Multiplicative Random Cascade Model 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.

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.

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.

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.

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.

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.

Multisite multivariate disaggregation of climate parameters using multiplicative random cascades

The Multiplicative Random Cascade (MRC) disaggregation model has been extensively used to disaggregate precipitation in many regions across the globe. In this study, it is adapted to disaggregate a range of climate variables (CV) relevant for hygrothermal modelling of building envelopes. This generalized MRC-G model is further improved by explicitly modelling the cross-correlation structure between CVs in the MRC-G-CV model. A thorough evaluation of MRC, MRC-G, and MRC-G-CV models is performed for five Canadian cities: Ottawa, Vancouver, Calgary, St. Johns, and Winnipeg. Results indicate that the MRC model is able to simulate grid-level statistics with >90% accuracy. Grid-level extreme magnitudes and spatial cross-correlation structures are also well simulated. Error magnitudes associated with hourly predictions indicate superior performance of the models in respect to thermal variables, followed by wind variables, and then moisture related variables. Finally, the performance of MRC-G model towards modelling cross-correlation among CVs is found to improve by >50% in terms of energy distance by explicitly modelling these relationships in the MRC-G-CV model. Results indicate that the MRC model variants demonstrated in this study have the potential to facilitate effective hygrothermal performance evaluation of building envelopes at locations where observational sub-daily climate records are unavailable.

TEMPORAL RAINFALL DISAGGREGATION BY A SIMPLE RANDOM CASCADE MODEL

This study evaluates the utility and suitability of a simple discrete multiplicative random cascade model for temporal rainfall disaggregation. Two of a simple random cascade model, namely log-Poisson and log-Normal models are applied to simulate hourly rainfall from daily rainfall at seven rain gauge stations in Peninsular Malaysia. The cascade models are evaluated based on the capability to simulate data that preserve three important properties of observed rainfall: rainfall variability, intermittency and extreme events. The results show that both cascade models are able to simulate reasonably well the commonly used statistical measures for rainfall variability (e.g. mean and standard deviation) of hourly rainfall. With respect to rainfall intermittency, even though both models are underestimated, the observed dry proportion, log-Normal model is likely to simulate number of dry spells better than log-Poisson model. In terms of rainfall extremes, it is demonstrated that log-Poisson and log-Normal models gave a satisfactory performance for most of the studied stations herein, except for Dungun and Kuala Krai stations, which both located in the east part of Peninsula.

A Multiplicative Cascade Model for High‐Resolution Space‐Time Downscaling of Rainfall

AbstractDistributions of rainfall with the time and space resolutions of minutes and kilometers, respectively, are often needed to drive the hydrological models used in a range of engineering, environmental, and urban design applications. The work described here is the first step in constructing a model capable of downscaling rainfall to scales of minutes and kilometers from time and space resolutions of several hours and a hundred kilometers. A multiplicative random cascade model known as the Short‐Term Ensemble Prediction System is run with parameters from the radar observations at Melbourne (Australia). The orographic effects are added through multiplicative correction factor after the model is run. In the first set of model calculations, 112 significant rain events over Melbourne are simulated 100 times. Because of the stochastic nature of the cascade model, the simulations represent 100 possible realizations of the same rain event. The cascade model produces realistic spatial and temporal patterns of rainfall at 6 min and 1 km resolution (the resolution of the radar data), the statistical properties of which are in close agreement with observation. In the second set of calculations, the cascade model is run continuously for all days from January 2008 to August 2015 and the rainfall accumulations are compared at 12 locations in the greater Melbourne area. The statistical properties of the observations lie with envelope of the 100 ensemble members. The model successfully reproduces the frequency distribution of the 6 min rainfall intensities, storm durations, interarrival times, and autocorrelation function.

Cascade Rainfall Disaggregation Application in U.S. Central Plains

Long-term observations of daily rainfall are common and routinely available for a variety of hydrologic applications. In contrast, observations of 10 or more years of continuous hourly rainfall are rare. Yet, sub-daily rainfall data are required in rainfall-runoff models. Rainfall disaggregation can generate sub-daily time-series from available long term daily observations. Herein, the performance of Multiplicative Random Cascade (MRC) model at disaggregating daily-to-hourly rainfall was investigated. The MRC model was parameterized and validated with 15 years of continuous observed daily and hourly rainfall data at three weather stations in Oklahoma. Model performance, or degree to which the disaggregated rainfall time series replicated observations, was assessed using 46 variables of hourly rainfall characteristics, such as longest wet spell duration, average number of rainfall hours per year, and largest hourly rainfall. Findings include: a) average-type hourly rainfall characteristics were better replicated than single value characteristics such as longest, maximum, or peak hourly rainfall; b) the large number of sub-trace hourly rainfall values (<0.254 mm h-1) generated by the MRC model were not supported by observations; c) the random component of the MRC model led to a variation under 15% of the average value for most rainfall characteristics with the exceptions of the “longest wet spell duration” and “maximum hourly rainfall”; and d) the MRC model produced fewer persistent rainfall events compared to those in the observed rainfall record. The large number of generated trace rainfall values and difficulties to replicate reliably extreme rainfall characteristics, reduces the number of potential hydrologic applications that could take advantage of the MRC disaggregated hourly rainfall. Nevertheless, in most cases, the disaggregated rainfall generated by the MRC model replicated observed average-type rainfall characteristics well.

Temporal rainfall disaggregation using a multiplicative cascade model for spatial application in urban hydrology

Rainfall time series of high temporal resolution and spatial density are crucial for urban hydrology. The multiplicative random cascade model can be used for temporal rainfall disaggregation of daily data to generate such time series. Here, the uniform splitting approach with a branching number of 3 in the first disaggregation step is applied. To achieve a final resolution of 5 min, subsequent steps after disaggregation are necessary. Three modifications at different disaggregation levels are tested in this investigation (uniform splitting at Δt = 15 min, linear interpolation at Δt = 7.5 min and Δt = 3.75 min). Results are compared both with observations and an often used approach, based on the assumption that a time steps with Δt = 5.625 min, as resulting if a branching number of 2 is applied throughout, can be replaced with Δt = 5 min (called the 1280 min approach). Spatial consistence is implemented in the disaggregated time series using a resampling algorithm. In total, 24 recording stations in Lower Saxony, Northern Germany with a 5 min resolution have been used for the validation of the disaggregation procedure. The urban-hydrological suitability is tested with an artificial combined sewer system of about 170 hectares. The results show that all three variations outperform the 1280 min approach regarding reproduction of wet spell duration, average intensity, fraction of dry intervals and lag-1 autocorrelation. Extreme values with durations of 5 min are also better represented. For durations of 1 h, all approaches show only slight deviations from the observed extremes. The applied resampling algorithm is capable to achieve sufficient spatial consistence. The effects on the urban hydrological simulations are significant. Without spatial consistence, flood volumes of manholes and combined sewer overflow are strongly underestimated. After resampling, results using disaggregated time series as input are in the range of those using observed time series. Best overall performance regarding rainfall statistics are obtained by the method in which the disaggregation process ends at time steps with 7.5 min duration, deriving the 5 min time steps by linear interpolation. With subsequent resampling this method leads to a good representation of manhole flooding and combined sewer overflow volume in terms of hydrological simulations and outperforms the 1280 min approach.

Temporal Rainfall Disaggregation with a Cascade Model: From Single-Station Disaggregation to Spatial Rainfall

Temporal rainfall disaggregation is an important tool to obtain high-resolution rainfall data, which are needed in many fields of hydrology and water resources management. The multiplicative random cascade model can be used for temporal rainfall disaggregation of daily time series. A resampling algorithm is introduced to implement spatial consistence in disaggregated time series. Spatial consistence is assumed to be represented by four bivariate and distance-dependent rainfall characteristics that complement each other. Relative diurnal cycles of the disaggregated time series are resampled with the aim to reproduce these spatial characteristics while preserving the structure generated by the cascade model. Also, to achieve a final resolution of 1 h the traditional cascade model has been modified. A modification called uniform splitting with a branching number of 3 in the first step is introduced. Results are compared with observations and an approach called diversion. In total 22 recording stations in Northern Germany with hourly resolution were used for the validation of the disaggregation procedure, starting with daily values. Investigation areas are two catchments considering different station densities. The results show that for the disaggregation, errors of time series characteristics between 3 and 12% occur. The nonexceedance curves of rainfall intensities are slightly overestimated. Extreme values are well represented. The uniform splitting method outperforms the diversion method. Spatial rainfall characteristics can be reproduced by the simulating annealing algorithm. However, with an increasing number of stations the reproduction performance declines for some rainfall characteristics. Nonexceedance curves of areal rainfall based on disaggregated and not resampled time series are generally underestimated. By application of the resampling algorithm, a better performance regarding the spatial characteristics can be achieved. The presented resampling algorithm also has the potential to be used for implementing spatial consistence for time series generated by other disaggregation models.

Stochastic temporal disaggregation of monthly precipitation for regional gridded data sets

AbstractWeather generators are used for spatiotemporal downscaling of climate model outputs (e.g., precipitation and temperature) to investigate the impact of climate change on the hydrological cycle. In this study, a multiplicative random cascade model is proposed for the stochastic temporal disaggregation of monthly to daily precipitation fields, which is designed to be applicable to grids of any spatial resolution and extent. The proposed method uses stationary distribution functions that describe the partitioning of precipitation throughout multiple temporal scales (e.g., weekly and biweekly scale). Moreover, it explicitly considers the intensity and spatial covariance of precipitation in the disaggregation procedure, but requires no assumption about the temporal relationship and spatial isotropy of precipitation fields. A split sampling test is conducted on a high‐resolution (i.e., 4 × 4 km2 grid) daily precipitation data set over Germany (≈357,000 km2) to assess the performance of the proposed method during future periods. The proposed method has proven to consistently reproduce distinctive location‐dependent precipitation distribution functions with biases less than 5% during both a calibration and evaluation period. Furthermore, extreme precipitation amounts and the spatial and temporal covariance of the generated fields are comparable to those of the observations. Consequently, the proposed temporal disaggregation approach satisfies the minimum conditions for a precipitation generator aiming at the assessment of hydrological response to climate change at regional and continental scales or for generating seamless predictions of hydrological variables.

On temporal stochastic modeling of precipitation, nesting models across scales

We analyze the performance of composite stochastic models of temporal precipitation which can satisfactorily reproduce precipitation properties across a wide range of temporal scales. The rationale is that a combination of stochastic precipitation models which are most appropriate for specific limited temporal scales leads to better overall performance across a wider range of scales than single models alone. We investigate different model combinations. For the coarse (daily) scale these are models based on Alternating renewal processes, Markov chains, and Poisson cluster models, which are then combined with a microcanonical Multiplicative Random Cascade model to disaggregate precipitation to finer (minute) scales. The composite models were tested on data at four sites in different climates. The results show that model combinations improve the performance in key statistics such as probability distributions of precipitation depth, autocorrelation structure, intermittency, reproduction of extremes, compared to single models. At the same time they remain reasonably parsimonious. No model combination was found to outperform the others at all sites and for all statistics, however we provide insight on the capabilities of specific model combinations. The results for the four different climates are similar, which suggests a degree of generality and wider applicability of the approach.

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.

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.