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[1] Synthetic time series generators of signal attenuation due to rain are an important tool for radio-communication system simulations. These generators must reflect both the long- and short-term dynamics of signal attenuation due to rain, and have to rely on prolonged experimental observations of attenuation time series. Such prolonged observations are not always feasible, especially if a large number of different climates have to be covered. Fortunately, rain attenuation is well correlated with the rain rate above all when long term statistics are considered. Only for particular events the correlation may be lower in some cases, especially for intense rain events which have a limited spatial expansion. The advantage of using rain rate data instead of rain attenuation data as a basis for synthetic time series generators of signal attenuation is the following: there exist consistent sets of rain rate time series from numerous locations around the globe. In this paper, enhancements to an existing synthetic rain rate generative model are presented and discussed. These enhancements include (1) the modeling of fast variations of rain rate, (2) the grouping of rain events into episodes, and (3) the extension to the spatial domain (space-time model).

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[2] In this paper enhancements to the UVIGO-DLR synthetic rain rate generative model [Alasseur et al., 2005] are presented and discussed. This model has already been used to describe the dynamics of rain rate in four different climatic areas in Europe with separations between sites larger than 500 km. More specifically, the model has been successfully used to model data from Vigo and Bilbao in Spain, Wessling in Germany and Milan in Italy. New data sets from other parts of Europe, and elsewhere, are expected to be analyzed in the future.

[3] In this paper, in addition to reviewing the original model and showing results, several enhancements are discussed which are based on further analyses carried out on 1 second tipping bucket rain gage data from Bilbao, Spain. The data was acquired with a data-logger attached to the rain gage that recorded the second when each tip occurred. The enhancements proposed here deal with the following issues:

[4] 1. Modeling of the fast variations of rain rate. The current UVIGO-DLR model generates 5-minute integration-time series (or 3-minute depending on the data set). A two-stage approach is proposed here in which, to the output of the current model (5-min rain rate series), faster rates are superposed. A model for these fast variations (1-minute integration time) has been developed consisting of both a family of conditional rain rate distributions, f(R_{1-min} ∣R_{5-min}), and an autocorrelation model for characterizing the actual variation rate of the faster rain rates.

[5] 2. Grouping of rain events. A two-level approach is proposed for grouping rain events into episodes and, thus, achieving a more accurate reproduction of interevent and interepisode durations.

[6] 3. Extension to the spatial domain. The UVIGO-DLR model is combined with the space-time model discussed and implemented by Goldhirsh [2000] which is based on the Excell model [Capsoni et al., 1987]. This combination allows reproduction of realistic simultaneous rain rate time series at different locations while preserving spatial cross-correlation properties.

2. UVIGO-DLR (Two-Rain Type) Model

[7] In the UVIGO-DLR model, a low to medium-rate type of rain (steady rain) is considered to occur in most cases and most of the time during the rainy periods. This rain type corresponds, in general, to widespread or stratiform rain.

[8] In some cases, intense, higher-rate rain can be observed both forming part of (embedded in) steady rain events, and in stand-alone intense rain events. Intense rain, in general, corresponds to local, convective phenomena. Intense events are described with a different model to that of steady events. The threshold between these two types of rain was set at (R_{th}) 13 mm/h (the integration time was 5 min) for the data set (Bilbao, Spain) analyzed here. The separation criteria also took into account the slope (Sp) which, in the case of the analyzed data, was set at 6 mm/h/5 min interval. A simple pattern recognition software was developed to separate out steady from intense events, and embedded intense events within steady events.

[9] The criteria for selecting both the level and slope thresholds, i.e., 13 mm/h and 6 mm/h/5 min were based on visual inspection of the available measured time series. It could be clearly observed how a marked change in behavior took place for rain intensities on the order of 13 mm/h, below such level, intensity variations were slowly varying with gentle slopes, while for intensities above the indicated value, variations were fast and abrupt. The selected thresholds turned out to be almost identical in all four data sets under investigation.

[10] One additional element in the model is the definition of transition probabilities. These describe the probabilities of switching from one to the other rain type. The approach followed in the model is illustrated with the help of Figure 1a. Figure 1b illustrates the states and transitions assumed, and Figure 1c outlines the flow diagram of a synthetic rain rate generator based on this model. The sample spacing is 5 minutes. Figures 2a and 2b show examples of measured rain events of the two types.

[11]Figure 1b gives an overview of the assumptions made in the model: time is split into rainy periods, to the right, and dry periods, to the left. Within the rainy periods, two possibilities exist: steady or intense rain. Dry periods are controlled by a given distribution, discussed below and, at the end of each dry period, a new rain event starts. This rain event can either start in a steady rain condition or in an intense rain condition. Transitions between these two conditions are possible and are driven by their corresponding distributions, explained below. Finally, the rain event can finalize from either the steady or the intense condition returning to a dry period, and so on.

[12] As indicated above, two submodels are needed, one for each rain type. Steady rain can be modeled by means of conditional distributions: Next-Rain Conditional Distributions, NRCDs, f^{NR} (R_{t+1} ∣R_{t}) With this approach an underlying discrete Markov process is assumed as illustrated in Figure 3. Such NRCDs have been fitted to theoretical distributions, more specifically, to Rayleigh distributions. The fitting has been found to yield a smooth evolution of the theoretical distribution parameters as a function of the current rain rate, R_{t}, except for the first- and the last-rains. This led to including in the model a First Rain Distribution, FRD, f^{FR}(R_{t+1}) = f^{NR} (R_{t+1} ∣R_{t} = 0), which has been fitted to a Gamma law. The last rain is discussed later.

[13] In the case of intense rain, the recorded occurrences were very brief, with typical durations on the order of 10–15 minutes, i.e., two-three samples. So far, due to the limited number of intense events observed, it has not been possible to fit theoretical distributions to the measured rates and duration distributions. Thus, the actual measured distribution of rates during intense events, f_{Spiky}(R), and the measured distribution of intense event durations, f_{Spiky}(D), are considered as a part of the model. To simulate intense rain, first, the number of samples is drawn according to the corresponding duration distribution, and then, the necessary rain intensity samples are generated using f_{Spiky}(R).

[14] To complete the model, transition probabilities to intense rain behavior, P_{Peak}(R_{t}), have been defined. These were extracted both for the case where the rain event starts directly with an intense rain value or for the case where the intense rain starts from a steady rain sample. Similarly, end of event probabilities, P_{0} (D_{t}), have been computed from the measured data. It has been observed that these probabilities are dependent on the duration up to that point in time of the current rain event. A single set of P_{0} (D_{t}) probabilities has been extracted for both rain types.

[15] Finally, interrain durations can be modeled with an appropriate Inter-Rain Duration Distribution, IRDD, f^{IR}(D). It has been found, however, that two different IRDDs were needed depending on the duration of the preceding rain event. The separation threshold was found to be of approximately 90 minutes for the analyzed data. Table 1 summarizes the model parameter set.

[16] In this section, the set of extracted parameters for the model just described are presented for rain rate time series corresponding to 8 years of recordings of 5-min integrated rain rates taken in Bilbao, Spain. Figure 4a shows the fitting of the measured First-Rain distribution to a Gamma distribution. The fitted parameters were α = 2.8926 and β = 1.8310 where the Gamma pdf is given by

[17]Figure 4b shows a fitting example for one member of the family of NRCDs, more specifically, for a current rate range in the interval R_{t} = (8, 9]. Rayleigh distributions have been used in this case, where its pdf is given by

[18] The calculated Rayleigh parameters, σ, are listed in Table 2. The evolution of the Rayleigh parameter has also been studied and the following polynomial was fitted (Figure 5) to describe its evolution, σ(R_{t}), with the current rate value, R_{t},

Table 2. Evolution of σ(R_{t})

R, mm/h

σ(R)

R, mm/h

σ(R)

(0, 1]

1.5577

(10, 11]

4.9659

(1, 2]

1.4331

(11, 12]

5.3088

(2, 3]

1.7276

(12, 13]

5.6867

(3, 4]

2.1552

(13, 14]

5.6098

(4, 5]

2.7068

(14, 15]

5.5632

(5, 6]

3.2972

(15, 16]

5.5951

(6, 7]

3.7478

(16, 17]

6.6993

(7, 8]

3.9021

(17, 18]

6.6007

(8, 9]

4.2893

(18, 19]

6.4123

(9, 10]

4.7401

[19] The above parameters are for the steady rain type. For intense rain, two additional parameters are required: the distribution of rates, f_{Spiky}(R_{t}), and of durations, f_{Spiky}(D). It has not been possible to find theoretical distributions that fit the measured ones. These are shown in Figures 6a and 6b.

[20] Finally, Figure 7 shows extracted P_{0}(D_{t}) values (end-of-event probabilities) as a function of the current duration of the current event, D_{t}. This figure shows a lack of data in some cases, i.e., not enough ends of event for given long events have been recorded. In any case, the trend seems to clearly indicate that the probability that the event ends increases as its current duration, D_{t}, increases.

[21]Figure 8 shows extracted transition probabilities from zero rain rate or from steady rain rate samples to intense values. This figure shows gaps which are due to a lack of sufficient intense events. Values for those gaps can be interpolated from neighboring readings. Finally, Figure 9 illustrates the two interrain event duration distributions, after rain events of long (≥90 min) and short (<90 min) durations.

[22] To test the model, a long simulation covering a period of over ten years has been run. The simulated time series have been compared with the measured ones in terms of (1) the overall cumulative distribution and (2) durations (second order statistics) for different rates: R = 0, 1, 5, 10, 20 mm/h.

[23]Figures 10a and 10b show comparisons between measurements and simulations. First-order statistics could be reproduced quite accurately. As for the second-order statistics, fairly good agreement has been achieved for some of the thresholds considered.

4. Model Improvements

[24] In this paper, enhancements to the current version of the UVIGO-DLR model are presented dealing with the following issues: (1) modeling of fast variations of rain rate, (2) grouping of rain events and (3) extension to the spatial domain.

4.1. Improvement 1: Introducing Superposed Fast Rain Rate Variations

[25] The available rain data was obtained using a tipping bucket gage. The sampling rate of the data-logger attached to it was of one sample per second, this means that the second of a “tip” occurrence was recorded.

[26] The UVIGO-DLR model is based, for historical reasons, i.e., first sets of analyzed data, on data accumulated each 5 minutes. This type of information is consistent with many available databases and thus has been kept as the model's baseline. However, faster, more intense variations in real rain rates are missed using this technique, i.e., they are low-pass filtered. If data with a sufficient resolution is available as it is the case for the Bilbao dataset, it is possible to develop a model describing these fast variations which could be superposed on the slower synthetic series generated with a sampling rate/integration time of 5 minutes.

[27]Figure 11 shows rain rate data processed in two different ways, one by accumulating the rain fallen within each 5-minute interval (slow variations) and by using a 1- and 5-minute sliding window (running mean) over linearly interpolated accumulated rain [Fiser, 2002].

[28]Figure 12 illustrates how the one-second rain data was processed. On the top part of the figure, actual tips are shown with dots. The 5-minute accumulation process means that all tips occurring in one 5-min window are added and converted to a rain rate value. For instance, if a 0.1 mm gage is used, one tip in 5 minutes is equivalent to a rate of 1.2 mm/h (0.1 mm/5-min period × 12 5-min periods/hour), two tips mean 2.4 mm/h, and so on. The series calculated in this way are represented by circles on the top part of the figure and correspond to the type of output produced by the original UVIGO-DLR model. To compute the fast variations in the measured data set, a different approach has been followed: first the accumulated rain is calculated with each new tip occurrence, then this series is linearly interpolated, lower part of the figure and, finally, its derivative is calculated. This processing allows that the faster rates are not lost in the time-windowing (low pass filter) processing.

[29] The fast rain rates have been fitted to a family of left-truncated Gaussian distributions as illustrated in Figure 13 (see A. C. Johnson and N. T. Thomopoulos, Characteristics and tables of the Left-Truncated Normal Distribution., working paper available at http://www.stuart.iit.edu/faculty/workingpapers/thomopoulos/char-left.pdf). Figure 14 provides the fitted parameters for this distribution which has been chosen given that rain rates cannot have negative values. The left-truncated Gaussian distribution is described by the expression,

where f(z) is its associated Gaussian distribution. The mean and standard deviation of the left-truncated Gaussian distribution can be approximated through the following fitted expressions,

where R_{5-min} indicates rain rate integrated over 5 min.

[30] One final step for fully describing the characteristics of the fast rain rate variations is the calculation of its autocorrelation function (Figure 15a) or, equivalently, its power spectral density (Figure 15b). This spectrum closely resembles the results reported, for example, in [Paulson, 2002]. This last parameter can be used to spectrally shape, by means of a low-pass filter, the rate of change of the fast variations which show correlation times in the order of 2–3 minutes. A block diagram for a synthetic rain rate time series generator capable of superposing fast variations (1-min integration time) over slower variations (5-min integration time) produced by the original UVIGO-DLR model is shown in Figure 16.

4.2. Improvement 2: Defining Rain Events and Episodes

[31] As discussed in a section above, it has been observed that two different interrain duration distributions between rain events were needed depending on the length of the previous event. Further observations indicated that, in many cases, short interevent times occurred, like there was a short rebound in the rain rate, a momentary zero-intensity value being actually part of a longer event.

[32] These observations have led to the definition of a cluster of events with short interevent times. Such cluster is called here an episode. Figure 17 illustrates the original and new approaches. The underlying model is a semi-Markov one [Bråten and Tjelta, 2002] where event durations are not directly drawn from a set of transition probabilities but from specific distributions. The arrowed loops in Figure 17 do not represent self-transitions but periods, long or short, of no-rain. Figure 18 shows the new generator's flow diagram including the new parameters whose distributions are given in Figures 19a–19d. These new parameters are: the distribution of the number of events in an episode, the distributions of interrain event durations within an episode and the distribution on interepisode durations. A redundant parameter, not used in the new model, is that in Figure 19a: the distribution of episode durations, which directly results from the other three.

[33] The question that needed answering was where to place the breakpoint to tell between events belonging or not to the same episode. To answer the above question, a scan for different maximum interevent durations (or minimum interepisode durations) was carried out (Figures 19a–19d). Stabilization of the various cumulative distributions was achieved for a maximum interevent time within the same episode of 8 hours. This value seems consistent with real life observations of rain events belonging to the same frontal system, for example. This value was thus selected and is shown with a thicker trace in Figures 19a–19d.

4.3. Improvement 3: Extending the Model to the Spatial Domain

[34] While existing spatial models [e.g., Capsoni et al., 1987; Féral et al., 2003] represent the average spatial behavior of rain, and provide a good match for the long term statistics, the instantaneous, local or point behavior is represented in more detail by dynamic models such as the UVIGO-DLR model. Thus, the idea of mixing the UVIGO-DLR model with the one discussed by Goldhirsh [2000] is presented here. Goldhirsh's implementation is based on the Excell model [Capsoni et al., 1987]: synthetic scenarios with exponential rain cells are generated according to given rules while fitting the yearly cumulative distributions corresponding to the former ITU-R climatic regions. Figure 20 shows this approach: an ensemble of rain cells is created and spread out according to the rules in the model. The various colors in the figure represent rain intensities, with the background color representing the no-rain areas. As discussed below, two rain intensity levels corresponding to intense and steady rain (threshold 13 mm/h) are defined to drive the UVIGO-DLR part of the combined approach.

[35] A given constant advection velocity is assumed so that the previously generated rain cell ensemble moves accordingly. This idea follows the approach proposed in Matricciani [1996] where the so called Synthetic Storm Technique, SST, is described. The SST assumes a constant advection velocity on the order of 10 m/s. No change in the original cell ensemble structure occurs as it moves (at least in the short-term). Furthermore, the area of each of the synthetic, moving cells is sliced into two distinct zones with different behaviors: steady and intense rain. Thus, a linkage can be established between the spatial model and the finer detail dynamics in the UVIGO-DLR model (Figure 21). This can be done for each point in the simulated scenario. A resolution grid of 1 × 1 km^{2} has been used in the figures. For each grid element, rain rates are produced using the UVIGO-DLR model according to its two rain types: steady and intense. Figure 22 illustrates how a synthetic multi-rain cell scenario such as the one in Figure 20, in this case covering an area of at least 200 × 200 km^{2}, is made to drift according to a constant velocity of advection. A window of 20 × 20 km^{2} is used to show the space-time dynamics simulated with the proposed approach. Figures 23a and 23b illustrate two snapshots where also the rain rates observed on the central line in the figure along the advection direction are shown.

[36] By following this mixed approach, some of the elements in the UVIGO-DLR model are no longer required such as the interrain distributions or the end-of-event probability. Moreover, things could even work the other way around: some of the distributions in the UVIGO-DLR model could be used to drive, in part, the generation of synthetic spatial scenarios.

5. Summary

[37] In this paper, three types of improvements to the current synthetic rain rate time series generator developed at UVIGO and DLR have been proposed. The improvements encompass: (1) modeling of fast variations of rain rate, (2) grouping of rain events and (3) extension to the spatial domain. These improvements have been shown to considerably increase the accuracy of the UVIGO-DLR model. These advantages include a higher time resolution due to a higher sampling rate, a better fit to the occurrence of sequences of rainy periods of short duration, and the opportunity to address spatial issues. Further work might still be required to incorporate issues such as diurnal and seasonal variations. However, the UVIGO-DLR model with the enhancements proposed here is already well suited to generate accurate rain rate time series which can easily be converted into rain attenuation time series.

Acknowledgments

[38] This work has been partially funded by EU's Network of Excellence SatNex, the Spanish Ministry of Education and Science, and the Regional Government of Galicia (Spain), Xunta de Galicia.