A New Approach for Rainfall Rate Field Space-Time Interpolation for 1 Western Europe

The prediction of rainfall rate characteristics at small space-time scales is currently an important topic, particularly within the context 10 of the planning and design of satellite network systems. A new comprehensive interpolation approach is presented in this paper to deal with 11 such an issue. There are three novelties in the proposed approach: 1) the proposed interpolation approach is not directly applied to measured 12 rain precipitation (either radar or raingauge-derived data) but focuses on the coefficients of the fitted statistical distributions and/or computed 13 rain characteristics at each location; 2) the parameter databases are provided and the contour maps of coefficients spanning Western Europe 14 have been created. It conveniently and efficiently provides the rain parameter for any location within the studied map; 3) more speculatively, 15 the space-time interpolation approach can extrapolate to rain parameters at space-time resolutions shorter than those in the NIMROD 16 databases. 17


Introduction
The spatial and temporal variation of point rainfall rates is important for the detailed planning and performance prediction for satellite and terrestrial networks (a group of links) (Yang, 2016).It is increasingly evident that models and/or approaches are needed in order to predict rainfall rate variation at smaller space-time scales than currently available from wide area coverage measured rainfall rate databases.
Extensive studies of rain have been carried out in the last few decades.After several generations many interesting rain models have been published.A model of particular interest was developed by Bell (Bell, 1987).His work showed that rainfall intensities in a field exhibit lognormal distribution and this was confirmed by Crane (Crane, 1996) and Jeannin et al (Jeannin et al., 2008).
The traditional rain models (e.g.stochastic models, Markov chain models) can be used to aid the planning of satellite networks.
However, there are some limitations inherent in such models and the two major ones are: 1) Data availability.The models are only applicable to areas/locations where rainfall precipitation with the necessary integration volume has been observed and the accuracy of the models in areas where no data is available is difficult to verify.
2) Integration volume.The application of the traditional models is limited by the integration length.The modelling of rain and simulated rainfall fields can only be limited to the space-time resolution derived from rain radar/gauge measurements.
Rainfall fields simulation at finer space-time scales is often possible but cannot be verified.
Based on this information, it is clear that the application range of stochastic models is limited by the above problems.
Improvements, thus, are needed to compensate, enhance and extend the performance of stochastic models.In particular, an increase in the use of high frequency over short communication links has led to an increase in the need to predict rainfall rates at finer resolutions.Current stochastic models cannot satisfy this demand.As a result, interpolation techniques have attracted a lot of attention in recent decades.For example, Drozdov and Sephelevskii (Drozdov and Shepelevskii, 1946)  called Kriging was developed based on the theory of regionalized variables to estimate area averages considered as realizations of a stochastic process introduced by Matheron (Matheron, 1971) Since then significant progress has been made and twodimensional ( ) space rainfall rate interpolation models have been developed, e.g.(Deidda,1999 andMenabde et al., 1997).The Random Midpoint Displacement algorithm (RMD) developed by Voss (Voss, 1985) in 1985 is one of the most popular interpolation algorithms.The basic idea of the technique is to introduce new rain rate samples with the same underlying distribution as existing measurements at new locations or times.The one-dimensional ( ) time interpolation is also of interest as network planners and designers of physical layer fade mitigation techniques (Gremont et al., 1999) require knowledge of rain variation over much shorter time scales (of the order of seconds or less).Some excellent models have been published like (Pathirana et al., 2003 andVeneziano et al., 1996).One of such models proposed by Kevin Paulson (Paulson, 2004) is a stochastic numerical model that can interpolate the point rain rate for short time durations down to .
The downscaling model, based on the space-time averaging theory, is another model that has also attracted significant attention.
According to (Deidda et al., 1999), there are two fundamental requirements for precipitation downscaling models, which are: 1) understanding of the statistical properties and scaling laws of rainfall fields, and 2) validation of downscaling models that are able to preserve statistical characteristics observed in real precipitation.Typically, based on the information given in (Rebora et al., 2006), downscaling algorithms can generally be grouped into three main families with some simplification: 1) point process based on the random positioning of a given number of rain bands and rain cells (Cowpertwait et al., 2006); 2) autoregressive processes passed through a static nonlinear transformation (Guillot and Lebel, 1999), and; 3) fractal cascades (Kiely and Ivanova, 1999).In particular, the theory of fractals, which was first introduced by Mandelbrot in 1967 (Mandelbrot, 1967) has attracted great attention.This theory was not applied to rainfall study until the mid-1980s (Lovejoy and Mandelbrot, 1985).Rain has been shown to hold fractal properties over a range of scales.The intermittence and discontinuous nature of rain is reproduced by the fractal based models, which are strongly favoured for rainfall modelling.Many studies have been carried out to interpolate the radar/raingauge measurement data to finer scales using the fractal theory, such as (Svensson, 1996).Multifractal models, which may be simulated using random cascades, can easily capture any moment of the observed signal; especially higher order moments have attracted a lot of attention.Because of their link with multifractal theory, multiplicative cascade models first proposed by Yaglom (Yaglom, Jul 1966), appeal to rainfall simuláations.The rainfall series have been shown to exhibit scaling invariance properties over a large range of space (Olsson, 1996) and time (Olsson et al., 1993) steps.Some multifractal models use discrete cascade algorithms to produce data at finer scales from original sparse observations, for example (Olsson, 1998).A classic work is given by Menabde (Menabde, 1997) who used a discrete random cascade to generate a rain field with the desired statistical structure and then applied a power law filter, thereby removing some of the blockiness resulting in a more realistic looking rain field.In addition, synthesis of rain field at high resolution is also important to the rain study especially devised for applications related to EM wave propagation.Many contributions have been done in this area, such as (Jeannin, 2012 andLuini, 2011) The prediction at finer space-time resolution however, has long been a challenging issue in rainfall field modeling.Results from interpolation studies are quite poor (Yang, 2016 andDeidda, 2000) as it is very difficult to consider both space and time variability and irregularity of rainfall in an appropriate way.The basic idea of published models is to try to find the underlying principle of how the space-time transformation can be achieved.A representative model was developed by Deidda (Deidda, 2000) based on the assumption that Taylor's hypothesis (Taylor, 1938) can be applied.The space-time rainfall field is assumed to be a three-dimensional ( space and time) homogeneous and isotropic process.An advection velocity parameter is introduced to connect the space scale and time scale.With the help of a velocity parameter, the statistical properties of rain at finer scales can be deduced from larger ones.Similar studies can be found in (Venugopal et al, 1999, Deidda, 2006 Venugopal et al., 1999) in which rain has been studied in a range of space-time scales to define the transformation parameter.In particular, Kundu and Bell (Kundu and Bell, 2006) developed a model that can provide the correlation function of rain in space-time domain but in a very complicated form.
The absence of high resolution rainfall data at desired space and time scales is the main knowledge gap.Deidda in (Deidda, 2000) pointed out that most of the existing rainfall studies at finer scales are purely focused on either space modeling (Hubert et al., 1993) or time modeling (Paulson, 2004).However, both of these approaches have limitations.For example, the statistical behavior of rain in time has implicit consideration of the spatial distribution and extension of the rain field itself; and the study in space is normally based on fixed time duration whilst the evolution in time of spatial patterns is ignored.Accurate rainfall field simulation requires knowledge of rainfall rate variability in both space and time domains.There is not enough research in the area of space-time interpolation apart from a few works, such as (Deidda, 2006).Thus, an appropriate space-time interpolation model that can preserve the underlying statistical properties at finer scales is needed.The absence of knowledge of rain characteristics at high space and time resolution is another important gap and is the second objective of this study.Kundu in (Kundu and Bell, 2006) showed that the characteristics of rain depend on the space and time scales over which rain data is averaged.However, all the existing interpolation and/or multifractal models directly focus on rain precipitation and no work has been found that studied the characteristics of rain at scales better than the one provided by rain radars.The study in this paper, therefore, will look into this issue to investigate the variability of rain characteristics at arbitrary space-time integration length.
To further the development of rain-induced radio-wave attenuation models, and to provide more accurate performance prediction of satellite links over wide areas, there is an increasing need for a good understanding of the space-time characteristics of rainfall rate at finer scales.As extension of our previous work (Yang, 2011), this paper presents a simple but accurate space-time interpolation approach that can interpolate the key studied properties of rain in both space and time domain simultaneously.We present a series of European maps superimposed with each parameter at different space-time resolution which is novel.In particular, a simple but accurate approach for interpolating the rain characteristics has been proposed.It can predict the coefficient values of the statistical model in both space and time with reasonable accuracy.
The rest of this paper is organized as follows: Section 2 describes the data used in this study.Section 3 reviews the statistical model proposed in previous work and describes the proposed approach how to interpolate the measurements into 3-dimensional space-time domain.The detailed results, including the 2D contour map of rain characteristics across Western Europe, as well as the 3D space-time predictions at each location, are presented in Section 4. Section 5 validates the results achieved from the proposed interpolation approach.Conclusions are drawn in Section 6.

Data Description
Five complete years of NIMROD rain radar data (from 2005 to 2009) have been analyzed for the development of a generic interpolation approach.The NIMROD radar system produces a series of composite rain field map by every .The measured rain rate samples are distributed on a squared Cartesian grid covering Western Europe.Each NIMROD map contains data cells, but only the data available points have been analysed, see the outline is Fig. 1(a).The study area ranges from to in latitude and to in longitude.In addition, NIMROD system also holds the database for the British Isles.This database has better resolution of rain rate measurement, which is in space and in time.The example radar map is given in Fig. 1(b).The performance of any model or approach needs to be validated through comparing with observational data from apparatus (e.g.raingauge or rain radar).UK data, which has better resolution than EU NIMROD data, can be utilized to implement the validation.for the British Isles.

Stochastic Model
The empirical equations that can accurately provide the estimates of the studied characteristics of rain have been discussed in 0.
The proposed model for the four key rain characteristics is described briefly here for completeness.
It is well accepted that rainfall rate in mm/h at one location is modeled as a lognormal process with mixed probability density function (pdf).According to (Filip and Vilar, 1990), the general formula for a straight line fit is given by: where * + is the set of lognormal parameters that are used to study the statistics of rainfall rate at a location of interest.
Research reported in (Yang, 2016) has produced a single general empirical equation that fits both the space correlation and the time correlation functions.The common function is given by: where can either be which represents the distance in or which is the time lag in .
An empirical equation has been proposed in (Yang, 2011) that can give an excellent estimate of the probability of rain occurrence ( ) throughout the whole range of integration length.The mathematical equation is described by: where , and are experimental constants which can be determined from study and denotes either spatial integration length or temporal integration length .

Data Integration
Following previous work (Yang, Oct 2011), the rainfall rate data can be up-scaled from short integration length to longer one using: ) where ( ) is the rain rate at position ( ) derived from a spatial integration region of linear size and temporal integration time .is known as the scale parameter.More generally, the spatial and temporal regions could have different scale parameters e.g.: The radar-derived rain rate data can be upscaled to coarser resolution based on above equations.It is important to highlight that each grid point will be used only once for each integration and no overlapping regions are considered.The integrated data will be tiled up without changing the size of original rain map but new dataset with larger integration scale will be achieved.Note that the larger the integration length the smaller number of data samples will be.Particularly, it requires and must be integer to enable this procedure.Therefore, it is notable that the integration length of the new data is the integral times of original radar data, and it will be and , here and .

Approach for the Implementation of 3D Interpolation
According to our previous work (Yang, 2011), we found that the rain characteristics regularly changing with increasing integration length both in space and time domains.This interesting finding indicates that the studied rain characteristics at other spatial or temporal integration lengths can be reasonably predicted using such regularity.More speculatively, it enables the 3D interpolation to be achievable if there are enough measurements with different space-time resolution combinations.

Contour Map of Rain Characteristics
The proposed statistical model can provide estimates of key rain characteristics (including the first order statistics of rain, the spatial and temporal correlation of rain rate, as well as the probability of rain/no rain) in two dimensions.Considerable computation is required to extract these summarizing statistics from the NIMROD databases.Based on the proposed model, however, the rain characteristics at any data available locations within the Western Europe can be achieved.The work in this paper has produced a multi-resolution database of parameters and contour maps that cover the whole of Western Europe.With the help of this database, the user can easily obtain the characteristics of rain (or the distribution coefficients) at any location within the studied area.and stored in the database, but not presented in this paper.In addition, the rain characteristics at other integration length combinations between * + and * + have been computed and stored in the database.Given this database, the prediction of the rain characteristics at some finer space-time resolutions can be estimated by interpolation.

Prediction of Rain Characteristics in Space-Time
The existing NIMROD radar maps have been integrated to some integration length combinations from * + to * +.The key characteristics of rain were then analysed to see how they vary with integration length.Table 1 gives an example of the probability of rain ( ) with a range of integration length combinations, at Portsmouth (UK).It shows that the value changes with increasing spatial-temporal integration length.Similar results can be found for other studied parameters.These data allow the prediction of parameters at other space-time resolutions.The top-left hand corner of the table is the computed value with the shortest available spatial-temporal integration length (* +) derived from EU NIMROD radar, and the right-hand bottom corner is the coarsest one (* +) after integration.From Table 1, one can see that the characteristics of rain change systematically with increasing integration length.Given this finding the predictions at finer resolution can be estimated by interpolation.
In this study, the cubic spline interpolation algorithm has been chosen to implement this task.The cubic spline is a function that is constructed by piecing together cubic polynomial on different intervals (Keys, 1981).It has the form where is a third degree polynomial defined by: ) Cubic spline is often used for 1D interpolation.The data in each row and column of the database (see the example in Table 1) can be treated as samples in one dimension.It enables the use of cubic spline interpolation to estimate parameter values at other scales, based on the measured parameters.The first step is to extract the multi-scale parameters for a desired location from the database.Cubic spline interpolation is then used to interpolate to a different spatial or temporal integration sizes.In this study, the "bicubic" interpolation algorithm in MATLAB was used.Mathematically, the bicubic interpolation, which is an extension of 1D cubic interpolation, is used to interpolate data points on a two dimensional regular grid.It can be accomplished using cubic spline algorithm (we provide part of the software program in Appendix B to show the approach of 3D space-time interpolation).
The software proposed in this work uses the produced parameters' database.It contains the fitted rain parameters for a range of integration lengths between * + and * + for the whole of the studied area (Western Europe).The software extracts the rain characteristics with all available integration lengths at the location of interest.Taking the extracted data as input values, the interpolation algorithm then processes the data and gives the prediction at other space-time resolutions.Note that this is true only for the locations for which radar measurements data is available (the black area in Fig. 1).with these data.The multi-scale data are regularly spaced, which reduces the complexity of the interpolation algorithm.
Interestingly, the results show that values increase systematically with increasing spatial-temporal integration length.In addition, by interpolation the values at resolutions smaller than * + can also be predicted.The exterpolation can be constrained by the assumption that as either or .This enables the predictions to be plotted smoothly to form a surface.The resolution of the studied key characteristics of rain offers significant improvements over previous methods (e.g.Bell, 1987) and it is these that are important for rainfall field simulation studies in future.The salient point of the proposed interpolation approach is that the best estimate can be obtained with high accuracy for the space and time resolutions up to and , respectively.Predictions finer than this threshold are unacceptable as negative data is produced.This is impossible due to the should not less than 0. Other interpolation technique might give better results but this is not covered in this paper.The validity of the interpolated parameters needs to be tested, and this is limited by the availability of data at small spatial and temporal integration volumes.One test that can be performed is to use * + EU NIMROD data to predict the distribution and correlation functions of * + UK NIMROD data.

Validation
The absence of measured data at the smaller space-time scales causes great difficulties in validating the proposed method.
However, the * + UK NIMROD radar measurements can be used to address this issue to some extent.In this paper, the key rain characteristics at Portsmouth have been estimated at scales of * + and these were compared with interpolations from the EU NIMROD data.Although the * + values of both are marginally different ( , , differences for , respectively), the associated , and exceeded rain rates are similar, this can be seen in Fig. 5.In particular, the proposed model gives excellent approximation for the first-order rainfall rate statistics, especially for the rain rate lower than for which the accuracy is higher than .The probability of heavy rain event is extremely low so that there is no sufficient data is available.This results in the higher bias for the range where .Fig. 6(a) shows that the spatial correlation using the predicted values is in agreement with the computed values.There is a small difference between the temporal correlation functions of rain rate using predicted data and measured data at short time lags up to roughly , see Fig. 6(b).However, the result is still acceptable as the trend is similar, especially for large time lags.This shows that the approach proposed in this unavailable); or something to do with the data at the edge of the radar network.The contour map of exceeded rain rate of the average year given by ITU-R P 837-6 (ITU, 2013) is presented in Fig. 8.These two figures (Fig. 7 and Fig. 8), illustrate that the results of the statistics in Fig. 7 are very similar.This indicates that the proposed model can give a reasonable estimation of rain parameters that can be used to produce rain rates with exceedance.However, the rain rate statistics given by ITU-R P 837-6 seems quite larger compared with the results from EU NIMROD data interpolate from to and estimated directly for data.This suggests that the ITU.Rec tends to over-estimates rain.Indeed, the overestimation of ITU-R P.837-6 is likely also due to the overestimation of the rain amounts over oceans as obtained from the ERA-40 data produced by the ECMWF (i.e. the input maps on which the ITU-R rain rate models relies on).This is why the ITU.Rec recommends users to use their own data in order to produce better results.).However, the difference is acceptable as it is in the range for most areas.For some areas, the difference can up to or higher, but this is rare.Figure 9: Contour map of 0.01% exceeded rain rates difference between the prediction from proposed approach and the measurements from UK NIMROD.Figure 10: Contour map of 0.01% exceeded rain rates difference between the prediction from proposed approach and ITU-R P 837-6.Fig. 10 presents the difference between the prediction from the proposed approach and ITU-R P 837-6 (EU predicted rain rates minus ITU predicted rain rates).The contour map shows that the ITU-R P 837-6 tends to over-estimate rain rate compare to the proposed approach for most areas.The difference can up to 40 for some regions.This indicates that the proposed approach gives more plausible estimates than ITU-R P 837-6, although it is restricted to Western Europe.However, it is necessary to highlight that for the Grand Massive alpine area of France, the proposed approach gives larger rain rates exceedance than ITU-R P 837-6.This indicates that it is hard to give accurate rainfall rate measurements or prediction over mountain area due to the difficulties associated with obtaining accurate rain radar readings (Johansson and Chen, 2003).
Fig. 9 and Fig. 10 present the visual comparison of 0.01% exceeded rain rates difference between the prediction from proposed approach and the measurements from UK NIMROD and ITU-R P 837-6.However, the error function can give more information to the model performance validation.According to (Paulson et al., 2015 andITU, 2013), the error function can be defined as: where and are the measured and predicted rainfall rate with 0.01% exceedance, respectively.The error at each individual location therefore can be calculated by Eq. ( 8).
Fig. 11 shows the error contour maps at exceeded rain rate over the UK for both the proposed approach and ITU-Rec model.Theoretically, the smaller the error value, the more accurate the model prediction will be.Fig. 11(a) shows that the error of the proposed approach is between and .It indicates that the approach proposed in this paper can produce reasonable prediction.However, the error from ITU-R model can be up to nearly , see Fig.In particular, the Root-Mean-Squared Error (RMSE) has been applied to measure the goodness of fit between measured lognormal parameter * + obtained from radar-derived statistics and predicted values.The RMSE is defined as developed a spatial interpolation technique to analyze the spatial variations of a process over an area.Then later, a modified interpolation technique Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-343Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 2 August 2018 c Author(s) 2018.CC BY 4.0 License.

Fig. 2
Fig.2shows the grid of available rain data points from the NIMROD radar measurements.The dots represent the available spacetime integration length combination where the rain characteristics can be computed based on available NIMROD data.The lines represent the range of integration length where rain characteristics can be calculated from equation (1) to (3).It is notable that the proposed statistical model in our previous work can only produce estimation of rain characteristics along the line but not the blank area.Taking advantage of the regular distribution of the measurements, the key rain characteristics at other spatialtemporal integration lengths can be predicted using any existing interpolation technique.

Figure 2 :
Figure 2: Grid of available rain data points from the NIMROD radar measurements.The dots represent the available space-time integration length combination where the rain characteristics can be computed based on available NIMROD data.

Figure 3 :
Figure 3: Contour maps of rain distribution coefficients with spatial integration length of and temporal integration length : (a) a plot of values and (b) a plot of values.

Figure 4 :
Figure 4: An example of space-time interpolation of at Portsmouth.

Fig. 4
Fig.4presents the example of predicted probability of rain occurrence at other spatial-temporal integration lengths, along with the measured data in Table1, for Portsmouth.It is clear that the outcome of the interpolation is a surface constructed from many curves both in space and time domains.The dots are the measured values at a range of spatial-temporal integration lengths that are multiples of the data resolution, whilst the surface is produced by the interpolation algorithms to be consistent

Figure 5 :
Figure 5: A comparison exceedance distribution of rainfall rate estimated by interpolation from data to data and estimated directly for data.

Figure 6 :Figure 7 :
Figure 6: A comparison of correlation function of rainfall rate estimated by interpolation from data to data and estimated directly for data: a) spatial correlation function; b) temporal correlation function.
Figure8: Contour map of 0.01% exceeded rain rates of the average year given by ITU-R P 837-6.The differences between EU contour map, UK contour map and ITU contour map have been studied to show how accurate the proposed approach is.Fig.9presents the contour map of the difference of rain rates with 0.01% exceedance based on the EU data minus UK data.It shows that the proposed approach tends to overestimate the rain rates over land (see the example in middle area of Fig.7(b), but under-estimates over the ocean/sea areas (see the left-bottom area of Fig. 7(b)).However, the Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-343Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 2 August 2018 c Author(s) 2018.CC BY 4.0 License.

Figure 11 :
Figure 11: Contour map of error at exceeded rain rate: (a) error distribution of proposed model, and (b) error distribution of ITU-R P 837-6.