variograms over durations 0, 0.4, 0.8, 1.2, 1.6, 2 . The time accumulationD 0 represents the elemental space-time variogram with 0. When the distance reaches the range 1, the variograms , 0 reach their sill, equal to the variance of the cumulated process. Three main observations can be made. First, the variance is equivalent to a dispersion variance. It decreases when the regularization time increases. Appendix 2 develops the expression of the dispersion variance of a regularized RF , within a domain :

/ ./ ./ (15)

The term ./ is constant and represents the dispersion variance of the RF , over the domain .

The term ./ is equivalent to , . It represents the dispersion variance of the RF , within a -

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block. As illustrated in Figure 1, the term ./ increases when the regularization time increases. Thus,

/ decreases when increases, like the variance . Second, the range is independent of the accumulation time . This property is named spatial scale invariance by Lebel and Le Barbé (1997) and Ali et al. (2003). Third, the shape of the variograms at short distance evolves with the regularization. Below the linear slope of the elemental spherical variogram, the slope at the origin of the regularized variograms diminishes with increasing accumulation times. This effect of homogenization is consistent with previous studies by Krajewski et al. (2003) and Ciach and Krajewski (2006). This result invites using a regularized spherical variogram to model accumulated processes. Although the shape of the regularized variograms suggests the well-known Gaussian model as a possible candidate, we avoid its use because the Gaussian model analytically propagates smoothing properties at long range.

Figure 1: D-block spatial variograms computed from an elemental space-time variogram based on a spherical model, with unit variance and ranges in time and space ( ) and for six accumulation durations , . , . , . , . , .

4.2 With advection

Figure 2 displays spatial variograms , , 0 , for the same set of values as above but now in the case of

an advection velocity equal to 1 in normalized time and space units. The dispersion variance / decreases with increasing regularization time as without advection but it also depends on the advection .

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Adapting expression (21) to the presence of advection, the dispersion variance of the regularized RF , within the domain is expressed by:

/ ./ ./ (16)

The first term ./ is independent of and remains equal to . The term ./ is equivalent

to , given by expression (14), it depends on a factor 1/ and it increases with an

increasing advection velocity. In turn, the dispersion variance / decreases either when the regularization time increases or when the advection decreases. As also shown in Figure 2, the range of the regularized variograms increases with increasing time durations.

According to expression (13), the term , depends on and influences the shape of regularized variogram. When the advection and the displacement are in the same direction, the range of the -block spatial variogram increases by the value| . | over the space range of the elemental variogram, which reads:

| . | (17)

Figure 2: Same set of D-block variograms as in Figure 1 with a unit advection velocity ( )

5 Inference of the elemental space-time variogram

In practice, the availability of rainfall data regularized over different durations gives access to a twofold set of -block spatial and temporal variograms. The -block spatial variograms are merely climatologic omni-directional experimental variograms:

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1 12

, , (18)

where represents the number of pair of rain gauges belonging to the distance class , represents the number of time steps within the available climatological time series, and represents the rainfall accumulation over a time step computed using a moving temporal window. Assuming that these -block spatial variograms are derived from a generating elemental variogram, we propose to infer the parameters , , , of this elemental variogram through a minimization procedure. We chose minimizing the mean squared relative errors (MSE) between experimental and theoretical variograms for a set of durations , between the spatial variograms , , 0 and :

1 1, , 0

(19)

where represents the number of durations, and represents the number of distance classes. We applied the simplex method to minimize the product and we used adequate monotone mappings to control the positiveness of the parameters. The regularized variograms , , 0 have been numerically integrated. One set of five spatial experimental variograms (aggregation durations of 1 to 3 hours) were available. Having four parameters to identify, the question of the eventual under-determination of the problem is open and a minimal sensitivity analysis is presented below, playing on the link between the “velocity” parameters L/T and U.

6 Case study

6.1 Dataset

Dataset comes from observation measured between 2006 and 2010 (5years) over the Rissa radar area by a raingauges network gathering tools from different stakeholders. The network consists of 28 raingauges providing data at hourly time-step. The main characteristics of the gauges are presented in Appendix 3, Table1. The raingauge network is located on a210 220 km² subarea (Figure 3) Nineteen raingauges have been selected inside this 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 non-homogeneous signal to the one measured by the other tools (#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 4, 5, 6. 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.

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. Figure 3: location of the different raingauges used in the study. Statkraft (filled triangle), Orkla (filled circle), met.no (filled square), Rissa radar location (blank circle)

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Figure 4: Variance of the non-zero rainfall

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Figure 5: Expectation of the rainfall indicator

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Figure 6: Mean of the non-zero rainfall

6.2 Inference of spatio-temporal variogram using spatial experimental variograms

Using the set of parameters given in Table 1, Figure 7 shows that the proposed coherence framework represents well the effect of time aggregation on rainfall variography until 3 hours. Using variographic values aggregated over 4 and 5 hours faces to an estimation of unrealistic parameters and an alteration of the good fit of the model. As mentioned by Koeltzsch (1999) and Skøien et al. (2003), the "external" movement of the precipitation system can only be seen in actual fields when it overgrows the "internal" dynamics. Remind that the advection is symbolized by and the Taylor velocity by (=L/T). The estimation gave an advection equal to 42.601 km/hour, and a Taylor advection equal to 21022.86 km/hour. The effect of advection is thus preponderant to those due to dissipation which is coherent with events observed with radar every 15 min. The value of the advection is quite high in comparison to a typical advection velocity for perturbations over Western Europe (500km/day = 21km/hour).

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Table 1 Parameters of the elemental space-time variogram inferred according to the spatial variograms at different time aggregation.

(mm²/hour²) km hour km/hour 70.604 29.432 0.0014 42.601

Figure 7: Three experimental spatial variograms for durations =1 to 3 hours (points) is jointly fitted by a set of theoretical D-block spatial variograms (block lines) derived from the optimized elemental space (-time) variogram (dashed line).

6.3 Limitations due to algorithm

Unlike theory explained in part 4, no smoothing behaviour appears at small distance on the spatial variogram with increasing time aggregation. This problem is due to the discrete integration established within (18). Several conditions have to be respected in order to keep an evolution coherent with the theory:

K ≫ D/T (20)

h U. D/T. 1/K (21) where K represents the number of iteration within the sum which replaces the integration. In our program K is equal to 500. According to the time aggregation D, (26) is never respected. According to (27), spatial variograms can be trust respectively from 60.85 km, 121.71 and 182.57 km for time aggregation of one to three hours. A convenient trustable distance would be under 10 km. The number of iteration should be then equal to 100 000 iteration which would increase dramatically the computation time. As an example, with 500 iterations, time computing lasts 2 hours.

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6.4 Limitations due to statistical heterogeneities

Different homogeneity issues underpin the above analysis. The large-scale meteorological conditions that govern the direction and velocity of prevailing winds as well as the rainfall type are not constant throughout our dataset. As a consequence, the statistical properties of the rain fields at local scale are not homogeneous either. Advection fluctuates from eastward to northward and could be used either to define weather types or to condition the variogram computation to filter out the wind orientation of the day. The rainfall variability also fluctuates from one rain event to the other as illustrated by calculating the free variance (0.232 mm2) as the sum of inter-event variance (0.060 mm2) and the intra-event variance (0.172 mm2), which is higher which increases with the time aggregation. Again, the use of weather types would enable to reduce the inter-event variance making the free variance closer to the intra-event variance, the space and time variograms having thus closer sill values. The problem we face is the absence of weather type at hourly time step. In this case study there is certainly a progression margin in filtering the meteorological heterogeneity. Unfortunately, when hourly data are used, the difficulty then comes the reduced size of the samples.

7 Conclusion

The classical formulation of the regularization of a random variable allied to a simple model of rain field advection allowed us to propose a coherence framework to fit a set of experimental rainfall variograms at different time steps using a unique theoretical space-time variogram. This elemental variogram is supposed to describe the rainfall variability at unobserved small scales. The number of parameters needed by the framework is limited, namely the space and time range of the elemental variogram, its sill and the advection velocity of the rain fields. The identification of these parameters through a simple optimization tool appears to work well in the case of a very elementary hourly rainfall database. On the one hand, the quality of fit of the experimental variograms is fairly good until a 3-hour. On the other hand, the identified parameters look physically sound, providing typical velocities, sizes and lifetimes. Elementary manipulations of the proposed framework allowed reproducing some well-known properties of experimental sets of variograms. The variogram sill decrease at increasing time steps is well reproduced by the framework. The variogram range increase at increasing time-steps cannot be obtained with a strictly homogeneous scheme but is well reproduced in the presence of advection. In spite of its obvious meteorological and statistical importance, the advection concept is delicate to implement. In a practical study, it seems difficult to make the distinction between the renewal of the rainfall cells inside the precipitation area (Taylor velocity) and the movement of the precipitation area itself (advection velocity). This statement that looks afterward logical, invites to introduce external advection information when building the data samples. The cumulative properties used in this study make no distinction between rainy or non-rainy areas in the elemental random field. Such a distinction could be made later on if one can solve the delicate question of intermittency aggregation. Although the conclusion is not the good place to motivate a study, we can recall in a last remark the main advantages expected from the proposed scheme. On the one hand, we believe that the shared view of hydrologic regimes as the multi-scale transformation of rainfall into runoff invites to have a coherent view of the rainfall structure across a reasonable scale window. On the other hand, the conceptual idea of getting the space-time structure of the underlying rainfall process is needed for a stochastic rainfall simulation having the ambition to be dynamically realistic.

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Part 2: inference of a space-time marginal distribution related to an

elemental process, a theoretical approach

1 Introduction

More general than only a space-time variogram, a rainfall field is a space-time distribution involving of course its space-time variability but also its punctual distribution. This part enables to not forget the punctual distribution, and then to generalized the previous part. Furthermore, estimated parameters will afterwards be used as input into a rainfall generator. The rainfall generator simulates a Gaussian field which afterwards has to be transformed into a realistic punctual distribution. Anamorphosis is used to establish this transformation. However, a transcription of a field from a Gaussian to a non-Gaussian cumulated probability density (cpd) has a consequence on the space-time variogram. A correction is therefore established. In our case we use the Hermite polynomial expansion. Nonetheless this correction is not perfect and has several issues, especially in the combined choice of the cpd model and the space-time variogram model. Non-realistic results are reached when using Inverse Gaussian or a Gamma cpd model and a Spherical or Exponential space-time variogram: the resulting space-time variogram after the correction is characterized by negative values. Working directly together with both the space-time variogram and the punctual distribution enables to avoid the step of variogram correction and (a priori) to choose the most adaptable punctual distribution model and space-time variogram model. This part is a theoretical approach. It describes theory that could be adapted for first a raingauge time series with the estimation method, and then for a radar time series.

2 A space time random process representing elemental rainfall

2.1 Elemental rainfall as a stationary process of order 2

We reconsider the rainfall field as one realization of a random function (RF) in a 3D domain: two dimensions in space and one dimension in time. The RF is written:

, , ∈ , ∈ (22) where is the location in (the geographical) space, and the location in time We assume the rainfall as a Gaussian process that is homogeneous in both space and time. Formally we choose it to be member of a second order stationary RF family. For such a RF the expected value (μ) and the variance ( ) exist and are constant over the space-time domain. The structure will be described by the space-time variogram , . Then, according to the space and time cpd of the raingauge network, one can describe the raingauges network values as the following marginal distribution:

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∀ , , , … , , ~ ⋮ , 1 , 0

(23)

where G is the Gaussian function.

3 Marginal distribution of a time-regularized random variable

Given the amount of temporal information within a raingauge network time series, it is more robust to aggregate those data over time instead of over space. It is then possible to aggregate those data over long duration and to get more information how rainfall data evolves with time aggregation.

3.1 Notation

A family of second order stationary RF , in both spatial and temporal domains is obtained by regularizing over a chosen duration the RF , representing the elemental rainfall. The marginal distribution is then read:

, , … , , (24)

Then, is related to the elemental structure by:

∀ , ~ ⋮ , 1 , , 0

(25)

4 Marginal distribution encompassing several regularized variables at different time

aggregation

4.1 Notations

Let be Γ a set of in both spatial and temporal domains that is read:

Γ ,… , (26)

A strong assumption is then taken; any are independent from each other. Then,Γ is related to the elemental process by:

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∀ , Γ~ ⋮ , 1 , , 0

(27)

where , is an extension of (12) over two different time regularization and :

, , 0122. , , ,

(28)

with:

,1

′ ′ / ′∈∈

(29)

and

,1

1/∈∈

(30)

and

,1

1/∈∈

(31)

5 Inference of the elemental space-time marginal distribution

Assuming that Γ is derived from a generating elemental marginal distribution, we propose to infer the parameters , , , of this elemental marginal distribution through a minimization procedure. We chose minimizing the mean squared errors (MSE) between the modeled cpd and the empirical frequency between two points:

, , , , 1 , , 0 , ,

(32)

where represents the number of time steps within the available time series, represents the durations,

represents the number of raingauges, , , represents the rain value at time step , time aggregation , and

at location , represents the distance between the two raingauges and , , , represents

the empirical frequency value at time step , time aggregation , and at location .

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6 Marginal distribution encompassing several regularized variables at different spatial

aggregation and the inference of the elemental space-time marginal distribution

Contrarly to the raingauge data, radar data get an higher density if information within the spatial dimension. It is then more robust to aggregate those data over space. It is then possible to aggregate those data over wide area and to have more information how rainfall data evolves with space aggregation. Similarly to parts 3, 4 and 5, one can extend the method to spatial aggregation. This approach will be more relevant for radar data whose spatial information is denser.

6.1 Notation

Let be Λ a set of second order stationary RF , in both spatial and temporal domains that is read: Λ ,… , (33)

Again, a strong assumption is taken; any are independent from each other. Then,Λ is related to the elemental process by:

∀ , Λ~ ⋮ , 1 , 0,

(34)

where , is an temporal adaptation of (28) over two different spatial regularization and :

, 0,122. , , ,

(35)

with:

,1

′ / ′∈∈

(36)

and

,1

′ ′∈∈

(37)

and

,1

′ ′∈∈

(38)

6.2 Inference of the elemental space-time marginal distribution

Assuming that Λ is derived from a generating elemental marginal distribution, we propose to infer the parameters , , , of this elemental marginal distribution through a minimization procedure. We chose minimizing the mean squared errors (MSE) between the modeled cpd and the empirical frequency between two points:

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, , , , 1 , 0, , ,

(39)

where represents the number of time steps within the available time series, represents the spatial aggregation, represents the number of pixels, , , represents the rain value at time step , spatial aggregation , and at location , represents the distance between the two time steps and , , , represents the empirical frequency value at time step , spatial aggregation , and at location .

7 Conclusion, limitations of the method and future adaption

This method is based on both space-time variability and punctual distribution. It enables to generalized the method presented in the part one. This method will help also to pass by the problem a the space-time variogram correction and to get directly the parameters values to use as input of the rainfall generator. However, this parameters estimation method is still a theoretical approach. Indeed, expression (32) involves so many loops within the optimization method that one need more than one day to estimate parameters of a simple 3-timestep, 16-raingauge dataset using 3 time duration aggregation and with only 5 terms in each integral to compute , . Due to double integral, optimization using the expression (39) is even much longer. Improvement in the computing has to be established both by using parallel cpu computing and gpu computing and by coupling R-scripts with C-routines. Another improvement within the approach would be to use a Bayesian method in order to locate the probable parameter values and thus to make the computing faster. Rainfall is not described by a Gaussian distribution but either by the mixture of an atom at zero and a distribution function representing the non-zero rainfall cpd, or independently by a rainfall indicator distribution and a non-zero rainfall cpd. Thus, anamorphosis or better copula method has to be taken into account in order to work with real rainfall fields.

8 References

Ali, A., Lebel, T., Amani, A., 2003. Invariance in the spatial structure of Sahelian rain fields at

climatological scales, Journal of Hydrometeorology, 4(6), 996-1011

Ciach, G.J., Krajewski, W.F., 2006. Analysis and modeling of spatial correlation structure of small-

scale rainfall in Central Oklahoma, Advances in Water Resources, 29, 1450–1463.

Gneiting, T., 2002. Nonseparable, stationary covariance functions for space-time data, Journal of

the American Statistical Association, 97(458), 590-600

Journel, A.G. and Huijbregts, C.J., 1978. Mining Geostatistics, Academic Press, London, 600p.

Krajewski, W.F., Ciach, G.J. Habib, E., 2003. An analysis of small-scale rainfall variability in

different climatological regimes, Hydrologic Sciences Journal, 48(2), 151-162.

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Kyriakidis, P.C., Journel, A.G., 1999. Geostatistical space-time models A review, Mathematical

Geology, 31( 6), 651-684

Lebel, T., Bastin, G., Obled, C. and Creutin, J.D., 1987. On the accuracy of areal rainfall estimation

: A case study, Water Resources Research, 23(11): 2123-2134.

Ma, C., 2008. Recent developments on the construction of spatio-temporal covariance models.,

Stoch. Environ. Res. Risk. Asses. 22 39–47.

Matheron, G., 1965. Les Variables Régionalisées et leur estimation. Masson, Paris, 305p

Mellor, D., 1996. The Modified Turning Bands (MTB) model for space-time rainfall. I. Model

definition and properties, Journal of Hydrology, 175(1-4), 113-127

Paquet, E., Gailhard, J. and Garcon, R., 2006. Evolution of the GRADEX method : improvement by

atmospheric circulation classification and hydrological modelling. La Houille Blanche, 5 : 80-

90.

Wackernagel, H., 1995. Multivariate Geostatistics : an Introduction with Applications., Springer-

Verlag, Berlin, 256 p

Zawadzki, I.I., 1973. Statistical Properties of Precipitation Patterns, Journal of Applied

Meteorology, 12(3), 459-472.

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A Extension variance

We consider a RF respecting the second order stationarity assumption, and characterized by a mean and a variance . The extension variance of the RF from a location to another location ’ is defined by twice the variogram:

, ′ ′ 2 ′ (40)

Let us consider two RFs, and ′ , obtained by regulating over the duration :

1

∈

(41)

and by regulating ′ over :

′1′

′ ′∈

(42)

The extension variance is then written:

, ′ (43)

Incorporating (29) and (30) in (31) becomes the expectation of the quadratic difference between all pairs of

values located at every within and every ’ within . Then, involving the variogram, expression (30)

becomes:

,2

∈∈

1

∈∈

1

∈∈

(44)

It is usual to use this notation:

,1

∈∈

(45)

Then (33) is written:

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, 2 , , , (46)

The extension variance depends thus of the integrated variograms , . According to the theoretical model of the variogram, their values will be obtained, either by reading Journel and Huijbregts (1978) for the simple case, either analytically by a numerical integration.

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B Dispersion variance

We consider a duration formed by identical sub-duration . The two RFs, and ′ respects the expressions (29) and (30). The dispersion variance of within the duration is written:

/1

′ (47)

where represent the i-th duration within . The theoretical expression of the dispersion variance is obtained by calculating the expectation of (35).

/ / (48)

From (31), the expression (36) can be written as:

/1

′

1,

(49)

Thus, using (32), the expression (37) can be developed as:

⁄1

2 , , ,

, ,2 1

∈∈

, ,2 1

∈∈

, , 2 , , ,

(50)

From (38) and considering the duration as instantaneous, it yields: ./ , (51)

Similarly, taking the duration instantaneous, it yields: ./ , (52)

The dispersion variance is then written: / ./ ./ (53)

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C 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

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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

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