Rain Fields Research Articles

A geostatistical approach to multisensor rain field reconstruction and downscaling

Abstract. A rain field reconstruction and downscaling methodology is presented, which allows suitable integration of large scale rainfall information and rain-gauge measurements at the ground. The former data set is assumed to provide probabilistic indicators that are used to infer the parameters of the probability density function of the stochastic rain process at each pixel site. Rain-gauge measurements are assumed as the ground truth and used to constrain the reconstructed rain field to the associated point values. Downscaling is performed by assuming the a posteriori estimates of the rain figures at each grid cell as the a priori large-scale conditioning values for reconstruction of the rain field at finer scale. The case study of an intense rain event recently observed in northern Italy is presented and results are discussed with reference to the modelling capabilities of the proposed methodology. Keywords: Reconstruction, downscaling, remote sensing, geostatistics, Meteosat

Verification of precipitation in weather systems: determination of systematic errors

An object-oriented verification procedure is presented for gridded quantitative precipitation forecasts (QPFs). It is carried out within the framework of “contiguous rain areas” (CRAs), whereby a weather system is defined as a region bounded by a user-specified isopleth of precipitation in the union of the forecast and observed rain fields. The horizontal displacement of the forecast is determined by translating the forecast rain field until the total squared difference between the observed and forecast fields is minimized. This allows a decomposition of total error into components due to: (a) location; (b) rain volume and (c) pattern. Results are first presented for a Monte Carlo simulation of 40,000 synthetic CRAs in order to determine the accuracy of the verification procedure when the rain systems are only partially observed due to the presence of domain boundaries. Verification is then carried out for operational 24-h forecasts from the Australian Bureau of Meteorology LAPS numerical weather prediction model over a four-year period. Forty-five percent of all rain events were well forecast by the model, with small location and intensity errors. Location error was generally the dominant source of QPF error, with the directions of most frequent displacement varying by region. Forty-five percent of extreme rainfall events (>100 mm d −1) were well forecast, but in this case the model's underestimation of rain intensity was the most frequent source of error.

A conditional simulation model of intermittent rain fields

Abstract. The synthetic generation of random fields with specified probability distribution, correlation structure and probability of no-rain areas is used as the basis for the formulation of a stochastic space-time rainfall model conditional on rain gauge observations. A new procedure for conditioning while preserving intermittence is developed to provide constraints to Monte Carlo realisations of possible rainfall scenarios. The method addresses the properties of the convolution operator involved in generating random field realisations and is actually independent of the numerical algorithm used for unconditional simulation. It requires only the solution of a linear system of algebraic equations the order of which is given by the number of the conditioning nodes. Applications of the methodology are expected in rainfall field reconstruction from sparse rain gauge data and in rainfall downscaling from the large scale information that may be provided by remote sensing devices or numerical weather prediction models. Keywords: Space-time rainfall; Conditioning; Stochastic models

Small-scale structure of rain field — Preliminary results basing on a digital gauge network and on MRL-5 legionowo radar

Spatial correlation functions of rain fields have been determined for accumulation time ranging from 10 minutes to 24 hours and for distances from 10 m to several tens of kilometres.A special digital gauge network including intra-gauge distances from 10 m to 5 km and MRL-5 radar data with spatial resolution of 2 km have been used for the study. The radar has possibility to recognise automatically between stratiform and convective storms and to determine storms motion.The gauge network has been used to determine the spatial correlation function of rain fields for scales ranging from 10 m to 5 km, radar data has been used for scales from 2 to several tens of kilometres.The spatial correlation functions obtained basing on radar data have shown strong anisotropy with respect to the direction of advection, especially for convective rains (showers) and for not very long accumulation time (below 12 hours).Applying rain field simulation in a form of random selection from correlated log-normal distribution, approximations of the spatial correlation functions “basing on gauges” have been determined till the distance 30 km both in isotropic and anisotropic version.Temporal correlation function functions have been determined for accumulation time ranging from 10 minutes to 1 hour (gauges) or from rain intensity through 20 minutes to 1 hour accumulation time (radar).

Sampling errors in radar estimates of rainfall

The relatively slow rate of application of radar rainfall to operational hydrology is partially due to concerns about the measurement errors. The errors that result in a bias in the field mean have been studied extensively and can to some extent be treated, but errors that manifest themselves as more or less white noise become important when considering what scale is appropriate for spatially distributed hydrological modeling. This paper evaluates the errors that arise in radar estimates of rainfall as a result of temporal sampling, spatial averaging, measuring the field at some distance above the ground, and recording the reflectivity data with a limited radiometric resolution. By far the most significant source of error was found to be due to measuring the field at some height above the ground. The mean standard difference in rainfall rate between fields separated by 1 km in height at 1 km spatial resolution was found to be of the order of 100% of the mean rainfall rate. When the spatial resolution is reduced to 5 km the mean standard difference between the fields with the same 1 km vertical separation fell to about 50% of the mean rainfall rate. Temporal sampling was found to be quite sensitive to the intermittency of the rain field being sampled. The mean standard error caused by 2‐min sampling for 10‐min accumulations decreased from 14% for scattered rainfall to 8% for widespread rainfall.

Disaggregation of Sahelian mesoscale convective system rain fields: Further developments and validation

Bridging the gaps between the GCM scales and the hydrological scales is a key issue when studying the impacts of potential climate changes on the water resources and, more generally, the links between climate variability and hydrological variability. This is especially true in tropical regions where (1) rainfall is highly variable in space and time due to its convective nature, and (2) measurements are scarce. Using high‐resolution data collected in the semi arid region of Niamey, Niger, Lebel et al. [1998] have proposed a space‐time model (the LBC model), allowing the disaggregation of large‐scale estimates produced either from satellite images or general circulation model (GCM) outputs. The behavior of this model was shown to be globally satisfying when tested on a small number of selected Sahelian mesoscale convective complexes (SMCCs). However, to be of use in simulation studies of the impact of climate changes as predicted by GCMs or in an operational context where only satellite data are readily available, a more systematic validation was required. Also, the initial version of the model was intended at dealing with SMCCs only, leaving aside the other convective systems displaying a less coherent spatial structure. This led to develop a new version of the LBC model, presented here, characterized by the following improvements: (1) a more precise modeling of the spatial structure of the total storm rain fields by taking into account their anisotropy and using a nested covariance, (2) a better representation of the storm kinematic by dealing with arrival times of rain rather than with speeds of movement, (3) a revision of the parameters used to define the standard hyetograph which is the basis of the time disaggregation algorithm. This new version of the model has two main advantages as compared to the older one: (1) the capacity of dealing with every kind of Sahelian mesoscale convective systems (SMCSs), which account for more than 90% of the total annual rainfall in the region; (2) the possibility of using the model both in simulation and in disaggregation modes. The validation of the model is carried out by comparing the rainfall statistics at various scales of aggregation, for a set of 170 observed SMCSs (corresponding to the 1990–1993 operating period) and a set of 170 simulated SMCSs.

A space and time model for design storm rainfall

Realistic rainfields that represent storms with a known return period are required as input to design calculations for hydrological projects that cover a wide range of hydrological scales. The current standard practice is to assume either that the storm is uniform in time and space or that it varies in some very simple manner. Multiaffine models of rainfall, based on the concept of a multiplicative cascade, provide the possibility of generating a stochastic series of space and time rainfall that reproduces the observed behavior. The spatial distribution of a field of instantaneous rain rates is modeled using the multiplicative cascade approach. The temporal development of the cascade weights at each level in the cascade is modeled with a simple autoregressive ARMA(1,1) model where the parameters vary in a systematic manner with scale. The model is verified using rain fields produced by a monsoonal depression that passed over a weather radar at Darwin, Australia. Radar data for the event were used to estimate the model parameters. The model was able to reproduce the observed temporal and spatial correlation functions over a range of scales, and the probability distributions over a range of scales, for both the instantaneous and the hourly accumulations.

The Relation between the Area-Average Rain Rate and the Rain Cell Size Distribution Parameters

Abstract A review of the literature on the rain cell diameter distribution (RCDD) is first presented. It shows that RCDD, for a given rain-rate threshold τ, is accurately described by an exponential distribution with an exponential parameter λ weakly variable from one place to another and almost independent of τ. Then it is demonstrated that integrating the RCDD enables the writing of interesting and simple relations between the moments of the RCDD and useful area-averaged rain field parameters, notably the average cell diameter and F(τ), the fractional area occupied by the rain cells having a rain rate above τ. From F(τ), an equation relating the area-average rain rate 〈R〉 to the second moment of the RCDD is obtained. Now, an explicit knowledge of the RCDD is not necessary to obtain 〈R〉. RCDD can be replaced by the distribution of the chords of the rain cells. Indeed, the moments of the rain cell chord distribution are analytically related to the moments of the RCDD. Thus, an equation showing that 〈R〉 is...

A short term, radar rainfall forecasting procedure for high resolution radar data

During the last few years, different methods to predict the development of rainfall have been developed and tested. Different scales in time and space as well as different radar data sets produced by different radar types have been used. Based on observations of an X-band radar automated methods have been developed to track and predict on a short-term basis (15 minutes – 2 hours lead time) the movement, direction and the evolution of rain fields in time and space. The study has been conducted as part of a research project undertaken by the Emschergenossen-schaft and Lippeverband (EG/LV, autonomous public bodies, responsible for wastewater treatment and maintenance of watercourses) in cooperation with the University of Hanover. The overall goal of the project is to minimise the total emissions from sewer systems by improving the use of the storage capacities and the performance of the wastewater treatment plant. This paper briefly describes the technique used to predict and calculate the future rainfall - an important input for the simulation and control models.

Multifractal analysis of hourly precipitation

The multifractal nature of the rainfall field is analysed using the methodology of singular measures. The analysis is applied to a long time series (54 years) of hourly rainfall intensities recorded at Valentia on the South-West Coast of Ireland. The empirical probability distribution function suggests a hyperbolic intermittency with the divergence of the statistical moments being higher than the second order. The latter is in agreement with findings of other authors for similar climatic regions (e.g. Sweden). The Fourier transform statistics of the data are used to obtain the scaling range in which the data obey a power law with a coefficient of ∼0.5. The scale invariance as identified by the spectral power law, ranges from 2 hours to about 24 hours. This is a narrower range than has been found for similar studies using continental sites where the range was found to be from 2 hours to about 3 days. Studies of Valentia rainfall using conventional statistics suggest that two distinct periods, (1940–1975 and 1976–1993) are clearly present. The second period is characterised by a greater annual rainfall depth than the first, and the increased depth was found to be concentrated primarily in the months of March and October. The intermittency analysis of the rain field of the two periods reveals two different K(q)-functions. The curvature of the K(q) convexity has been found to be larger for the second period suggesting lower intermittency or more frequent rain events. The intermittency function C1 for the period 1976–1993 is shown to be quantitatively less than C1 for the period 1940–1975, for the annual, March and October time series verifying increasing precipitation since 1975.

Spectral multiscaling of spatially averaged rain rate: a hint for spatio-temporal modelling

Recent research has shown that, conditionally on rain, probability moments of instantaneous spatial averages of rain rate, over differently scaled subregions of a rain field, possess an interesting scaling property called wide sense multiscaling, with respect to magnifying spatial scales. This empirical fact, along with some theoretical considerations, have led to implementation of the so called multiplicative random cascades in order to model the instantaneous spatial variability of rain rate, given that it rains. However, random cascades do not account for the evolution of spatial variability through time. In this article an alternative notion of multiscaling is introduced, providing new theoretical basis for further consideration of multiplicative random cascades, as a class of models which potentially capture not only the spatial variability of spatially averaged rain rate fields, but also accounting for its evolution in the course of time. The new notion of multiscaling introduced here is referred to as spectral multiscaling, is defined for second order stationary processes of spatially averaged rain rate, and (similar to probabilistic multiscaling) is discerned to that of strict and of wide sense, with regard to the normalized spectral distribution and its spectral moments, respectively. In fact, the validity of spectral multiscaling properties is verified statistically, using time series of spatially averaged rain rate data from a regularly observed (by radar) tropical rain field known with the acronym TOGA-COARE. The results of this empirical non-model-based analysis, point to the validity of multiscaling of the normalized spectral distribution across the entire spectrum of frequencies, and also of multiscaling of the corresponding spectral moments. Copyright © 1998 John Wiley & Sons, Ltd.

Breakdown coefficients and scaling properties of rain fields

Abstract. The theory of scale similarity and breakdown coefficients is applied here to intermittent rainfall data consisting of time series and spatial rain fields. The probability distributions (pdf) of the logarithm of the breakdown coefficients are the principal descriptor used. Rain fields are distinguished as being either multiscaling or multiaffine depending on whether the pdfs of breakdown coefficients are scale similar or scale dependent, respectively. Parameter estimation techniques are developed which are applicable to both multiscaling and multiaffine fields. The scale parameter (width), σ, of the pdfs of the log-breakdown coefficients is a measure of the intermittency of a field. For multiaffine fields, this scale parameter is found to increase with scale in a power-law fashion consistent with a bounded-cascade picture of rainfall modelling. The resulting power-law exponent, H, is indicative of the smoothness of the field. Some details of breakdown coefficient analysis are addressed and a theoretical link between this analysis and moment scaling analysis is also presented. Breakdown coefficient properties of cascades are also investigated in the context of parameter estimation for modelling purposes.

Uncertainty in satellite rainfall estimates: Time series comparison

We examined nine satellite rainfall algorithms and compared the rain fields produced from these algorithms for the period of August 1987 to December 1988. Preliminary results show algorithms which use the same satellite sensor data tend to be similar, suggesting the importance of sampling. Oceanic global mean rainfall ranges from 2.7 to 3.6 mm/d. The variability in zonal mean rain rate is about 1.5–2 mm/d for these algorithms.

A new microwave tomography approach for rainfall monitoring over limited areas

Exploiting the relationship between microwave attenuation and rainfall intensity in a microwave tomography approach was demonstrated in the past to be a valid alternative to weather radar measurements for the reconstruction of rainfall fields in limited areas. This is useful in areas (e.g. valleys or small basins), where radar data are not available due to beam blocking or, in general, for real time monitoring of rainfall events occurring over zones characterized by a high hydrological risk. At each time step, path-integrated attenuation measurements are provided by a set of microwave transmitter-receiver pairs, and two-dimensional Gaussian basis functions are utilized to reconstruct the space-time distribution of the rainfall intensity field. The inversion problem to be solved is highly ill-conditioned, due to the practical (and economical) impossibility to set up an adequate network for performing classical tomography. Therefore a Global Optimization Stochastic Technique has been developed to obtain valid results in quasi real time. Some search parameters of the algorithm must be chosen depending on the network characteristics: this allows a kind of calibration depending on the adopted quality (complexity) of the network. Some investigations have been performed both on simulated rain fields and on radar monitored rainfall events. Some significant examples are presented, showing computational speed and flexibility of the technique, and that a network based on 10 stations can provide a estimate of the mean rainfall intensity over a basin of nearly 400 Km 2 with an error less than 10% even with a monitoring time of 30 seconds. Under the same conditions, it is possible to guess the rainfall distribution pattern. We show that, utilizing a longer observation time, more reliable estimates are obtained.

Causal space‐time multifractal processes: Predictability and forecasting of rain fields

Building on earlier cascade models of rainfall, we propose a model of space‐time rain fields based on scaling dynamics. These dynamics are indeed related to the space‐time symmetries of the turbulent medium within which rainfall occurs: the underlying phenomenology corresponds to a cascade of structures with lifetimes depending only on the scale of the structures. In this paper we clarify two major issues: the scaling anisotropy between space and time, and the need to respect causality, i.e., a fundamental asymmetry between past and future. We detail how this ”arrow of time” breaks the mirror symmetry with respect to the spatial hyperplane, and how it can be introduced in continuous multiplicative cascade models so as to remove the artificial temporal mirror symmetry of earlier models. We show that such a causal multifractal field can be understood as the result of an anomalous diffusion acting on the singularities of the field. Finally we will exploit and test these models through (1) a succinct analysis of rainfall data, (2) numerical simulations of the temporal decorrelation of two initially similar fields (accounting for the loss of predictability of the process), and (3) a forecasting method for multifractal rain fields.

Scalar multifractal radar observer's problem

The classical radar observer's problem in rain is to interpret the fluctuating radar echo from precipitation. Contrary to the usual homogeneity assumption involving Poisson statistics and incoherent scattering, we make a (scaling) heterogeneity assumption involving multifractal statistics and (some) coherent scattering. We consider the simplest problem, which is to relate the liquid water (σ) statistics to the (measured) effective radar reflectivity statistics (Ze) and to the (theoretical) radar reflectivity factor (Z; Ze =Z for incoherent scattering). We ignore polarization effects (that is, we use the scalar wave approximation), and denote the pulse length l, wavelength λw, the inner (homogeneity) scale of the rain field (η), and the outer (largest) scale of rain (L). For the simplest (conservative) multifractal σ the two main effects are 1) as in the standard theory, Z ≈ σ2; however, because of the strong subpulse volume gradients, there is a bias of (l/λww)Kσ(2);(Kσ(2) is the scaling exponent of σ2); 2) because of partial coherence, there is an enhancement: Ze/Z ≈ (λw/η(D‐Kσ(2) where D is the (effective) dimension of space. For nonconservative multifractals (parametrized by H) we obtain the overall bias in the means: <Ze>/<Z> ≈ (λw/η)D‐Kσ(2)(L/λw)−2H. Using available data, we estimated this as typically ≈10−3 which is ≪1; Z should therefore not be used as a proxy for Ze. New theories relating radar measurements to rain must therefore be developed. Finally, we show that radar “speckle” (the drop “rearrangement” problem) is a general consequence of multifractal liquid water/ drop correlations.

Multifractal characterization of rain fields with a strong orographic influence

Data from networks of high time resolution (15 second) rain gauges forming a transect from the base to the divide of a mountain range are used to study the multifractal characteristics of precipitation at different locations along the transect. In particular, scaling power spectra, multiscaling moments and power law exceedence probability tails are analyzed from rain gauge time series of ∼35–70 hours in length. The parameters discussed are specifically the spectral scaling exponent, β, the intermittency parameter, C1 (a fundamental parameter of the moment scaling function, K(q)) and qcr, the power law probability tail exponent. Results show a systematic trend in the exponents, indicating a decrease in intermittency, the frequency of extreme values and smoothness of the time series with increasing altitude along the transect. K(q) and qcr are briefly discussed and derived as natural properties of multifractal cascades. It is concluded that the parameters of multifractal cascade models of rainfall are related to the physical processes which are qualitatively discussed in the context of the observed changes in the statistics.

Adaptation and application of a quantitative rainfall forecasting model in a mountainous region

This work addresses operational forecasting of orographic rainfall for hydrologic modeling. The forecast model formulation builds on existing work and proposes a moisture source parameterization which accounts for the primary mechanisms of precipitation processes and conditions in mountainous regions. Specifically, the issue of orographic rainfall is addressed utilizing two components to represent the water vapor sources. One component is associated with the mesoscale rain system in the absence of orography. This component is defined during a particular rain event using radar observations and a discrete water budget analysis over a fixed region of the rain field. The second source term is orographic in nature and is a function of the average wind vector and the ground slope at the location of interest. The orographic moisture source term is additive to the contribution from the mesoscale source. The result is a rainfall forecasting model with an explicit physics-based orographic component and a relatively straightforward parameterization. Model performance is illustrated using two rainfall events observed by radar in the Cevennes region of France; 1 h lead-time rainfall forecasts for three watersheds in this mountainous region are evaluated. One event is used for calibration of the model and the second event is used for validation. The proposed model formulation is shown to perform either as well as or slightly better than the simplified approaches of persistence and advection forecasts. Additional evaluations of the approach with data from other regions remains necessary for making comprehensive conclusions about the proposed formulation. This work shows the importance of obtaining information about the moisture input sources so as to increase the information defining the rain water source. Observing and forecasting rainfall rate alone does not explicitly account for the source of moisture that is transformed into rainfall at a future time. This work shows that a limit on forecast lead-time may be related to the response time of the precipitating cloud system.

Radar rain field evaluation and possible use of its high temporal and spatial resolution for hydrological purposes

The purpose of this paper is to show how accurate radar-estimated rainfall, with good temporal and spatial resolution, can be used for hydrological purposes. A recent methodological advance in rainfall measurement using conventional weather radars has made it possible to account for much of the variation between the precipitation radar echo intensity and rain intensity. A method known as the window probability matching method (WPMM) was applied to radar measurements over several catchment areas in central Israel. Comparison of daily raingauge measurements with radar rainfall estimates demonstrated good agreement. The standard error of radar-estimated rainfall was only 7% for a storm with a total average accumulation of 328 mm. Several case studies are provided which demonstrate the advantage of having an accurate rain field for calculating excess rainfall for each of the area grid squares in the watershed. Storm excess rainfall for different time durations depends strongly not only on the area size, but also on the differences in the rainfall intensity sequences. Accurate radar rain fields can permit dynamic calculations to be made along the storm path.

A discussion of rain erosion testing at the United States Air Force rain erosion test facility

For the past twenty-five years the University of Dayton Research Institute has operated the Rain Erosion Test Facility for the United States Air Force Wright Laboratory Materials Directorate at Wright-Patterson Air Force Base, Ohio. This is a commercially available, whirling-arm facility that produces a 25.4 mm per h rain field and is capable of providing specimen velocities up to 290 m s −1 (650 mile h −1, 0.87 Mach). Various attack angles of specimen-to-rainfall are also possible ranging from 15° to normal. The average rainfall drop size is 2 mm (0.079 in). The facility is in use on a daily basis and serves the aerospace community on virtually a world-wide basis. Almost 40 000 test specimens have been evaluated as of January 1994. This paper will describe the facility in some detail. The method of testing and operation will be described and discussion will include the many types of specimens and materials that can be tested—ranging from metals, composites, plastics, honeycomb constructions, glass/optical, ceramics—and their allowable size and configurations. The mode of operation and the evaluation procedures used will be detailed. Failure mechanisms as erosion and adhesion loss will be discussed and related to materials performance and applications. Work in this facility has become the cornerstone for testing to many Military Specification coatings' requirements. It has been referred to as a ‘national resource’ by USAF personnel.