Multiplicative Random Cascade Research Articles

Development of an inclusive, scalable, and flexible hydrologic modeling system: Establishing integrated flood simulation system at agricultural watersheds

In this study, we developed a comprehensive hydrological modeling system to address the diverse needs of hydrologists and researchers. The system comprised nine modules, each serving a specific purpose. These modules include a multiplicative random cascade model, frequency analysis, inflow simulation, Hydrologic Engineering Center – 5, Hydrologic Engineering Center – River Analysis System, and farmland drainage simulations. The system follows a structured sequence within a large framework, beginning with user inputs for the initial condition information and period specifications for frequency analysis. It then calculates the time-disaggregated precipitation data utilizing the Dask distributed server for efficient computation. Subsequent module computations were conducted on dedicated mask workers. The Integrated Database serves as a comprehensive repository for simulation studies, encompassing historical precipitation data, shared socioeconomic pathways, climate change scenario data, and reservoir and farmland survey data. This system has been used in several studies and has provided cohesive and reliable results for flood simulations.

Machine-learning classification with additivity and diverse multifractal pathways in multiplicativity

Evidence of multifractal structures has spread to a wider set of physiological time series supporting the intricate interplay of biological and psychological functioning. These dynamics manifest as random multiplicative cascades, embodying nonlinear relationships characterized by recurring division, branching, and aggregation processes implicating noise across successive generations. This investigation focuses on how well the diversity of multifractal properties can be specific to the type of cascade relationship between generation (i.e., multiplicative, additive, or a mixture) as well as to the type of noise (i.e., including additive white Gaussian noise, fractional Gaussian noise, and various amalgamations) among 15 distinct types of binomial cascade processes. Cross-correlation analysis of multifractal spectral features confirms that these features capture nuanced aspects of cascading processes with minimal redundancy. Principal component analysis using 13 distinct multifractal spectral features shows that different cascade processes can manifest multifractal evidence of nonlinearity for distinct reasons. This transparency of multifractal spectral features to underlying cascade dynamics becomes less amenable to machine-learning strategies. Fully connected neural networks struggled to classify the 15 distinct types of cascade processes based on the respective multifractal spectral features (45.5% accuracy) yet demonstrated improved accuracy when addressing single categories of cross-generation relationships, that is, additive (91.6%), multiplicative (75.4%), or additomultiplicative (70.6%). While traditional principal component analysis reveals distinct loadings attributed to individual noise processes, multiplicative relationships between generations effectively make the constituent noise processes less discernible to neural networks. Neural networks may lack sufficient hierarchical depth required to effectively distinguish among nonadditive cascading processes, recommending either elaborating multifractal geometry or using alternate architectures for machine-learning classification of cascades with multiplicative relationships. Published by the American Physical Society 2024

Applicability of statistical and deep-learning models for rainfall disaggregation at metropolitan stations in India

Study regionMetropolitan stations in India with distinct climatic conditions, namely, Delhi, Chennai, Kolkata, and Mumbai were selected for this study. Study focusRainfall disaggregation models were studied based on four models, namely Neyman-Scott Rectangular pulse (NSRP) process, Microcanonical Multiplicative Random Cascade (MMRC) process and its variant MMRC-K, and one Deep-Learning based process (ANN-K), using metrics like dry periods, event rainfall volumes, extreme rainfall characteristics, etc. New hydrological insights for the regionThe study successfully established individual rainfall volume, event rainfall volume, and event durations are all within 10% of each other for the four stations. Delhi due to its continental climate showed a higher percentage of dry periods and longer dry periods, which were most successfully modelled by MMRC and MMRC-K. Kolkata and Mumbai stations displayed a higher number of extremely intense and cloudburst types of rainfall, which were modeled effectively by the Deep-Learning Model. Chennai has a different rainfall pattern due to returning monsoon which was also captured by the models. Generally, for extreme rainfall parameters, the ANN-K model performs significantly better, successfully reproducing the characteristics at all quantiles, especially with rainfall above 100 mm/hour intensity or cloud bursts, while 50% of the models overestimated these. NSRP on the other hand performs reasonably well for most considered parameters, without being exceptional at any of them. MMRC and MMRC-K most accurately modeled the dry period parameters.

Evaluation of future flood probability in agricultural reservoir watersheds using an integrated flood simulation system

In agricultural reservoir watersheds, the presence of various agricultural hydraulic structures combined with the influence of climate change necessitates effective flood risk analysis. This study aims to develop a flood risk evaluation technique for agricultural basins by incorporating the low temporal resolution of climate change scenarios. The approach utilizes flood probability as a key component and incorporates the Multiplicative Random Cascade model as a probability-based rainfall time decomposition model to capture the inherent uncertainty in climate change data. Reliability analysis with the limit state equation is employed for probability-based risk evaluation. Given that agricultural watersheds typically consist of agricultural reservoirs and farmland, the study develops a hydraulic and hydrological linkage model to comprehensively calculate flood probability, simulating from the agricultural reservoir in the upper side to the farmland located at the bottom edge of the watershed. The flood risk assessment is applied to the Yedang Reservoir watershed in Korea, using the Shared Socio-economic Pathways climate change scenario. Results indicate that flood probability is the highest in river levees during future precipitation events, converging to nearly 100% under 24-hour duration rainfall, irrespective of the rain frequency. Despite the hydraulic structures at the farmland being designed for a lower frequency of rain events, the average probability of flooding in the farmland have reached 50%. The agricultural reservoir exhibits a low risk of flooding even under climate change conditions. The proposed method not only enables the establishment of flood countermeasures but also facilitates the evaluation and prediction of future flood risk for each hydraulic structure located in rural areas affected by climate change.

Accounting for precipitation asymmetry in a multiplicative random cascade disaggregation model

Abstract. Analytical multiplicative random cascades (MRCs) are widely used for the temporal disaggregation of coarse-resolution precipitation time series. This class of models applies scaling models to represent the dependence of the cascade generator on the temporal scale and the precipitation intensity. Although determinant, the dependence on the external precipitation pattern is usually disregarded in the analytical scaling models. Our work presents a unified MRC modelling framework that allows the cascade generator to depend in a continuous way on the temporal scale, precipitation intensity and a so-called precipitation asymmetry index. Different MRC configurations are compared for 81 locations in Switzerland with contrasted climates. The added value of the dependence of the MRC on the temporal scale appears to be unclear, unlike what was suggested in previous works. Introducing the precipitation asymmetry dependence into the model leads to a drastic improvement in model performance for all statistics related to precipitation temporal persistence (wet–dry transition probabilities, lag-n autocorrelation coefficients, lengths of dry–wet spells). Accounting for precipitation asymmetry seems to solve this important limitation of previous MRCs. The model configuration that only accounts for the dependence on precipitation intensity and asymmetry is highly parsimonious, with only five parameters, and provides adequate performances for all locations, seasons and temporal resolutions. The spatial coherency of the parameter estimates indicates a real potential for regionalisation and for further application to any location in Switzerland.

Introduction of k-means clustering into random cascade model for disaggregation of rainfall from daily to 1-hour resolution with improved preservation of extreme rainfall

Temporal disaggregation of rainfall has been of particular focus because of the non-availability of higher-resolution rainfall data for a long-duration period. Fine temporal resolution rainfall is used in a multitude of hydrological applications. Researchers have proposed various disaggregation models to disaggregate coarse temporal resolution rainfall. In this paper, firstly, the microcanonical multiplicative random cascade (MMRC) model is applied for disaggregation from daily rainfall to a one-hour scale. The model is applied in four different rainfall stations for disaggregation having varying rainfall patterns and characteristics. It is observed that the MMRC model can generate statistically reliable rainfall time series; however, the extreme rainfall characteristics are not well conserved by the model for all the stations.This paper describes a new model based on a random multiplicative cascade process where classification and parameter generation is done by k-means clustering such that it can better conserve extreme rainfall conditions and generate a reliable rainfall time series (MMRC-K). K-means clustering is a vector quantization method that divides the observations into a particular number of clusters based on the nearest mean called cluster centroid. The novel approach is tested with the same four Indian cities. The use of k-means clustering has made the classification and parameter generation of the model robust such that it can work with data sets of varying characteristics. It is found that MMRC-K provides improved conservation of extreme rainfall characteristics compared to the MMRC model for all four stations. The MMRC-K model reproduces the IDF curves of Delhi and Mumbai stations quite well; however, a little discrepancy was observed at higher resolution and larger return periods in Kolkata and Chennai stations. Extreme rainfall at finer resolution is used in various hydrological analyses and design problems like urban drainage design, stormwater management, etc. The overall superior conservation of the extreme rainfall characteristics in the model-generated rainfall time series by the MMRC-K model compared to the MMRC model supports the potential applicability of the model for temporal disaggregation.

Assessing the performance of daily to subdaily temporal disaggregation methods for the IDF curve generation under climate change

AbstractGiven the short concentration time in urban watersheds, the design of municipal water infrastructures often requires knowledge of sub-daily precipitation intensity. Sub-daily time series can be directly used in a rainfall–runoff model or to derive intensity–duration–frequency (IDF) curves and calculate the design precipitation. Given that precipitation projections are typically at a daily time scale, temporal disaggregation using techniques of variable complexity is often needed to evaluate the risk/performance of urban infrastructure in the future. This paper proposes a simple steady-state stochastic disaggregation model that generates wet/dry day occurrence using a binomial distribution and precipitation intensity using an exponential distribution. Daily precipitation data from four regional climate models (RCMs) forced with the high-emission scenario representative concentration pathway (RCP 8.5) were downscaled using the quantile mapping (QM) method. The performance of the developed method is compared to widely used temporal disaggregation methods, namely, the multiplicative random cascade model (MRC), the Hurst–Kolmogorov process (HKP), and three versions of the K-nearest neighbour (KNN) model, using the Kolmogorov–Smirnov (KS) test. The six disaggregation techniques were assessed at four stations in the South Nation River Watershed in Eastern Ontario, Canada. Results indicate that, despite its simplicity, the proposed method performed well compared to other temporal disaggregation methods when resampling the observed extreme precipitation.

Generating a missing half of multifractal fields with a blunt extension of discrete cascades

ABSTRACT Despite strong limitations, discrete random multiplicative cascades are often used to address scale issues that are ubiquitous in geosciences. A blunt extension based on the parsimonious framework of universal multifractals has recently been suggested. It preserves its simplicity and intuitiveness while overcoming its non-stationarity features. It relies on smoothing through a geometrical moving average the increments at each cascade step. Here, a space-time extension is suggested. Theoretically expected multifractal behaviour is retrieved on numerical simulations for typical rainfall parameters. A new algorithm to generate the missing half of multifractal fields in one, two or three dimensions is developed and tested on rainfall fields and numerical simulations. It consists in stochastically generating half of the increments and deterministically iteratively reconstructing the others to retrieve the available data and ensure a smooth transition with the unknown portion while preserving the multifractal behaviour. Potential applications to nowcasting of hydro-meteorological extremes are discussed.

A local fitting based multifractal detrend fluctuation analysis method

In this study, we propose a one-dimensional multifractal local fitting detrending fluctuation analysis (MF-LF-DFA) algorithm using traditional multifractal detrended fluctuation analysis (MF-DFA). The classical MF-DFA uses the least squares technique to fit the local trend of the time series, and the fitting order is fixed. However, the proposed MF-LF-DFA algorithm sets the fitting order reasonably for different local intervals according to the fluctuation characteristics of the sequence. We examine the performances of MF-DFA and MF-LF-DFA using the time series made using p-model multiplication cascade. By comparing the numerical results and analytical values of the generalized Hurst exponent H(q), the Scaling exponent τ(q) and multifractal spectrum f(α), the empirical tests demonstrate that H(q), τ(q) and f(α) calculated by MF-LF-DFA algorithm are closer to the analytical values. Furthermore, we compute the relative errors Herr and τerr of the two methods, and the relative errors calculated by MF-LF-DFA are smaller. Besides, we examine the monofractal structure of MF-LF-DFA by constructing a random sequence and calculating the value of H(2). In addition, we generate and test shuffled and phase Fourier randomized sequences of the original random time series and multiplicative cascade time series. We also study different length sequences within the same model, and verify the stability of the results. Finally, we change the range of segmentation window s and the fitting order of MF-DFA model to check the robust property of the proposed MF-LF-DFA algorithm. The computational results indicate that MF-LF-DFA outperforms the traditional MF-DFA in all situations.

Box-Counting Dimension in One-Dimensional Random Geometry of Multiplicative Cascades

We investigate the box-counting dimension of the image of a set E subset mathbb {R} under a random multiplicative cascade function f. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for the box-counting dimension. However, we show that this is far from true in general, and we compute explicitly a formula of a very different nature that gives the almost sure box-counting dimension of the random image f(E) when the set E comprises a convergent sequence. In particular, the box-counting dimension of f(E) depends more subtly on E than just on its dimensions. We also obtain lower and upper bounds for the box-counting dimension of the random images for general sets E.

Forecasting green roof detention performance by temporal downscaling of precipitation time-series projections

Abstract. A strategy to evaluate the suitability of different multiplicative random cascades to produce rainfall time series, taking into account climate change, inputs for green infrastructures models. The multiplicative random cascades reproduce a (multi)fractal distribution of precipitation through an iterative and multiplicative random process. In the current study, the initial model, a flexible cascade that deviates from multifractal scale invariance, was improved with (i) a temperature dependency and (ii) an additional function to reproduce the temporal structure of rainfall. The structure of the models with depth and temperature dependency was found to be applicable in eight locations studied across Norway and France. The resulting time series from both reference period and projection based on RCP 8.5 were applied to two green roofs with different properties. The different models led to a slight change in the performance of green roofs, but this was not significant compared to the range of outcomes due to ensemble uncertainty in climate modelling and the stochastic uncertainty due to the nature of the process. The hydrological dampening effect of the green infrastructure was found to decrease in most of the Norwegian cities due to an increase in precipitation, especially Bergen (Norway), while slightly increasing in Marseille (France) due to decrease in rainfall event frequency.

Noncommutative Probability and Multiplicative Cascades

Various aspects of standard model particle physics might be explained by a suitably rich algebra acting on itself, as suggested by Furey (2015). The present paper develops the asymptotics of large causal tree diagrams that combine freely independent elements in such an algebra. The Marčenko–Pastur law and Wigner’s semicircle law are shown to emerge as limits of normalized sum-over-paths of nonnegative elements assigned to the edges of causal trees. These results are established in the setting of noncommutative probability. Trees with classically independent positive edge weights (random multiplicative cascades) were originally proposed by Mandelbrot as a model displaying the fractal features of turbulence. The novelty of the present work is the use of noncommutative (free) probability to allow the edge weights to take values in an algebra. An application to theoretical neuroscience is also discussed.

Multiplicative random cascade models of multifractal urban structures

Spatial structures of urban systems are best described as multifractals. To provide insight on a hidden mechanism responsible for such scaling we investigate the multiplicative random cascade (MRC) model of urban spatial structure. MRC is a multiplicative process capable of producing multifractal 2D patterns, which is expressed in a closed analytical form. We use street intersection points pattern (SIPP) data as a proxy of urban structure. Renyi’s generalized dimensions (RGD) spectra, probability distributions of intersections densities, and urban layouts generated by the MRC model are compared to the same descriptors extracted from SIPP data for six U.S. metropolitan statistical areas (MSAs). We found that the RGD spectrum of the best-fit MRC model closely matches the observed spectrum in all cases. We also found that CDF of models’ density distributions do not match closely observed density distributions, although, in some cases, differences are not large. Observed density distributions have forms that are close to the log-normal distribution which suggests that a hidden process is multiplicative. Finally, we found that the MRC model is broadly compatible with the observed urban layouts. Given our findings, we suggest that the hidden process of urban scaling is the preferential attachment dynamics — development is attracted to areas in proportion to the degree of their existing development.

Statistical model for filamentary structures of molecular clouds

We propose a new statistical model that can reproduce the hierarchical nature of the ubiquitous filamentary structures of molecular clouds. This model is based on the multiplicative random cascade, which is designed to replicate the multifractal nature of intermittency in developed turbulence. We present a modified version of the multiplicative process where the spatial fluctuations as a function of scales are produced with the wavelet transforms of a fractional Brownian motion realisation. This simple approach produces naturally a log-normal distribution function and hierarchical coherent structures. Despite the highly contrasted aspect of these coherent structures against a smoother background, their Fourier power spectrum can be fitted by a single power law. As reported in previous works using the multiscale non-Gaussian segmentation (MnGSeg) technique, it is proven that the fit of a single power law reflects the inability of the Fourier power spectrum to detect the progressive non-Gaussian contributions that are at the origin of these structures across the inertial range of the power spectrum. The mutifractal nature of these coherent structures is discussed, and an extension of the MnGSeg technique is proposed to calculate the multifractal spectrum that is associated with them. Using directional wavelets, we show that filamentary structures can easily be produced without changing the general shape of the power spectrum. The cumulative effect of random multiplicative sequences succeeds in producing the general aspect of filamentary structures similar to those associated with star-forming regions. The filamentary structures are formed through the product of a large number of random-phase linear waves at different spatial wavelengths. Dynamically, this effect might be associated with the collection of compressive processes that occur in the interstellar medium.

A new discrete multiplicative random cascade model for downscaling intermittent rainfall fields

Abstract. Spatial downscaling of rainfall fields is a challenging mathematical problem for which many different types of methods have been proposed. One popular solution consists of redistributing rainfall amounts over smaller and smaller scales by means of a discrete multiplicative random cascade (DMRCs). This works well for slowly varying homogeneous rainfall fields but often fails in the presence of intermittency (i.e., large amounts of zero rainfall values). The most common workaround in this case is to use two separate cascade models, namely one for the occurrence and another for the intensity. In this paper, a new and simpler approach based on the notion of equal-volume areas (EVAs) is proposed. Unlike classical cascades where rainfall amounts are redistributed over grid cells of equal size, the EVA cascade splits grid cells into areas of different sizes, with each of them containing exactly half of the original amount of water. The relative areas of the subgrid cells are determined by drawing random values from a logit-normal cascade generator model with scale and intensity-dependent standard deviation (SD). The process ends when the amount of water in each subgrid cell is smaller than a fixed-bucket capacity, at which point the output of the cascade can be resampled over a regular Cartesian mesh. The present paper describes the implementation of the EVA cascade model and gives some first results for 100 selected events in the Netherlands. Performance is assessed by comparing the outputs of the EVA model to bilinear interpolation and to a classical DMRC model based on fixed grid cell sizes. Results show that, on average, the EVA cascade outperforms the classical method, producing fields with more realistic distributions, small-scale extremes and spatial structures. Improvements are mostly credited to the higher robustness of the EVA model in the presence of intermittency and to the lower variance of its generator. However, both approaches have their advantages and weaknesses. For example, while the classical cascade tends to overestimate small-scale variability and extremes, the EVA model tends to produce fields that are slightly too smooth and block shaped compared to the observations. The complementary nature of the two approaches, and the fact that they produce errors of opposite signs, opens up new possibilities for quality control and bias corrections of downscaled fields.

Scale Symmetry in the Universe

Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a truly scale invariant distribution of matter, namely it gives rise to a fractal geometry. A suitable explanation of the features of the fractal cosmic mass distribution is provided by the nonlinear Poisson–Boltzmann–Emden equation. An alternative interpretation of this equation is connected with theories of quantum gravity. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades, which originated in the theory of fluid turbulence. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution.

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.

Temperature-Driven Rise in Extreme Sub-Hourly Rainfall

AbstractEstimates of present and future extreme sub-hourly rainfall are derived from a daily spatial followed by a sub-daily temporal downscaling, the latter of which incorporates a novel, and crucial, temperature sensitivity. Specifically, daily global climate fields are spatially downscaled to local temperature T and precipitation P, which are then disaggregated to a temporal resolution of 10 min using a multiplicative random cascade model. The scheme is calibrated and validated with a group of 21 station records of 10-min resolution in Germany. The cascade model is used in the classical (denoted as MC) and in the new T-sensitive (MC+) version, which respects local Clausius–Clapeyron (CC) effects such as CC scaling. Extreme P is positively biased in both MC versions. Observed T sensitivity is absent in MC but well reproduced by MC+. Long-term positive trends in extreme sub-hourly P are generally more pronounced and more significant in MC+ than in MC. In units of 10-min rainfall, observed centennial trends in annual exceedance counts (EC) of P > 5 mm are +29% and in 3-yr return levels (RL) +27%. For the RCP4.5-simulated future, higher extremes are projected in both versions MC and MC+: per century, EC increases by 30% for MC and by 83% for MC+; the RL rises by 14% for MC and by 33% for MC+. Because the projected daily P trends are negligible, the sub-daily signal is mainly driven by local temperature.

Multiplicative random cascades with additional stochastic process in financial markets

Multiplicative random cascade model naturally reproduces the intermittency or multifractality, which is frequently shown among hierarchical complex systems such as turbulence and financial markets. As described herein, we investigate the validity of a multiplicative hierarchical random cascade model through an empirical study using financial data. Although the intermittency and multifractality of the time series are verified, random multiplicative factors linking successive hierarchical layers show strongly negative correlation. We extend the multiplicative model to incorporate an additional stochastic term. Results show that the proposed model is consistent with all the empirical results presented here.

An hourly‐scale scenario‐neutral flood risk assessment in a mesoscale catchment under climate change

AbstractReliable assessments of flood risks are essential to formulate robust adaptation policies to climate change. Although the scenario‐neutral flood assessments have reduced the dependency on uncertain climate predictions, coarse temporal resolutions of rainfall–runoff modelling adopted for the stress tests may introduce appreciable bias to flood risks driven by climatic stresses. To refine the temporal scale of flow estimates, this study proposes to incorporate a multiplicative random cascade (MRC) scheme and a simple catchment model in the bottom‐up flood risk assessment. Results showed that use of a daily flow indicator for the stress tests could considerably underestimate the impact of climatic change on flood risks. The non‐linearity between daily and hourly peak flows could increasingly amplify flood risks as the mean and variance of daily precipitation become greater in the study catchment. The first‐order catchment model combined with the MRC could acceptably estimate hourly peak flows and the catchment recession behaviours at high flows. This study suggests that subdaily flood indicators should be used in the scenario‐neutral assessments for small or mesoscale catchments to prevent underprediction of flood risks. To expand the applicability of the bottom‐up framework, we also suggest developing efficient tools that can perturb high‐resolution weather time series for the stress tests.