Encoding daily rainfall records via adaptations of the fractal multifractal method

A deterministic geometric approach, the fractal–multifractal (FM) method, already found useful in modeling storm events, is adapted here in order to encode, for the first time, highly intermittent daily rainfall records gathered over a water year and containing many days of zero rain. Through application to data sets gathered at Laikakota in Bolivia and Tinkham in Washington, USA, it is demonstrated that the modified FM approach can represent erratic rainfall records faithfully, while using only a few FM parameters. It is shown that the modified FM approach, by capturing the rain accumulated over the season, ends up preserving other statistical attributes as well as the overall “texture” of the records, leading to FM sets that are indistinguishable from observed sets and certainly within the limits of accuracy of measured rainfall. This fact is further corroborated comparing 20 consecutive years at Laikakota and a modified FM representation, via common statistical qualifiers, such as histogram, entropy function, and inter-arrival times.     

A physical interpretation of the deterministic fractal–multifractal method as a realization of a generalized multiplicative cascade

In this study, we attempt to offer a solid physical basis for the deterministic fractal–multifractal (FM) approach in geophysics (Puente, Phys Let A 161:441–447, 1992; J Hydrol 187:65–80, 1996). We show how the geometric construction of derived measures, as Platonic projections of fractal interpolating functions transforming multinomial multifractal measures, naturally defines a non-trivial cascade process that may be interpreted as a particular realization of a random multiplicative cascade. In such a light, we argue that the FM approach is as “physical” as any other phenomenological approach based on Richardson’s eddies splitting, which indeed lead to well-accepted models of the intermittencies of nature, as it happens, for instance, when rainfall is interpreted as a quasi-passive tracer in a turbulent flow. Although neither a fractal interpolating function nor the specific multipliers of a random multiplicative cascade can be measured physically, we show how a fractal transformation “cuts through” plausible scenarios to produce a suitable realization that reflects specific arrangements of energies (masses) as seen in nature. This explains why the FM approach properly captures the spectrum of singularities and other statistical features of given data sets. As the FM approach faithfully encodes data sets with compression ratios typically exceeding 100:1, such a property further enhances its “physical simplicity.” We also provide a connection between the FM approach and advection–diffusion processes.     

Geometric harnessing of precipitation records: reexamining four storms from Iowa City

Complex geometries often present in hydrologic data sets such as precipitation records have been difficult to model in their totality using classical stochastic methods. In recent years, we have developed extensions of a deterministic procedure, the fractal-multifractal (FM) method, whose patterns share fine details and textures of individual data sets in addition to the usual key statistical properties. This work discusses our latest efforts at encoding four geometrically distinct storms gathered in Iowa City with parameters found running a modified particle swarm optimization procedure. The results reaffirm the capabilities of the FM method as all storms are closely fitted within measurement errors. All sets may be encoded with a compression ratio exceeding 350:1, have a maximum error in cumulative distribution less than 2.5 %, and closely preserve the autocorrelation, power spectrum, and multifractal spectrum of the records.     

Rainfall data simulation by hidden Markov model and discrete wavelet transformation

In many regions, monthly (or bimonthly) rainfall data can be considered as deterministic while daily rainfall data may be treated as random. As a result, deterministic models may not sufficiently fit the daily data because of the strong stochastic nature, while stochastic models may also not reliably fit into daily rainfall time series because of the deterministic nature at the large scale (i.e. coarse scale). Although there are different approaches for simulating daily rainfall, mixing of deterministic and stochastic models (towards possible representation of both deterministic and stochastic properties) has not hitherto been proposed. An attempt is made in this study to simulate daily rainfall data by utilizing discrete wavelet transformation and hidden Markov model. We use a deterministic model to obtain large-scale data, and a stochastic model to simulate the wavelet tree coefficients. The simulated daily rainfall is obtained by inverse transformation. We then compare the accumulated simulated and accumulated observed data from the Chao Phraya Basin in Thailand. Because of the stochastic nature at the small scale, the simulated daily rainfall on a point to point comparison show deviations with the observed data. However the accumulated simulated data do show some level of agreement with the observed data.     

Improving rainfall–runoff modelling through the control of uncertainties under increasing climate variability in the Ouémé River basin (Benin,West Africa)

ABSTRACT

The objective of this paper is to understand how the natural dynamics of a time-varying catchment, i.e. the rainfall pattern, transforms the random component of rainfall and how this transformation influences the river discharge. To this end, this paper develops a rainfall–runoff modelling approach that aims to capture the multiple sources and types of uncertainty in a single framework. The main assumption is that hydrological systems are nonlinear dynamical systems which can be described by stochastic differential equations (SDE). The dynamics of the system is based on the least action principle (LAP) as derived from Noether’s theorem. The inflow process is considered as a sum of deterministic and random components. Using data from the Ouémé River basin (Benin, West Africa), the basic properties for the random component are considered and the triple relationship between the structure of the inflowing rainfall, the corresponding SDE that describes the river basin and the associated Fokker-Planck equations (FPE) is analysed.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR D. Gerten     

A new magnitude category disaggregation approach for temporal high‐resolution rainfall intensities

Simulation of quick runoff components such as surface runoff and associated soil erosion requires temporal high‐resolution rainfall intensities. However, these data are often not available because such measurements are costly and time consuming. Current rainfall disaggregation methods have shortcomings, especially in generating the distribution of storm events. The objectives of this study were to improve point rainfall disaggregation using a new magnitude category rainfall disaggregation approach. The procedure is introduced using a coupled disaggregation approach (Hyetos and cascade) for multisite rainfall disaggregation. The new procedure was tested with ten long‐term precipitation data sets of central Germany using summer and winter precipitation to determine seasonal variability. Results showed that dividing the rainfall amount into four daily rainfall magnitude categories (1–10, 11–25, 26–50, >50 mm) improves the simulation of high rainfall intensity (convective rainfall). The Hyetos model category approach (Hyetos_Cat) with seasonal variation performs representative to observed hourly rainfall compared with without categories on each month. The mean absolute percentage accuracy of standard deviation for hourly rainfall is 89.7% in winter and 95.6% in summer. The proposed magnitude category method applied with the coupled Hyetos_Cat–cascade approach reproduces successfully the statistical behaviour of local 10‐min rainfall intensities in terms of intermittency as well as variability. The root mean square error performance statistics for disaggregated 10‐min rainfall depth ranges from 0.20 to 2.38 mm for summer and from 0.12 to 2.82 mm for the winter season in all categories. The coupled stochastic approach preserves the statistical self‐similarity and intermittency at each magnitude category with a relatively low computational burden. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation and spatial interpolation of rainfall intensity distribution from the effective rate of precipitation

Great emphasis is being placed on the use of rainfall intensity data at short time intervals to accurately model the dynamics of modern cropping systems, runoff, erosion and pollutant transport. However, rainfall data are often readily available at more aggregated level of time scale and measurements of rainfall intensity at higher resolution are available only at limited stations. A distribution approach is a good compromise between fine-scale (e.g. sub-daily) models and coarse-scale (e.g. daily) rainfall data, because the use of rainfall intensity distribution could substantially improve hydrological models. In the distribution approach, the cumulative distribution function of rainfall intensity is employed to represent the effect of the within-day temporal variability of rainfall and a disaggregation model (i.e. a model disaggregates time series into sets of higher solution) is used to estimate distribution parameters from the daily average effective precipitation. Scaling problems in hydrologic applications often occur at both space and time dimensions and temporal scaling effects on hydrologic responses may exhibit great spatial variability. Transferring disaggregation model parameter values from one station to an arbitrary position is prone to error, thus a satisfactory alternative is to employ spatial interpolation between stations. This study investigates the spatial interpolation of the probability-based disaggregation model. Rainfall intensity observations are represented as a two-parameter lognormal distribution and methods are developed to estimate distribution parameters from either high-resolution rainfall data or coarse-scale precipitation information such as effective intensity rates. Model parameters are spatially interpolated by kriging to obtain the rainfall intensity distribution when only daily totals are available. The method was applied to 56 pluviometer stations in Western Australia. Two goodness-of-fit statistics were used to evaluate the skill—daily and quantile coefficient of efficiency between simulations and observations. Simulations based on cross-validation show that kriging performed better than other two spatial interpolation approaches (B-splines and thin-plate splines).     

Continuous daily hydrograph simulation using duration curves of a precipitation index

The paper describes a parsimonious approach for generating continuous daily stream‐flow time‐series from observed daily rainfall data in a catchment. The key characteristic in the method is a duration curve. It is used to convert the daily rainfall information from source rain gauges into a continuous daily hydrograph at the destination river site. For each source rain gauge a time‐series of rainfall related ‘current precipitation index’ is generated and its duration curve is established. The current precipitation index reflects the current catchment wetness and is defined as a continuous function of precipitation, which accumulates on rainy days and exponentially decays during the periods of no rainfall. The process of rainfall‐to‐runoff conversion is based on the assumption that daily current precipitation index values at rainfall site(s) in a catchment and the destination site's daily flows correspond to similar probabilities on their respective duration curves. The method is tested in several small catchments in South Africa. The method is designed primarily for application at ungauged sites in data‐poor regions where the use of more complex and information consuming techniques of data generation may not be justified. Copyright © 2000 John Wiley & Sons, Ltd.     

A rapid method for simulating three-dimensional fluvial terrain

A rapid, computer-based method of simulating ‘geomorphologically-sensible’ three-dimensional terrain data by modelling landform morphology is presented. For some engineering applications such an approach, even in a simple form, is preferable to the synthesis of terrain data by purely stochastic methods, and additionally can be useful where real data sets are difficult to obtain. The approach adopted utilizes a computer program which models landscape evolution by establishing a stream network on a tilted surface, with accompanying fluvial downcutting and slope adjustment. This is achieved by an iterative mechanism that combines deterministic and stochastic processes with geomorphological theory. The end-product is a matrix of high-resolution altitude data that has been used as the terrain model for a vehicle simulation exercise.     

Daily hydrological modeling in the Amazon basin using TRMM rainfall estimates

Rainfall measurements by conventional raingauges provide relatively accurate estimates at a few points of a region. The actual rainfield can be approximated by interpolating the available raingauge data to the remaining of the area of interest. In places with relatively low gauge density such interpolated rainfields will be very rough estimates of the actual events. This is especially true for tropical regions where most rainfall has a convective origin with high spatial variability at the daily level. Estimates of rainfall by remote sensing can be very useful in regions such as the Amazon basin, where raingauge density is very low and rainfall highly variable. This paper evaluates the rainfall estimates of the Tropical Rainfall Measuring Mission (TRMM) satellite over the Tapajós river basin, a major tributary of the Amazon. Three-hour TRMM rainfall estimates were aggregated to daily values and were compared with catch of ground-level precipitation gauges on a daily basis after interpolating both data to a regular grid. Both daily TRMM and raingauge-interpolated rainfields were then used as input to a large-scale hydrological model for the whole basin; the calculated hydrographs were then compared to observations at several streamgauges along the river Tapajos and its main tributaries. Results of the rainfield comparisons showed that satellite estimates can be a practical tool for identifying damaged or aberrant raingauges at a basin-wide scale. Results of the hydrological modeling showed that TRMM-based calculated hydrographs are comparable with those obtained using raingauge data.     

A preliminary investigation on the scaling behaviour of rainfall observed in two different climates

Abstract

The problem of transformation of rainfall data from one scale to another has been gaining considerable importance in recent years. Though the application of the concept of fractal theory, in the studies conducted thus far, nearly unanimously points at the possibility of such a transformation, the suitability of the theory to the highly variable rainfall in time and space has very often been questioned. A preliminary attempt is made herein to address this issue by investigating the existence of temporal scaling behaviour in rainfall data observed in two different climatic regions: (a) a subtropical climatic region (Leaf River basin, Mississippi, USA) and (b) an equatorial climatic region (Singapore). Rainfall data of three different resolutions, six-hourly, daily, and weekly, observed over a period of 25 years, are investigated. A mono- or simple-scaling method (box dimension method) is employed. The results achieved for the different data sets clearly indicate the existence of temporal scaling in rainfall observed in the two regions, an encouraging news on the suitability of fractal theory in understanding and modelling the rainfall process. However, the insufficiency of a single dimension to characterize the rainfall behaviour is realized, as the dimension depends on the rainfall intensity level, which, in turn, may be related to the rainfall generating mechanisms. A comparison of the box-dimension results obtained for data of different resolutions, from each of the regions, seems to indicate a possible connection between them, a prospect of tremendous practical importance. Another interesting observation is the similarity between the box dimension results obtained for rainfall data from Leaf River basin and Singapore, but this is also clearly related to the intensity level. The dependence of the dimension on the intensity threshold suggests the use of a multi-dimensional fractal approach, where the process is characterized by more than one dimension (or a dimension function) instead of one single dimension. On the basis of the present results, some potential areas for further study are identified.     

Uncertainty analysis of urban sewer system using spatial simulation of radar rainfall fields: New York City case study

The goal of this study is to investigate the uncertainty of an urban sewer system’s response under various rainfall and infrastructure scenarios by applying a recently developed nonparametric copula-based simulation approach to extreme rainfall fields. The approach allows for Monte Carlo simulation of multiple variables with differing marginal distributions and arbitrary dependence structure. The independent and identically distributed daily extreme rainfall events of the corresponding urban area, extracted from nationwide high resolution radar data stage IV, are the inputs of the spatial simulator. The simulated extreme rainfall fields were used to calculate excess runoff using the Natural Resources Conservation Service’s approach. New York City is selected as a case study and the results highlight the importance of preserving the spatial dependence of rainfall fields between the grids, even for simplified hydrologic models. This study estimates the probability of combined sewer overflows under extreme rainfall events and identifies the most effective locations in New York City to install green infrastructure for detaining excess stormwater runoff. The results of this study are beneficial for planners working on stormwater management and the approach is broadly applicable because it does not rely on extensive sewer system information.     

A deterministic linearized recurrent neural network for recognizing the transition of rainfall–runoff processes

Characterizing the dynamic relationship between rainfall and runoff is a highly interesting modeling problem in hydrology. This study develops a deterministic linearized recurrent neural network (denoted as DLRNN) that deals with the system’s nonlinearity by recalibration at each time interval, and relates the weights of DLRNN to unit hydrographs in order to describe the transition of the rainfall–runoff processes. Case studies of 38 events, from 1966 to 1997, are implemented in the Wu-Tu watershed of Taiwan, where the runoff path-lines are short and steep. A comparison between the DLRNN and a feed-forward neural network demonstrates the advantage of DLRNN as a dynamic system model. It is concluded that DLRNN shows superiority in the performance of rainfall–runoff simulations and the ability to recognize transitions in hydrological processes.     

Construction of a high-resolution gridded rainfall dataset for Peru from 1981 to the present day

ABSTRACT

A new gridded rainfall dataset available for Peru is introduced, called PISCOp V2.1 (Peruvian Interpolated data of SENAMHI’s Climatological and Hydrological Observations). PISCOp has been developed for the period 1981 to the present, with an average latency of eight weeks at 0.1° spatial resolution. The merging algorithm is based on geostatistical and deterministic interpolation methods including three different rainfall sources: (i) the national quality-controlled and infilled raingauge dataset, (ii) radar-gauge merged precipitation climatologies and (iii) the Climate Hazards Group Infrared Precipitation (CHIRP) estimates. The validation results suggest that precipitation estimates are acceptable showing the highest performance for the Pacific coast and the western flank of the Andes. Furthermore, a meticulous quality-control and gap-infilling procedure allowed us to reduce the formation of inhomogeneities (non-climatic breaks). The dataset is publicly available at https://piscoprec.github.io/ and is intended to support hydrological studies and water management practices.     

Fractal analysis of rainfall observed in two different climatic regions

Abstract

The importance of high-resolution rainfall data to understand the intricacies of the dynamics of hydrological processes and describe them in a sophisticated and accurate way has been increasingly realized. The present study investigates the general suitability of fractal (or scaling) theory for understanding the rainfall behaviour and transforming rainfall data from one time scale to another. The study, employing a multi-fractal approach, follows the research undertaken earlier by the author (Sivakumar, 2000) employing a mono-fractal approach in which some preliminary indication as to the possibility of existence of (multi-) fractals was obtained. Rainfall data of three different resolutions, six-hourly, daily, and weekly, observed over a period of 25 years in two different climatic regions: a subtropical climatic region (Leaf River basin, Mississippi, USA); and an equatorial climatic region (Singapore) are analysed. The existence of multi-fractal behaviour in the rainfall data is investigated using (a) the power spectrum method; (b) the empirical probability distribution function (PDF) method; (c) the statistical moment scaling method; and (d) the probability distribution multiple scaling (PDMS) method. The results achieved from all these methods for the six different rainfall data sets considered indicate the existence of multi-fractal behaviour of rainfall observed in Leaf River basin and Singapore, providing further support to the results obtained using the mono-fractal approach (Sivakumar, 2000). The suitability of a multi-fractal framework to characterize the behaviour of rainfall observed in the above two significantly different climatic regions, subtropical and equatorial, seems to suggest the general suitability of the fractal theory for transforming rainfall from one time scale to another. Investigations with rainfall data from several other climatic regions are underway with a view to strengthening the above conclusions.     

A stochastic framework to assess the performance of flood warning systems based on rainfall‐runoff modeling

Nowadays, Flood Forecasting and Warning Systems (FFWSs) are known as the most inexpensive and efficient non‐structural measures for flood damage mitigation in the world. Benefit to cost of the FFWSs has been reported to be several times of other flood mitigation measures. Beside these advantages, uncertainty in flood predictions is a subject that may affect FFWS's reliability and the benefits of these systems. Determining the reliability of advanced flood warning systems based on the rainfall–runoff models is a challenge in assessment of the FFWS performance which is the subject of this study. In this paper, a stochastic methodology is proposed to provide the uncertainty band of the rainfall–runoff model and to calculate the probability of acceptable forecasts. The proposed method is based on Monte Carlo simulation and multivariate analysis of the predicted time and discharge error data sets. For this purpose, after the calibration of the rainfall–runoff model, the probability distributions of input calibration parameters and uncertainty band of the model are estimated through the Bayesian inference. Then, data sets of the time and discharge errors are calculated using the Monte Carlo simulation, and the probability of acceptable model forecasts is calculated by multivariate analysis of data using copula functions. The proposed approach was applied for a small watershed in Iran as a case study. The results showed using rainfall–runoff modeling based on real‐time precipitation is not enough to attain high performance for FFWSs in small watersheds, and it seems using weather forecasts as the inputs of rainfall–runoff models is essential to increase lead times and the reliability of FFWSs in small watersheds. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation of rainfall elasticity of streamflow in Australia

Abstract

Estimates of rainfall elasticity of streamflow in 219 catchments across Australia are presented. The rainfall elasticity of streamflow is defined here as the proportional change in mean annual streamflow divided by the proportional change in mean annual rainfall. The elasticity is therefore a simple estimate of the sensitivity of long-term streamflow to changes in long-term rainfall, and is particularly useful as an initial estimate of climate change impact in land and water resources projects. The rainfall elasticity of streamflow is estimated here using a hydrological modelling approach and a nonparametric estimator. The results indicate that the rainfall elasticity of streamflow (? _P ) in Australia is about 2.0–3.5 (observed in about 70% of the catchments), that is, a 1% change in mean annual rainfall results in a 2.0–3.5% change in mean annual streamflow. The rainfall elasticity of streamflow is strongly correlated to runoff coefficient and mean annual rainfall and streamflow, where streamflow is more sensitive to rainfall in drier catchments, and those with low runoff coefficients. There is a clear relation-ship between the ? _P values estimated using the hydrological modelling approach and those estimated using the nonparametric estimator for the 219 catchments, although the values estimated by the hydrological modelling approach are, on average, slightly higher. The modelling approach is useful where a detailed study is required and where there are sufficient data to reliably develop and calibrate a hydrological model. The nonparametric estimator is useful where consistent estimates of the sensitivity of long-term streamflow to climate are required, because it is simple to use and estimates the elasticity directly from the historical data. The nonparametric method, being model independent, can also be easily applied in comparative studies to data sets from many catchments across large regions.     

Effect of autocorrelation on temporal trends in rainfall in a valley region at the foothills of Indian Himalayas

This study examines the effect of autocorrelation on step and monotonic trends in seasonal and annual rainfall. Initially, for step change, modified-Pettitt test is applied in two ways. First, using the corrected and unbiased trend-free-pre-whitening (TFPWcu) approach. Second, using a new approach in which time series is modelled by intervention analysis for modified Pettitt test. Subsequently, for monotonic trends, Mann–Kendall (MK) and six approaches of modified Mann–Kendall (MMK) test are applied to NCDC data for period 1901–2012 and its sub-periods. Approaches of MMK include pre-whitening (PW), trend-free-pre-whitening (TFPW), TFPWcu, two Variance Correction Approaches (VCAs) based on empirical formula (VCA:CF1) and Monte-Carlo-Simulations (VCA:CF2) and long term persistence (MK-LTP). A single change point is identified in 1970 for annual and monsoon rainfall from original and modified-Pettitt’s test using TFPWcu, while time series modelling approach has not exhibited any change point. Process shift in rainfall series is also studied using CUSUM and multiple change points are identified using Segment-Neighbourhood method. Outcomes of MMK show that TFPWcu is able to efficiently limit the effect of autocorrelation and may be preferred over PW and TFPW. The VCA:CF2 is not dependent on whole autocorrelation structure and corrects variance of all data series using lag-1 autocorrelation and may be preferred over VCA:CF1. MK-LTP considers long term persistence and it has exhibited presence of weaker trends than exhibited by other approaches. VCA:CF2 and MK-LTP are used to study trends of rainfall in Dehradun.