Modelling Rain Rate By Means Of Arrival Processes

We try to address in this paper our current work toward the generation of synthetic time series of rain rate, R, in mm/h with an integration time of 1 min This is the ITU-R, standard for predicting rain induced attenuation [1]. For one, the availability of rain intensity records adapted to the ITU-R requirements is very limited given that the main users of rain information (weather prediction, water management, etc.) do not require such a short integration time and data with longer accumulation periods, even exceeding 1 hour, are commonly found. In this respect, there is recent published work where a method based on a physical model (EXCELL) [2] has been proposed for converting recordings with longer integration times to the standardized 1-min period.On the other hand, even using 1-second resolution rain gauge data, it is rather difficult to agree on how to produce R(t) series from which reliable yearly complementary cumulative distributions, CCDFs, are obtained. From such statistics, CCDFs of rain-induced attenuation, A (dB), are generated for use in link budgets. Several methods have been compared in [3] showing important discrepancies in critical outage probability levels.Moreover, if the interest is in generating instantaneous time series of A(t) for simulation purposes, for example making use of Matricciani's Synthetic Storm Technique [4], the resulting series would consist of frequent, unrealistic signal plateaus as a consequence of the somehow unrealistic rain rate series. The latter are produced with the current tipping bucket data processing methods [3], [5-6]. Fig. 1 illustrates an example of an R(t) series obtained from bucket tip series using [6].Note the presence of frequent plateaus, especially in the lower intensity sections. However, one would expect that the series of both R and A should contain an additional, superposed random component, not present using with the actual processing techniques.It is thus important to investigate the smaller scale nature of the precipitation phenomenon in order to being able to reproduce it more accurately. Additionally, we need to find a better approach toward the conversion of bucket tip data into rain intensity and then on to obtaining cumulative statistics.Our current approach to the above issues is developing a model capable of simulating the occurrence of rain gauge tips according to an it-state hidden Markov model representing it rainy regimes within which different arrival (tip) rates are defined. Moreover, the idea is to further complement the above approach with a similar model which derives rain drop arrivals of different sizes associated to each tip.Thus, we are exploring the application of a class of stochastic point process models, Markovian Arrival Process (1VIAPs) [7], in the analysis of fine time-scale rain rate. MAPs can model statistics of any order of inter-arrival times between measured events. We are using these models to generate synthetic tippingbucket rain gauge data at a specific site. Since R(t) series derived from tipping series are strongly dependent on the processing methodology, five methods are being used to calculate R(t) and their corresponding CCDFs for comparison with those provided by the ITU-R for the same location.In this application, the arrival pattern of rain gauge bucket tip times, {NO}, is viewed as a MAP whose rate of occurrence varies according to an unobserved finite state irreducible Markov process.An it-state MAP is specified by two nxn matrices: (1) a stable matrix Do with non-negative off-diagonal entries, and (2) a non-negative matrix D1 describing the transition rates between the it states. Each transition in D1 produces an arrival event (tip). Matrix Do describes the background transitions, not associated with arrivals. The matrix Q=Do+D-i is the infinitesimal generator of the underlying Markov process.Here, we are considering a special case of MAP where D-1 is a diagonal matrix: a Markov Modulated Poisson Process (MMPPs) [8]. In particular, for our initial work we are using the MMPP model parameters listed in Table 1 of [9] corresponding to 13 years of 0.2 mm rainfall tip time data for Heathrow, West of London, UK. Twelve parameters sets, one for each month of the year, are available. The model assumes three states: (1) intense rainy state, (2) wet state, and (3) dry state (little or no rain). A realization of this model is illustrated in Fig. 2, the state series on the top line and the actual tips on the bottom line, with the highest tipping rate corresponding to the state 1.The synthetic rain gauge data generated with the above 3-state MIVIPP model was used to generate R(t) series. Five methods were considered for producing R(t): Natural, Usual meteorological, First IAP, Novel IAP, Novel IAP-RM, and D'Amico et al. [3],[5-6]. Fig. 1 shows an example result using the Novel IAP-RM method [6]. As indicated earlier, the generated R(t) series does not provide the expected realism: lack of random variability. The same happens for the other processing methods. This is due to the fact that most of the above studied techniques correspond to some sort of convolution between a rectangular window with a train of Dirac deltas (tips), which are later accumulated in 1-min integration intervals. We would assume that an experimentally derived, random window should give rise to more realistic R(t) series. One of our current lines of work involves the derivation of one or a family of such windows from fine time resolution disdrometer data.After all R(t) series are obtained through the different data processing methods, the final step involves the analysis of results by comparing the yearly CCDFs with that for Heathrow according to ITU-R P.837-6 [1].It has been observed that the model parameters proposed in [9] give rise to an average CCDF that follows quite well that provided by the ITU-R for the lower intensities. However, the higher intensities produced seem to be underestimated. We suspect that this originates from the fact that the optimization process used for extracting the model parameters targeted matching the yearly mean amount of rain. We are working toward a parameter extraction method that overcomes this problem and that could be used in other climatic areas.In conclusion, a summary of our current work toward the study and characterization of rain intensity at a finer scale that can be used both for simulation purposes and for the generation of yearly or other periods (seasonal, monthly, ...) cumulative distributions for link budget applications has been presented. It is expected that through the study of arrivals: first of rain gauge tips and, possibly, drop arrivals, a more faithful reproduction of rain induced effects on radio links could be achieved.