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

An Approach for Modeling the Orographic–Forcing Effect via Random Cascades and the Long‐Term Statistics of Mexico City's Daily Precipitation

AbstractThe orographic effect on the spatial structure of precipitation is a fundamental problem in hydrometeorology that still requires a better understanding of the physical processes involved in the emergence of rainfall patterns and their complex statistical structure. In tropical regions, where meteorological measurements are notoriously sparse and data quality control is often poor or missing, the study of precipitation modeling and prediction is challenging. This research aims to show an innovative approach based on a random cascade downscaling method to generate high‐resolution precipitation products from coarse‐scale precipitation products. This approach also includes a topographic enhancement function for describing the altitudinal variability of precipitation and a numerical diffusion filter to lessen the blockiness problem of random cascades. The suggested approach was applied to analyze some long‐term precipitation statistics in the metropolitan area of Mexico City. The model result agrees closely with the temporal statistics of the selected precipitation products and reflects complex orographic constraints. The proposed downscaling approach becomes an alternative to expensive computational methods and allows urban hydrology applications and analysis of small watersheds to incorporate the effects of complex orography.

Statistical downscaling of GCMs wind speed data for trend analysis of future scenarios: a case study in the Lombardy region

Near-surface wind speed is a key climatic variable, affecting many sectors, such as energy production, air pollution, and natural hazard. Lombardy region of Italy is among the European areas with lowest average wind speed, leading generally to low air quality and wind energy potential. However, it is also one of the most affected area by tornadoes in Italy. Here we investigate possible changes in wind circulation as due to prospective global warming. We analysed wind speed WS under future scenarios (SSP1-2.6 and SSP5-8.5) from six Global Climate Models (GCMs) until 2100, tuned against observed WS data. We employed a statistical downscaling method, namely Stochastic Time Random Cascade (STRC) to correct locally GCMs outputs. Three statistical tests, i.e. Linear Regression, Mann Kendall, Moving Window Average, were carried out to analyse future trends of: annual WS averages, 95th quantile (as an indicator of large WS), and the number of days of calm wind per year (NWC). The proposed STRC algorithm can successfully adjust the mean, standard deviation, and autocorrelation structure of the GCM outputs. No strong trends are found for the future. The chosen variables would all display non-stationarity, and the 95th percentile display a positive trend for most of the stations. Concerning NWC, notable discrepancies among GCMs are seen. The STRC algorithm can be used to successfully adjust GCMs outputs to reflect locally observed data and to then generate credible long-term scenarios for WSs as a tool for decision-making.

Stability analysis of a loess landslide considering rainfall patterns and spatial variability of soil

Accurate analysis of the stability of rainfall-induced landslides is an important and urgent task. Under complex geological conditions, traditional stability analysis methods cannot meet stability evaluation requirements. To study the impact of the rainfall pattern on the probability of instability of loess slopes, the intensity-frequency-duration curve and breakdown coefficient methods were introduced to analyse the effects of the rainfall extremum distribution and time scale, and bounded random cascade simulation were used to generate random rainfall patterns (RRPs). The obtained RRPs under several classical rainfall patterns were compared in terms of the safety factor, rainfall threshold (IaS/IaF), and annual failure probability (AFP) of loess slopes. Finally, soil preferential flow and spatial variability combined with the RRPs were introduced to analyse the change in the stability of rainfall-induced loess landslides under different theoretical frameworks. It was revealed that the use of a uniform rainfall pattern produced conservative stability evaluation results. Considering the RRPs, the calculation of IaS/IaF and AFP should include the common characteristics of various rainfall patterns. By comprehensively evaluating slope stability considering soil spatial variability and rainfall pattern variability, this study revealed that neglecting certain factors may result in incorrect slope stability estimates. Incorporating multiple assumptions in failure probability evaluation could provide more comprehensive slope safety assessment.

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.

Development of an integrated model for assessing landslide susceptibility on vegetated slopes under random rainfall scenarios

Physically-based landslide susceptibility analysis has been extensively employed to predict the likelihood of landslides owing to its remarkable predictive capability. However, due to an insufficiently comprehensive analysis of the vegetation-disaster connection, this approach may face limitations in its application to regions where there is significant variation in vegetation diversity and topography. An integrated model was thus proposed for predicting landslide susceptibility with the consideration of the positive and negative effects of vegetation on slope stability. However, to accurately implement the slope failure process under the influence of vegetation, it is important to have a high temporal resolution rainfall scenario for simulating the interception process of vegetation crowns on rainfall. Analysis of vegetation hydrological processes is associated with difficulties in a regional area because of limited data availability. The stochastic simulation method for extreme rainfall scenarios, adopted in this study, is known to be effective in dealing with uncertainty caused by insufficient data. The method, which comprises the Gumbel model and Bounded Random Cascade Model (BRCM), was used to overcome the difficulties in integrating the comprehensive effects of vegetation, the results of which provide refined rainfall scenarios for landslide susceptibility prediction from the perspective of the failure mechanism of vegetated slopes. The application of the proposed method for landslide susceptibility zoning was validated using a confusion matrix and ROC curve analysis in a practical case. Results showed that the method performed effectively and reliably. Furthermore, this study investigated the influence of different extreme rainfall scenarios on slope stability. Results revealed that as the return period of extreme rainfall increased from 50 to 150 years, the overall proportion of susceptible areas for different rainfall durations exhibited an upward trend. Additionally, the study found that increasing vegetation biomass had a positive impact on slope stability in the study area. The findings of this study are significant for disaster prevention and mitigation in mountainous regions, providing scientific support for risk management strategies.

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.

Rgfc-Forest: An enhanced deep forest method towards small-sample fault diagnosis of electromechanical system

Deep forest models offer a promising alternative to traditional deep neural networks by demanding fewer training samples and hyperparameters. However, existing deep forest fault diagnosis models encounter persistent challenges such as insufficient representation of multi-grained spatial information and redundancy of cascaded forest features. To address the above challenges, an enhanced deep forest method called random multi-grained fusion cascade forest (rgfc-Forest) is presented for fault diagnosis of electromechanical systems with limited training samples. First, a random multi-grained scanning module is designed to improve feature information learning. Subsequently, a feature fusion cascade forest module is constructed to improve the representativeness of features in multi-grained scanning and cascade forest delivery while ensuring data diversity. Finally, a decision tree self-growth strategy is combined to refine the classification capability of the high-level forest. To evaluate the effectiveness of our proposed method, we applied it to experimental data related to motor system and gearbox faults. Our results demonstrate significant improvements over existing methods: With just 20 samples per class, our method achieved an average accuracy of 84.41% for motor System Diagnosis. Similarly, for the gearbox system, we attained an impressive accuracy of up to 92.72% with the same limited dataset. These outcomes underscore the superior feature representation and fault classification capabilities of our approach compared to both benchmark deep forest models and mainstream deep learning methods when confronted with small training datasets.

Applying feature selection and machine learning techniques to estimate the biomass higher heating value

The biomass higher heating value (HHV) is an important thermal property that determines the amount of recoverable energy from agriculture byproducts. Precise laboratory measurement or accurate prediction of the HHV is essential for designing biomass conversion equipment. The current study combines feature selection scenarios and machine learning tools to establish a general model for estimating biomass HHV. Multiple linear regression and Pearson’s correlation coefficients justified that volatile matter, nitrogen, and oxygen content of biomass samples have a slight effect on the HHV and it is better to ignore them during the HHV modeling. Then, the prediction performance of random forest, multilayer and cascade feedforward neural networks, group method of data handling, and least-squares support vector regressor are compared to determine the intelligent estimator with the highest accuracy toward biomass HHV prediction. The ranking test shows that the multilayer perceptron neural network better predicts the HHV of 532 biomass samples than the other intelligent models. This model presents the outstanding absolute average relative error of 2.75% and 3.12% and regression coefficients of 0.9500 and 0.9418 in the learning and testing stages. The model performance is also superior to a recurrent neural network which was recently developed in the literature using the same databank.

Scaling and intermittency in time of urban precipitation in Warsaw, Poland

Study regionA network of 25 rain gauges roughly covering an area of 517.2 km2 of Warsaw, in Poland. Study focusScaling and intermittency of data series of rainfall intensity are investigated for each of the network rain gauges using five multifractal framework methods: spectral density analysis, functional box-counting, trace moment, probability distribution/multiple scaling, and double trace moment. Synthetic precipitation series are then generated by means of continuous universal random cascade models. Cascade generators are defined by universal parameters calculated for selected Warsaw rain gauges, or rain gauge clusters displaying similarity of multifractal parameters. The generated synthetic precipitation time series are subject to statistical evaluation by comparing the complementary cumulative distribution function and the intermittency calculated for synthetic versus observed data. Finally, a novel filtering algorithm is proposed to correct the intermittency characteristics of synthetic precipitation time series. New hydrological insightsBased on our analyses, the temporal structure of the recorded Warsaw precipitation time series is found to be a multifractal set characterized by a scaling behaviour over a wide range of scales. Furthermore, it has been observed that most Warsaw rain gauges, have a distinct similarity of multifractal properties. It has also been demonstrated that, for the first time in Poland, the universal continuous cascades could be used in practice for the generation of synthetic rainfall series at fine (1-min) temporal resolution

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.

Self-Induced Mode-Locking in Electrically Pumped Far-Infrared Random Lasers.

Mode locking, the self-starting synchronous oscillation of electromagnetic modes in a laser cavity, is the primary way to generate ultrashort light pulses. In random lasers, without a cavity, mode-locking, the nonlinear coupling amongst low spatially coherent random modes,can be activated via optical pumping, even without the emission of short pulses. Here, by exploiting the combination of the inherently giant third-order χ(3) nonlinearity of semiconductor heterostructure lasers and the nonlinear properties of graphene, the authors demonstrate mode-locking in surface-emitting electrically pumped random quantum cascade lasers at terahertz frequencies. This is achieved by either lithographically patterning a multilayer graphene film to define a surface random pattern of light scatterers, or by coupling on chip a saturable absorber graphene reflector. Intermode beatnote mapping unveils self-induced phase-coherence between naturally incoherent random modes. Self-mixing intermode spectroscopy reveals phase-locked random modes. This is an important milestone in the physics of disordered systems. It paves the way to the development of a new generation of miniaturized, electrically pumped mode-locked light sources, ideal for broadband spectroscopy, multicolor speckle-free imaging applications, and reservoir quantum computing.

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.

For Brenda Dunne, Fond Memories and Deep Respect

In early 1980, I answered a circumspect ad in the Chronicle of Higher Education looking for a “cognitive scientist interested in the lesser-known aspects of perception.” After some correspondence, I traveled from northern Vermont to Princeton to interview for a job that would, as I learned, touch on truly rarefied aspects of consciousness. By “accident,” I encountered Brenda walking down the hallway toward Bob Jahn's office, where I was headed for an interview, recognizing her though we had never met. It was quite a first impression—she was wearing a long flowing green dress and looked magical, and needless to say, obviously memorable. She was then and always a notable presence. The Princeton Engineering Anomalies Research (PEAR) lab was taking shape in the basement of Princeton University's School of Engineering, and from the beginning it had an unusually human quality because Brenda saw how important being at ease would be for people willing to try our experiments. She made the lab comfortable and home-like, installing the great orange couch with all its stuffed animals in PEAR's living room, and Comforto the Incredible chairs to coddle our operators as they attempted improbable tasks like intending that our Random Event Generator (REG) should produce high (or low) numbers on demand, or attempting to add some order (negentropy) to the Random Mechanical Cascade (RMC) or the big and beautiful but randomly arhythmic Native American drum. More important by far than the furniture was Brenda's presence in the lab. She was warm and genuinely interested in the people who came by, and many of them became long-term friends. Her easy confidence about the phenomena we studied was infectious, and that probably accounted for a large part of the success we had in demonstrating that the improbable could happen, and the impossible, too, though it might take longer. Brenda was clear that our studies were of phenomena, not people, and she invited the folks we called operators to relax and have fun with the experiments. She set a tone of collaboration, and rather than telling people how to work their will on the REG, she asked what they thought and felt. Some of the lab's most instructive findings come from what they had to say. Typically, our operators told us it was a matter of developing a relationship with the machine. “I began to feel loving connection.” Then the scores would climb.

Regional-scale modeling of rainfall-induced landslides under random rainfall patterns

The importance of rainfall patterns in triggering landslides has been recognized worldwide. In this study, the triggering and evolution of rainfall-induced landslides were investigated by a regional-scale physical model based on one-dimensional Richard equation and infinite slope model, where a bounded random cascade model (BRCM) was adopted to generate random rainfall patterns (RRPs) for rainfall events derived from extreme value analysis. The results indicated that the RRPs affect the evolution of triggered landslide and the cumulative landslide area. The shape of RRPs determined the time evolution of triggered landslide. Due to the randomness of rainfall peaks under RRPs and limited hydraulic conductivity of soil, an upper bound existed for the cumulative landslide area under RRPs and the dispersion of the cumulative landslide area increased with an increase in the return period. This study aids in the understanding of the diversity and discipline of the evolution of rainfall-induced landslides.