Multiplicative Random Cascade Research Articles

Multiplicative cascade models for fine spatial downscaling of rainfall: parameterization with rain gauge data

Abstract. Capturing the spatial distribution of high-intensity rainfall over short-time intervals is critical for accurately assessing the efficacy of urban stormwater drainage systems. In a stochastic simulation framework, one method of generating realistic rainfall fields is by multiplicative random cascade (MRC) models. Estimation of MRC model parameters has typically relied on radar imagery or, less frequently, rainfall fields interpolated from dense rain gauge networks. However, such data are not always available. Furthermore, the literature is lacking estimation procedures for spatially incomplete datasets. Therefore, we proposed a simple method of calibrating an MRC model when only data from a moderately dense network of rain gauges is available, rather than from the full rainfall field. The number of gauges needs only be sufficient to adequately estimate the variance in the ratio of the rain rate at the rain gauges to the areal average rain rate across the entire spatial domain. In our example for Warsaw, Poland, we used 25 gauges over an area of approximately 1600 km2. MRC models calibrated using the proposed method were used to downscale 15-min rainfall rates from a 20 by 20 km area to the scale of the rain gauge capture area. Frequency distributions of observed and simulated 15-min rainfall at the gauge scale were very similar. Moreover, the spatial covariance structure of rainfall rates, as characterized by the semivariogram, was reproduced after allowing the probability density function of the random cascade generator to vary with spatial scale.

Temporal dependence structure in weights in a multiplicative cascade model for precipitation

We investigate the ability of the multiplicative random cascade model to accurately simulate temporal precipitation. Specifically, we explore the effect of the dependence structure in cascade weights due to clustering and within‐storm variability on the temporal correlation in simulated precipitation, and we compare the results with data at 69 stations with 10 min precipitation records in Switzerland. Correlation is quantified with the oscillation coefficient, which is a measure of patterns of fluctuations in data. Simulation results show that the assumption of temporal independence in cascade weights is generally not supported by observations of both rainfall and snowfall, which show generally higher correlation (lower fluctuations) at the hourly time resolution. Seasonal signatures are also apparent, with higher correlation in the cold season with dominant stratiform precipitation than in the warm season with convective precipitation. Measurement artifacts caused by the tipping bucket mechanism at high resolutions (10 min) are shown to play a significant role in the estimation of the correlation structure in cascade weights because of the quantization of precipitation intensity by the tip volume and sampling time resolution of the gauge. These effects are smoothed out at resolutions above 1 h when the oscillation coefficients become independent of resolution. Such measurement artifacts may have an important effect on the estimated scaling and correlation behavior in precipitation at high temporal resolutions.

Random cascade driven rainfall disaggregation for urban hydrology: An evaluation of six models and a new generator

Six variations of multiplicative random cascade models for generating fine-resolution (i.e., 5-minute interval) rainfall time series were evaluated for rainfall in Wroclaw, Poland. Of these variations, one included a new beta-normal generator for a microcanonical cascade. This newly proposed model successfully reproduces the statistical behavior of local 5-minute rainfalls, in terms of intermittency as well as variability. In contrast, both the canonical cascade models with either constant or time-scaled parameters and a microcanonical cascade model with a beta generator substantially underestimate 5-minute maximum rainfall intensities. The canonical models also fail to properly reproduce the intermittency of the rainfall process across a range of timescales. New observations are also made concerning the histograms of the breakdown coefficients (BDC). The tendency of the BDC histograms to have values exactly equal to 0.5 is identified and explained by the quality of pluviograph records. Moreover, the hierarchical evolution of BDC histograms from beta-like for long time steps to beta-normal histograms for short time steps is observed for the first time. The potential advantage is shown of synthetic high resolution rainfall time series generated by the revised microcanonical model for use in hydrology, especially hydrodynamic modelling of urban drainage networks.

Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models

Abstract. Discrete multiplicative random cascade (MRC) models were extensively studied and applied to disaggregate rainfall data, thanks to their formal simplicity and the small number of involved parameters. Focusing on temporal disaggregation, the rationale of these models is based on multiplying the value assumed by a physical attribute (e.g., rainfall intensity) at a given time scale L, by a suitable number b of random weights, to obtain b attribute values corresponding to statistically plausible observations at a smaller L/b time resolution. In the original formulation of the MRC models, the random weights were assumed to be independent and identically distributed. However, for several studies this hypothesis did not appear to be realistic for the observed rainfall series as the distribution of the weights was shown to depend on the space-time scale and rainfall intensity. Since these findings contrast with the scale invariance assumption behind the MRC models and impact on the applicability of these models, it is worth studying their nature. This study explores the possible presence of dependence of the parameters of two discrete MRC models on rainfall intensity and time scale, by analyzing point rainfall series with 5-min time resolution. Taking into account a discrete microcanonical (MC) model based on beta distribution and a discrete canonical beta-logstable (BLS), the analysis points out that the relations between the parameters and rainfall intensity across the time scales are detectable and can be modeled by a set of simple functions accounting for the parameter-rainfall intensity relationship, and another set describing the link between the parameters and the time scale. Therefore, MC and BLS models were modified to explicitly account for these relationships and compared with the continuous in scale universal multifractal (CUM) model, which is used as a physically based benchmark model. Monte Carlo simulations point out that the dependence of MC and BLS parameters on rainfall intensity and cascade scales can be recognized also in CUM series, meaning that these relations cannot be considered as a definitive sign of departure from multifractality. Even though the modified MC model is not properly a scaling model (parameters depend on rainfall intensity and scale), it reproduces the empirical traces of the moments and moment exponent function as effective as the CUM model. Moreover, the MC model is able to reproduce some rainfall properties of hydrological interest, such as the distribution of event rainfall amount, wet/dry spell length, and the autocorrelation function, better than its competitors owing to its strong, albeit unrealistic, conservative nature. Therefore, even though the CUM model represents the most parsimonious and the only physically/theoretically consistent model, results provided by MC model motivate, to some extent, the interest recognized in the literature for this type of discrete models.

Multifractal analysis in a mixed asymptotic framework

Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, the observed process can be modeled by a concatenation of independent binary cascades and statistics are estimated over a sample whose size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where, at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of “mixed” partitions functions, that is, the estimator of the cumulant generating function of the cascade generator distribution depends on some “mixed asymptotic” exponent χ, respectively, above and below two critical value pχ− and pχ+. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. Moreover, within the mixed asymptotic framework, we establish a “box-counting” multifractal formalism that can be seen as a rigorous formulation of Mandelbrot’s negative dimension theory. Numerical illustrations of our results on specific examples are also provided. A possible application of these results is to distinguish data generated by log-Normal or log-Poisson models.

Causality across rainfall time scales revealed by continuous wavelet transforms

Rainfall variability occurs over a wide range of time scales owing to processes initiated by cloud microphysics and sustained by atmospheric circulation. A central topic in rainfall research is to determine whether rainfall variability at a given scale is caused by dynamics acting at some other scales. Random multiplicative cascades (RMCs) are standard approaches for describing rainfall variability across such a wide range of time scales. Their popularity stems from their ability to reproduce rainfall self‐similarity and long‐range correlations as well as intermittency buildup at finer scales. However, standard RMCs only predict instantaneous flow of variance (energy or activity) from large to fine scales and cannot account for scale‐wise causal relationships. Such relationships reveal themselves through noninstantaneous cascade mechanisms, namely, large‐scale events influencing finer‐scale events at later times (i.e., forward causal cascade) or conversely (inverse causal cascade). The presence of causal cascade signatures within the rainfall process is explored here using both continuous wavelet decomposition (CWT) and scale‐by‐scale causality measures such as cross‐scale correlation and linearized transfer entropy. The causality hypothesis is further tested against results from toy models, surrogate data, and a scalar turbulence time series (water vapor) to ensure that rainfall causality is not an artifact of the estimation method or resulting from the redundancy in CWT. The analysis demonstrates the presence of causal cascades (mainly forward) in rainfall series when sampled at fine temporal resolutions (seconds). These causal relationships tend to vanish when rainfall is aggregated at coarser time scales (hours and longer).

Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation

Multiplicative random cascades (MRCs) can parsimoniously generate highly intermittent patterns similar to those in rainfall. The elemental MRC model parameter is the cascade weight, which determines how rainfall at one scale is partitioned at the next smallest scale in the cascade. While it is known that the probability density of these weights may vary with both time scale and rainfall intensity, nearly all previous studies have considered either time scale or intensity separately. We examined the simultaneous dependency of the weights on both factors and assessed the impacts of explicitly including these dependencies in the MRC model. On the basis of the observed relationships between cascade weights and time scale and intensity, four progressively more “dependent” models were constructed to disaggregate a long time series of daily rainfall to hourly intervals. We found that inclusion of the intensity dependency on the model parameters that generate dry intervals greatly improved performance. For the relatively small range of time scales over which the rainfall was disaggregated, varying model parameters with time scale resulted in minor improvement.

Memory effects in the statistics of interoccurrence times between large returns in financial records

We study the statistics of the interoccurrence times between events above some threshold Q in two kinds of multifractal data sets (multiplicative random cascades and multifractal random walks) with vanishing linear correlations. We show that in both data sets the relevant quantities (probability density functions and the autocorrelation function of the interoccurrence times, as well as the conditional return period) are governed by power laws with exponents that depend explicitly on the considered threshold. By studying a large number of representative financial records (market indices, stock prices, exchange rates, and commodities), we show explicitly that the interoccurrence times between large daily returns follow the same behavior, in a nearly quantitative manner. We conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets.

Log selfsimilarity of continuous soil Particle-size distributions estimated using random multiplicative cascades

AbstractParticle-size distribution (PSD) is a fundamental soil property usually reported as discrete clay, silt, and sand percentages. Models and methods to effectively generate a continuous PSD from such poor descriptions using another property would be extremely useful to predict and understand in fragmented distributions, which are ubiquitous in nature. Power laws for soil PSDs imply scale invariance (or selfsimilarity), a property which has proven useful in PSD description. This work is based on two novel ideas in modeling PSDs: (1) the concept of selfsimilarity in PSDs; and (2) mathematical tools to calculate fractal distributions for specific soil PSDs using few actual texture data. Based on these ideas, a random, multiplicative cascade model was developed that relies on a regularity of scale invariance called ‘log-selfsimilarity.’ The model allows the estimation of intermediate particle size values from common texture data. Using equivalent inputs, this new modeling approach was checked using soil data and shown to provide greatly improved results in comparison to the selfsimilar model for soil PSD data. The Kolmogorov-Smirnov D-statistic for the log-selfsimilar model was smaller than the selfsimilar model in 92.94% of cases. The average error was 0.74 times that of the selfsimilar model. The proposed method allows measurement of a heterogeneity index, H, defined using Hölder exponents, which facilitates quantitative characterization of soil textural classes. The average H value ranged from 0.381 for silt texture to 0.838 for sandy loam texture, with a variance of <0.034 for all textural classes. The index can also be used to distinguish textures within the same textural class. These results strongly suggest that the model and its parameters might be useful in estimating other soil physical properties and in developing new soil PSD pedotransfer functions. This modeling approach, along with its potential applications, might be extended to fine-grained mineral and material studies.

Spatial disaggregation of bias-corrected GCM precipitation for improved hydrologic simulation: Ping River Basin, Thailand

Abstract. Global Climate Models (GCMs) precipitation scenarios are often characterized by biases and coarse resolution that limit their direct application for basin level hydrological modeling. Bias-correction and spatial disaggregation methods are employed to improve the quality of ECHAM4/OPYC SRES A2 and B2 precipitation for the Ping River Basin in Thailand. Bias-correction method, based on gamma-gamma transformation, is applied to improve the frequency and amount of raw GCM precipitation at the grid nodes. Spatial disaggregation model parameters (β,σ2), based on multiplicative random cascade theory, are estimated using Mandelbrot-Kahane-Peyriere (MKP) function at q=1 for each month. Bias-correction method exhibits ability of reducing biases from the frequency and amount when compared with the computed frequency and amount at grid nodes based on spatially interpolated observed rainfall data. Spatial disaggregation model satisfactorily reproduces the observed trend and variation of average rainfall amount except during heavy rainfall events with certain degree of spatial and temporal variations. Finally, the hydrologic model, HEC-HMS, is applied to simulate the observed runoff for upper Ping River Basin based on the modified GCM precipitation scenarios and the raw GCM precipitation. Precipitation scenario developed with bias-correction and disaggregation provides an improved reproduction of basin level runoff observations.

Use of Scale Recursive Estimation for assimilation of precipitation data from TRMM (PR and TMI) and NEXRAD

The paper shows an application of Scale Recursive Estimation (SRE) used to assimilate rainfall rates estimated during a storm event from three remote sensing devices. These are the TMI radiometer and the PR radar, carried on board of the TRMM satellite and the KNQA Memphis Weather Surveillance radar, belonging to the NEXRAD network, each one providing rain rate estimates at a different spatial scale. The variability of rain rate process in scales is modeled as a multiplicative random cascade, including spatial intermittence. The observational noise in the estimates is modeled according to a multiplicative error. System estimation, including process and observational noise, is carried out using Maximum Likelihood Estimation implemented by a scale recursive Expectation Maximization (EM) algorithm. As a result, new rainfall rate estimates are obtained that feature decreased estimation error as compared to those coming from each device alone. The performance of the SRE-EM approach is compared with that of the latest methods proposed for data fusion of multisensor estimates. The proposed approach improves the current methods adopted for SRE and provides an alternative for data fusion in the field of precipitation.

Complete multiplier statistics explained by stochastic cascade processes

Abstract The prevalent understanding of fully developed turbulence is that of a cascade in which vortices successively break up into smaller ones. Based on this idea, theoretical developments were often concerned with multiplicative random cascade processes. We focus on the velocity fluctuations and derive a closed as well as complete statistical description of the underlying velocity multipliers using a Fokker–Planck equation, which is estimated directly from experimental data. This shed new light on the statistics of multipliers and their often assumed independence. For the heavy-tailed statistical features of the multipliers, close to a Cauchy distribution, no intermittency of turbulence is needed.

Rainfall stochastic disaggregation models: Calibration and validation of a multiplicative cascade model

The simulation of long time series of rainfall rates at short time steps remains an important issue for various applications in hydrology. Among the various types of simulation models, random multiplicative cascade models (RMC models) appear as an appealing solution which displays the advantages to be parameter parsimonious and linked to the multifractal theory. This paper deals with the calibration and validation of RMC models. More precisely, it discusses the limits of the scaling exponent function method often used to calibrate RMC models, and presents an hydrological validation of calibrated RMC models. A 8-year time series of 1-min rainfall rates is used for the calibration and the validation of the tested models. The paper is organized in three parts. In the first part, the scaling invariance properties of the studied rainfall series is shown using various methods ( q-moments, PDMS, autocovariance structure) and a RMC model is calibrated on the basis of the rainfall data scaling exponent function. A detailed analysis of the obtained results reveals that the shape of the scaling exponent function, and hence the values of the calibrated parameters of the RMC model, are highly sensitive to sampling fluctuation and may also be biased. In the second part, the origin of the sensivity to sampling fluctuation and of the bias is studied in detail and a modified Jackknife estimator is tested to reduce the bias. Finally, two hydrological applications are proposed to validate two candidate RMC models: a canonical model based on a log-Poisson random generator, and a basic micro-canonical model based on a uniform random generator. It is tested in this third part if the models reproduce faithfully the statistical distribution of rainfall characteristics on which they have not been calibrated. The results obtained for two validation tests are relatively satisfactory but also show that the temporal structure of the measured rainfall time series at small time steps is not well reproduced by the two selected simple random cascade models.

Extreme values and fat tails of multifractal fluctuations

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non-self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framework that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed "inverse cubic law" of the return pdf tails.

Are asset return tail estimations related to volatility long-range correlations?

We discuss a possible scenario explaining in what respect the observed fat tails of asset returns or volatility fluctuations can be related to volatility long-range correlations. Our approach is based on recently introduced multifractal models for asset returns that account for the volatility correlations through a multiplicative random cascade. Within the framework of these models, it can be shown that the sample size required for a correct estimation of the behavior of extreme return fluctuations is generally huge and outside the range of accessible size of data. Consequently, in many cases, the extreme tail probability appears as a power-law, with a rather small (underestimated) tail exponent. We point out that increasing the amount of data by using smaller and smaller (intraday) scales, does not contribute to reduce the bias and, as observed empirically, the tail exponent turns out to be rather stable across scales.

Multiplicative cascades applied to PDEs (two numerical examples)

Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined. The results show initial numerical approximations with errors less than 5% in the leading Fourier coefficients of the solution. This approximation is then improved substantially using a Picard iteration scheme on the integral equation associated with the representation of the respective PDE in Fourier space.

Statistical performances of various deterministic and stochastic models for rainfall series disaggregation

This paper compares the performance of seven disaggregation models, based on various approaches and/or concepts, for the generation of 10-min time step rainfall series from hourly rainfall series. The so-called constant disaggregation model and a linear model based on the external temporal pattern of rainfall are first considered. The other models are stochastic: the first ones are based on a given probability density function applying for the 10-min rainfall amounts of the hour to disaggregate. This probability density function is either uniform or derived from the external temporal pattern of rainfall. The other stochastic models are scaling models using canonical or microcanonical multiplicative random cascades. The comparison of the models is based on their ability to reproduce some important statistical characteristics of the observed time series: variance, skewness coefficient, wet/dry properties of 10-min rainfall amounts; rainfall amounts quantiles for different return periods; autocorrelation of 10-min rainfall amounts. A continuous hydrological simulation is next applied to produce for each generated rainfall series a continuous discharge series used afterwards for a retention design. The ability of the different disaggregation models to produce rainfall time series resulting in the same retention design than the one obtained with the observed rainfall series is finally analysed. Deterministic models as well as simpler stochastic models have rather bad performances when compared to the others. Because it is non-conservative, the model based on a microcanonical random cascade performs also very poorly. It significantly overestimates all studied statistics. Models based on microcanonical random cascades achieve the best performance. They perform reasonably well for the reproduction of rainfall statistics and almost perfectly for the reproduction of runoff and storage design variables. Results finally highlight the interest of including in the disaggregation scheme information related to the external temporal pattern of rainfall.

Preservation of rainfall properties in stochastic disaggregation by a simple random cascade model

We analyse properties of a simple discrete multiplicative random cascade model for rainfall disaggregation in urban hydrology. Two types of cascade models (canonical and microcanonical) are applied to the stochastic fine graining of rainfall observations into high resolution data. In particular, we analyse the disaggregation of 1280-min into 10-min data for a 20-year record period (Zurich raingauge, 1979–1998). Differences between the models and parameter estimation techniques are explored on simulated data with a special focus on three important properties of observed rainfall: distribution, intermittency, and extremes. The canonical models are overall better at preserving the distribution of rainfall at the 10-min scale. It is demonstrated that the growth of intermittency across scales is preserved well with all studied models. The ability of the models to reproduce rainfall extremes is a fundamental requirement in disaggregation. The studied models preserved annual rainfall maxima satisfactorily for short durations; however, the performance deteriorated for longer durations. The canonical models performed substantially better in capturing the variability in rainfall. The results are encouraging considering the parsimonious nature of the models and simple parameter estimation.

Finite-size scaling of two-point statistics and the turbulent energy cascade generators

Within the framework of random multiplicative energy cascade models of fully developed turbulence, finite-size-scaling expressions for two-point correlators and cumulants are derived, taking into account the observationally unavoidable conversion from an ultrametric to an Euclidean two-point distance. The comparison with two-point statistics of the surrogate energy dissipation, extracted from various wind tunnel and atmospheric boundary layer records, allows an accurate deduction of multiscaling exponents and cumulants, even at moderate Reynolds numbers for which simple power-law fits are not feasible. The extracted exponents serve as input for parametric estimates of the probabilistic cascade generator. Various cascade generators are evaluated.

Two correlation patterns as indicators for underlying dynamics of complex systems

Two spatial correlation patterns, fixed-to-arbitrary and neighboring bin correlation patterns, are suggested. It is demonstrated that these patterns present scale-independent and distinguishable measures for correlated and uncorrelated random multiplicative cascade processes. Their application to very high multiplicity events in relativistic heavy ion collisions is discussed.