Non-zero Rainfall Research Articles

Rainfall Generation Revisited: Introducing CoSMoS‐2s and Advancing Copula‐Based Intermittent Time Series Modeling

AbstractWhat elements should a parsimonious model reproduce at a single scale to precisely simulate rainfall at many scales? We posit these elements are: (a) the probability of dry and linear correlation structure of the wet/dry sequence as a proxy reproducing the distribution of wet/dry spells, and (b) the marginal distribution of nonzero rainfall and its correlation structure. We build a two‐state rainfall model, the CoSMoS‐2s, that explicitly reproduces these elements and is easily applicable at any timescale. Additionally, the paper: (a) introduces the Generalized Exponential () distribution system comprising six flexible distributions with desired properties to describe nonzero rainfall and facilitate time series generation; (b) extends the CoSMoS framework to allow simulations with negative correlations; (c) simplifies the generation of binary sequences with any correlation structure by analytical approximations; (d) introduces the rank‐based CoSMoS‐2s that preserves Spearman's correlations, has an analytical formulation, and is also applicable for infinite variance time series, (e) introduces the copula‐based CoSMoS‐2s enabling intermittent times series generation with nonzero values having the dependence structure of any desired copula, and (f) offers conceptual generalizations for rainfall modeling and beyond, with specific ideas for future improvements and extensions. The CoSMoS‐2s is tested using four long hourly rainfall records; the simulations reproduce rainfall properties at multiple scales including the wet/dry spells, probability of dry, characteristics of nonzero rainfall, and the behavior of extremes.

A note on the misuses of the variance test in meteorological studies

Stochastic modeling of rainfall data is an important area in meteorology. The gamma distribution is a widely used probability model for non-zero rainfall. Typically the choice of the distribution for such meteorological studies is based on two goodness-of-fit tests—the Pearson’s Chi-square test and the Kolmogorov–Smirnov test. Inspired by the index of dispersion introduced by Fisher (Statistical methods for research workers. Hafner Publishing Company Inc., New York, 1925), Mooley (Mon Weather Rev 101:160–176, 1973) proposed the variance test as a goodness-of-fit measure in this context and a number of researchers have implemented it since then. We show that the asymptotic distribution of the test statistic for the variance test is generally not comparable to any central Chi-square distribution and hence the test is erroneous. We also describe a method for checking the validity of the asymptotic distribution for a class of distributions. We implement the erroneous test on some simulated, as well as real datasets and demonstrate how it leads to some wrong conclusions.

Bayesian Modelling of Rainfall Data by Using Non-Homogeneous Hidden Markov Models and Latent Gaussian Variables

SummaryWe present a non-homogeneous hidden Markov model for the spatiotemporal analysis of rainfall data, within a subjective Bayesian framework. In this model, daily rainfall patterns are driven by a small number of unobserved states, interpreted as states of the weather, that evolve in time according to a first-order non-homogeneous Markov chain, with transition probabilities dependent on time varying atmospheric data. The weather states alone do not account for all the space–time structure in the data and so we introduce latent multivariate normal random variables in a flexible model for the probability of rain and the distribution of non-zero rainfall amounts. In the resulting hierarchical non-homogeneous hidden Markov model, rainfall occurrences and non-zero rainfall amounts are spatially dependent and conditionally Markov in time, given the weather state. We build a prior distribution that conveys genuine initial beliefs and apply the model and inferential procedures to data from a network of 12 sites located throughout the UK.

A Contrast in Precipitation Characteristics across the Baiu Front near Japan. Part I: TRMM PR Observation

Abstract Contrasts in precipitation characteristics across the baiu front are examined with Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR) data near Japan during June–July (1998–2011). The vertical structure of atmospheric stratification differs between the tropics and midlatitudes. On an average, the baiu front is found around the latitude that roughly divides the midlatitude atmosphere from the tropical atmosphere. Precipitation characteristics are compared between the southern and northern sides of the reference latitude of the baiu front, which is detected with equivalent potential temperature at 1000 hPa of 345 K in terms of the boundary between the tropics and midlatitudes. The results show that there are obvious differences in precipitation characteristics between the southern and northern sides. In the south, convective rainfall ratios (CRRs) are 40%–60%, which are larger than those in the north (20%–40%). Greater rainfall intensity and taller/deeper precipitation are also observed in the south. Moreover, the characteristics of precipitation features (PFs), which are contiguous areas of nonzero rainfall, differ between the southern and northern sides. In the north, wide stratiform precipitation systems with CRRs of 0%–40% and heights of 8–11 km are dominant. In the south, organized precipitation systems with heights of 12–14 km and CRRs of 30%–50% and those with very large heights (14–17 km) and CRRs of 50%–80% are dominant in addition to wide stratiform precipitation systems. These results suggest that the mechanisms to bring rainfall are different between the southern and northern regions of the baiu front.

Space-time simulation of intermittent rainfall with prescribed advection field: Adaptation of the turning band method

[1] Space-time rainfall simulation is useful to study questions like, for instance, the propagation of rainfall-measurement uncertainty in hydrological modeling. This study adapts a classical Gaussian field simulation technique, the turning-band method, in order to produce sequences of rainfall fields satisfying three key features of actual precipitation systems: (i) the skewed point distribution and the space-time structure of nonzero rainfall (NZR); (ii) the average probability and the space-time structure of intermittency; and (iii) a prescribed advection field. The acronym of our simulator is SAMPO, for simulation of advected mesoscale precipitations and their occurrence. SAMPO assembles various theoretical developments available from the literature. The concept of backtrajectories introduces a priori any type of advection field in the heart of the turning band method (TBM). TBM outputs transformation into rainfall fields with a desired structure is controlled using Chebyshev-Hermite polynomial expansion. The intermittency taken as a binary process statistically independent of the NZR process allows the use of a common algorithm for both processes. The 3-D simulation with a space-time anisotropy captures important details of the precipitation kinematics summarized by the Taylor velocity of both NZR and intermittency. A case study based on high-resolution weather radar data serves as an example of model inference. Illustrative simulations revisit some classical questions about rainfall variography like the influence of advection or intermittency. They also show the combined role of Taylor's and advection velocities.

Temporal and spatial variability of rainfall at the urban hydrological scale

The main objective of this paper is to characterize the spatial and temporal variability of different types of rain events at scales that are consistent with urban hydrological applications. In this way, a total of 24 rain periods have been analyzed according to a geostatistical approach. This analysis has focused on the non-zero rainfall variogram. The studied rain periods were recorded by the weather radar of Treillieres (10 km north of Nantes, France) in 2009. This radar device provides rainfall radar images with a high level of spatial resolution (250 m x 250 m) and instantaneous temporal resolution. Results indicated four different types of rainfall fields, which display very different variability scales, including double structures within the same field. These four types of rainfall fields can be characterized by their climatological variograms, as calculated for increasing time resolutions: instantaneous, 5 min, 15 min, and 30 min, thus making it possible to estimate the decorrelation distance (i.e. range). This study highlights the benefit of radar images featuring high temporal and spatial resolution, which in turn allow studying small-scale variability.

Simulation of monthly rainfall totals

The observed distribution of non-zero rainfall totals for a given month is often modelled using a maximum likelihood estimate for the Gamma distribution. In this paper we show that a Gamma distribution can be regarded as a zero order approximation to a density distribution constructed by a series of associated Laguerre polynomials. The coefficients of the series are easily calculated and used to improve the shape of the initial approximation by adjusting higher order moments. We show that this more general method models joint probability distributions for two or more months and in particular that the series model does not require an assumption of independence between months. Finally we explore how the series method generates simulated data that is statistically indistinguishable from the observed data. We illustrate our methods on a case study at Mawson Lakes and although monthly correlations may not be significant we note that the rainfall records at Koonamore Station show many significant correlations for successive months.

Correcting of real-time radar rainfall bias using a Kalman filtering approach

This paper presents a method to adjust the mean field bias of radar rainfall estimates in real time. The underlying philosophy for estimation of radar rainfall in real-time used in this study is that the estimated radar rainfall must first be corrected for the reflectivity measurement errors and the Z– R conversion errors based on the physical methods, and thereafter a statistical method is used to remove the average difference (mean field bias) between radar estimates at the rain gauge locations and the corresponding gauge rainfall amounts. Kalman filtering techniques are used as the basis for correcting the mean filed bias in real time. The logarithmic mean field radar rainfall bias is modelled as an Autoregressive order 1 process having a stationary variance. The observation error variance is estimated from a difference between a variance of the observed mean field bias and the stationary variance of the logarithmic bias process. The variance of the observed mean field bias is estimated using a radar rainfall error variance model which depends on (a) the number of rain gauges that measure non-zero rainfall at that hour; (b) the location of these rain gauges; (c) the conditional mean of rain gauge rainfall; and (d) the number of pulses used for reflectivity measurements. A 7-month radar and rain gauge data record from the Kurnell radar in Sydney, Australia are used to test the efficiency of the proposed method. The results shown that the Kalman filtering approach outperforms the use of sample bias correction method when rain gauge density is less than about 1 gauge per 90 km 2 for both climatological and stratiform rainfall, and has a higher accuracy compared to the case where no bias correction is performed, irrespective of the number of rain gauges used in the calibration.

Chaos in rainfall: variability, temporal scale and zeros

A recent study on rainfall observed at the Leaf River basin reports that the presence of a large number of zeros in the data significantly underestimates the correlation dimension. The present study attempts to verify such a claim, by making predictions and comparing the results with the correlation dimensions. A nonlinear prediction method, which uses the concept of reconstruction of a single-variable series in a multi-dimensional phase space to represent the underlying dynamics, is employed. Correlation dimension analysis of only the non-zero rainfall series is also carried out for further verification. Rainfall data of four different temporal resolutions (or scales), i.e. daily, 2-day, 4-day and 8-day, are analyzed. The predictions for the finer-resolution (i.e. higher-resolution) rainfall are found to be better than those obtained for the coarser-resolution (i.e. lower-resolution) rainfall and seem to be consistent with the variability vs. predictability logic in a deterministic sense, i.e. higher prediction accuracy for data with lower correlation dimension and vice versa. An important implication of this result is that the presence of (a large number of) zeros in the rainfall data may not always result in an underestimation of the correlation dimension. The correlation dimensions estimated for the non-zero rainfall series are not significantly different when compared to those obtained for series including zero values, supporting the above. These results suggest that the low correlation dimensions for rainfall (in particular finer-resolution ones that commonly have a large number of zeros), as reported by past studies, could well be, or at least closer to, the actual dimensions of the rainfall processes studied.