Significance of AIC differences for precipitation intensity distributions

Chain dependent models for daily precipitation typically model the occurrence process as a Markov chain and the precipitation intensity process using one of several probability distributions. It has been argued that the mixed exponential distribution is a superior model for the rainfall intensity process, since the value of its information criterion (Akaike information criterion or Bayesian information criterion) when fit to precipitation data is usually less than the more commonly used gamma distribution. The differences between the criterion values of the best and lesser models are generally small relative to the magnitude of the criterion value, which raises the question of whether these differences are statistically significant. Using a likelihood ratio statistic and nesting the gamma and mixed exponential distributions in a parent distribution, we show indirectly that generally the superiority of the mixed exponential distribution over the gamma distribution for modeling precipitation intensity is statistically significant. Comparisons are also made with a common-a gamma model, which are less informative.     

Stochastic multi-site generation of daily weather data

Spatial autocorrelation is a correlation between the values of a single variable, considering their geographical locations. This concept has successfully been used for multi-site generation of daily precipitation data (Khalili et al. in J Hydrometeorol 8(3):396–412, 2007). This paper presents an extension of this approach. It aims firstly to obtain an accurate reproduction of the spatial intermittence property in synthetic precipitation amounts, and then to extend the multi-site approach to the generation of daily maximum temperature, minimum temperature and solar radiation data. Monthly spatial exponential functions have been developed for each weather station according to the spatial dependence of the occurrence processes over the watershed, in order to fulfill the spatial intermittence condition in the synthetic time series of precipitation amounts. As was the case for the precipitation processes, the multi-site generation of daily maximum temperature, minimum temperature and solar radiation data is realized using spatially autocorrelated random numbers. These random numbers are incorporated into the weakly stationary generating process, as with the Richardson weather generator, and with no modifications made. Suitable spatial autocorrelations of random numbers allow the reproduction of the observed daily spatial autocorrelations and monthly interstation correlations. The Peribonca River Basin watershed is used to test the performance of the proposed approaches. Results indicate that the spatial exponential functions succeeded in reproducing an accurate spatial intermittence in the synthetic precipitation amounts. The multi-site generation approach was successfully applied for the weather data, which were adequately generated, while maintaining efficient daily spatial autocorrelations and monthly interstation correlations.     

Finding the most appropriate precipitation probability distribution for stochastic weather generation and hydrological modelling in Nordic watersheds

Six precipitation probability distributions (exponential, Gamma, Weibull, skewed normal, mixed exponential and hybrid exponential/Pareto distributions) are evaluated on their ability to reproduce the statistics of the original observed time series. Each probability distribution is also indirectly assessed by looking at its ability to reproduce key hydrological variables after being used as inputs to a lumped hydrological model. Data from 24 weather stations and two watersheds (Chute‐du‐Diable and Yamaska watersheds) in the province of Quebec (Canada) were used for this assessment. Various indices or statistics, such as the mean, variance, frequency distribution and extreme values are used to quantify the performance in simulating the precipitation and discharge. Performance in reproducing key statistics of the precipitation time series is well correlated to the number of parameters of the distribution function, and the three‐parameter precipitation models outperform the other models, with the mixed exponential distribution being the best at simulating daily precipitation. The advantage of using more complex precipitation distributions is not as clear‐cut when the simulated time series are used to drive a hydrological model. Although the advantage of using functions with more parameters is not nearly as obvious, the mixed exponential distribution appears nonetheless as the best candidate for hydrological modelling. The implications of choosing a distribution function with respect to hydrological modelling and climate change impact studies are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulated changes due to global warming in the variability of precipitation, and their interpretation using a gamma-distributed stochastic model

The interannual variability of monthly mean January and July precipitation and its possible change due to global warming are assessed using a five-member ensemble of climate for the period 1871–2100, simulated by the CSIRO Mark 2 global coupled atmosphere–ocean model. In the 1961–1990 climate, for much of the middle to high latitudes the standard deviation of precipitation for both months is roughly proportional to the mean, with the coefficient of variation (C) typically 0.3–0.5. The variability there is shown to be largely consistent with that from a first-order Markov chain model of the daily rainfall occurrence, with the distribution of wet-day amounts approximated by a gamma distribution. Global distributions of Mark 2-based parameters of this stochastic model, commonly used in weather generators, are presented. In low latitudes, however, the variability from the coupled model is typically double that anticipated by the stochastic model, as quantified by an ‘overdispersion ratio’. C often exceeds one at subtropical locations, where rain is less frequent, but sometimes relatively heavy.The standard deviation of monthly mean precipitation S generally increases as the global model warms, with the global mean S in 2071–2100 in January (July) being 9.0% (11.5%) larger than in 1961–1990. Decreases in some subtropical locations occur, particularly where mean precipitation decreases. The global pattern of overdispersion is largely unchanged, however, and the changes in S can be related to those in the stochastic model parameters. Much of the increase in S is associated with increases in the scale parameter of the gamma distribution of wet-day amounts. Changes in C, which is unaffected by this parameter, are generally small. Increases in C in several subtropical bands and over northern midlatitude land in July are related to a decreased frequency of precipitation, and (to a lesser degree) changes in the gamma shape parameter. Some potential applications of the results to downscaling are discussed, and illustrated using observed rainfall from southeast Australia.     

A stochastic model describing the impact of daily rainfall depth distribution on the soil water balance

Simplified, vertically-averaged soil moisture models have been widely used to describe and study eco-hydrological processes in water-limited ecosystems. The principal aim of these models is to understand how the main physical and biological processes linking soil, vegetation, and climate impact on the statistical properties of soil moisture. A key component of these models is the stochastic nature of daily rainfall, which is mathematically described as a compound Poisson process with daily rainfall amounts drawn from an exponential distribution. Since measurements show that the exponential distribution is often not the best candidate to fit daily rainfall, we compare the soil moisture probability density functions obtained from a soil water balance model with daily rainfall depths assumed to be distributed as exponential, mixed-exponential, and gamma. This model with different daily rainfall distributions is applied to a catchment in New South Wales, Australia, in order to show that the estimation of the seasonal statistics of soil moisture might be improved when using the distribution that better fits daily rainfall data. This study also shows that the choice of the daily rainfall distributions might considerably affect the estimation of vegetation water-stress, leakage and runoff occurrence, and the whole water balance.     

A global survey on the seasonal variation of the marginal distribution of daily precipitation

To characterize the seasonal variation of the marginal distribution of daily precipitation, it is important to find which statistical characteristics of daily precipitation actually vary the most from month-to-month and which could be regarded to be invariant. Relevant to the latter issue is the question whether there is a single model capable to describe effectively the nonzero daily precipitation for every month worldwide. To study these questions we introduce and apply a novel test for seasonal variation (SV-Test) and explore the performance of two flexible distributions in a massive analysis of approximately 170,000 monthly daily precipitation records at more than 14,000 stations from all over the globe. The analysis indicates that: (a) the shape characteristics of the marginal distribution of daily precipitation, generally, vary over the months, (b) commonly used distributions such as the Exponential, Gamma, Weibull, Lognormal, and the Pareto, are incapable to describe “universally” the daily precipitation, (c) exponential-tail distributions like the Exponential, mixed Exponentials or the Gamma can severely underestimate the magnitude of extreme events and thus may be a wrong choice, and (d) the Burr type XII and the Generalized Gamma distributions are two good models, with the latter performing exceptionally well.     

Simulation of daily precipitation in the Pacific Northwest using a weather classification scheme

A weather classification scheme was coupled with a semi-Markov model to represent the coincident occurrence of rain/no rain states at a single rain gauge and classes representing regional atmospheric circulation patterns, as identified from National Meteorological Center gridded observations for a large area of the North Pacific. Weather classes were identified from daily observations of surface pressure and 850 mb pressure height at five selected ten degree latitude by ten degree longitude cells using a K-means clustering algorithm, which was applied on a month-by-month basis. The number of climate classes, K, for each month was chosen based on a preliminary analysis of the model's ability to describe statistics of observed precipitation occurrences at the Stampede Pass, Washington weather station. The length of stay distributions within each precipitation occurrence/weather class were assumed to be geometric, and the precipitation amounts for each class and season were fitted with a mixed exponential distribution. Parameters of the length of stay distributions, transition probabilities, and precipitation amounts were estimated from the period of record 1975–84.The fitted model was used to simulate a ten year sequence of daily precipitation. It was found that the semi-Markov model of climate class/wet-dry states preserved the length of wet and dry day runs reasonably well, with the exception of months with long average run lengths. Likewise, the occurrence frequencies of the climate classes were reasonably well preserved with a few exceptions. An exploratory analysis of the properties of wet and dry period runs for those classes and months whose run frequencies were poorly preserved showed that the log survivor functions and variance time curves were also poorly preserved, which suggests that more complex distributions may be required for some of the run length distributions.     

Extreme value theory for precipitation: sensitivity analysis for climate change

Extreme value theory for the maximum of a time series of daily precipitation amount is described. A chain-dependent process is assumed as a stochastic model for daily precipitation, with the intensity distribution being the gamma. To examine how the effective return period for extreme high precipitation amounts would change as the parameters of the chain-dependent process change (i.e., probability of a wet day, shape and scale parameters of the gamma distribution), a sensitivity analysis is performed. This sensitivity analysis is guided by some results from statistical downscaling that relate patterns in large-scale atmospheric circulation to local precipitation, providing a physically plausible range of changes in the parameters. For the particular location considered in the example, the effective return period is most sensitive to the scale parameter of the intensity distribution.     

Modelling streamflow trends for a watershed with limited data: case of the Litani basin,Lebanon

Abstract

Streamflow variability in the Upper and Lower Litani basin, Lebanon was modelled as there is a lack of long-term measured runoff data. To simulate runoff and streamflow, daily rainfall was derived using a stochastic rainfall generation model and monthly rainfall data. Two distinct synthetic rainfall models were developed based on a two-part probabilistic distribution approach. The rainfall occurrence was described by a Markov chain process, while the rainfall distribution on wet days was represented by two different distributions (i.e. gamma and mixed exponential distributions). Both distributions yielded similar results. The rainfall data were then processed using water balance and routing models to generate daily and monthly streamflow. Compared with measured data, the model results were generally reasonable (mean errors ranging from 0.1 to 0.8?m³/s at select locations). Finally, the simulated monthly streamflow data were used to investigate discharge trends in the Litani basin during the 20th century using the Mann-Kendall and Sen slope nonparametric trend detection methods. A significant drying trend of the basin was detected, reaching a streamflow reduction of 0.8 and 0.7 m³/s per decade in January for the Upper and Lower basin, respectively.

Editor D. Koutsoyiannis; Associate editor Sheng Yue

Citation Ramadan, H.H., Beighley, R.E., and Ramamurthy, A.S., 2012. Modelling streamflow trends for a watershed with limited data: case of the Litani basin, Lebanon. Hydrological Sciences Journal, 57 (8), 1516–1529.     

A hybrid stochastic-weather-generation method for temporal disaggregation of precipitation with consideration of seasonality and within-month variations

Linking atmospheric and hydrological models is challenging because of a mismatch of spatial and temporal resolutions in which the models operate: dynamic hydrological models need input at relatively fine temporal (daily) scale, but the outputs from general circulation models are usually not realistic at the same scale, even though fine scale outputs are available. Temporal dimension downscaling methods called disaggregation are designed to produce finer temporal-scale data from reliable larger temporal-scale data. Here, we investigate a hybrid stochastic weather-generation method to simulate a high-frequency (daily) precipitation sequence based on lower frequency (monthly) amounts. To deal with many small precipitation amounts and capture large amounts, we divide the precipitation amounts on rainy days (with non-zero precipitation amounts) into two states (named moist and wet states, respectively) by a pre-defined threshold and propose a multi-state Markov chain model for the occurrences of different states (also including non-rain days called dry state). The truncated Gamma and censored extended Burr XII distributions are then employed to model the precipitation amounts in the moist and wet states, respectively. This approach avoids the need to deal with discontinuity in the distribution, and ensures that the states (dry, moist and wet) and corresponding amounts in rainy days are well matched. The method also considers seasonality by constructing individual models for different months, and monthly variation by incorporating the low-frequency amounts as a model predictor. The proposed method is compared with existing models using typical catchment data in Australia with different climate conditions (non-seasonal rainfall, summer rainfall and winter rainfall patterns) and demonstrates better performances under several evaluation criteria which are important in hydrological studies.     

A comparison of three methods for downscaling daily precipitation in the Punjab region

Many downscaling techniques have been developed in the past few years for projection of station‐scale hydrological variables from large‐scale atmospheric variables simulated by general circulation models (GCMs) to assess the hydrological impacts of climate change. This article compares the performances of three downscaling methods, viz. conditional random field (CRF), K‐nearest neighbour (KNN) and support vector machine (SVM) methods in downscaling precipitation in the Punjab region of India, belonging to the monsoon regime. The CRF model is a recently developed method for downscaling hydrological variables in a probabilistic framework, while the SVM model is a popular machine learning tool useful in terms of its ability to generalize and capture nonlinear relationships between predictors and predictand. The KNN model is an analogue‐type method that queries days similar to a given feature vector from the training data and classifies future days by random sampling from a weighted set of K closest training examples. The models are applied for downscaling monsoon (June to September) daily precipitation at six locations in Punjab. Model performances with respect to reproduction of various statistics such as dry and wet spell length distributions, daily rainfall distribution, and intersite correlations are examined. It is found that the CRF and KNN models perform slightly better than the SVM model in reproducing most daily rainfall statistics. These models are then used to project future precipitation at the six locations. Output from the Canadian global climate model (CGCM3) GCM for three scenarios, viz. A1B, A2, and B1 is used for projection of future precipitation. The projections show a change in probability density functions of daily rainfall amount and changes in the wet and dry spell distributions of daily precipitation. Copyright © 2011 John Wiley & Sons, Ltd.     

DISTRIBUTION FOR k-DAY RAINFALL TOTALS

Abstract

A lot of different distribution functions have been proposed to represent the precipitation totals collected in k days, and in particular a transformed incomplete gamma distribution aimed to be the basis of the searched law (Formula 3.).

This distribution contains as particular cases, or as limit-cases the distributions usually employed for the daily, weekly, monthly and annual precipitations.

Periods even shorter than a day, as well as the intensity may be presented by the same law.

The general validity of the proposed law is confirmed by applying it at the rainfall data coliected at the Observatory of Ghent.     

Rainfall distribution function for Libya and rainfall prediction

Abstract

Monthly rainfall amounts are distributed according to different frequency distribution functions in different parts of the world. However, in extremely arid regions gamma probability distribution functions are most often found to fit the existing data well. Libyan monthly rainfall distributions are found to abide by gamma probability distribution function which is confirmed on the basis of chi-square tests. Almost all the rainfall sequences recorded for at least the last 20 years in Libya are investigated statistically and gamma distribution parameters are calculated at existing stations. The shape and scale parameters are then regionalized and hence it becomes possible to find the parameter values at any desired location within the study area and then to generate synthetic sequences according to the gamma distribution. Predictions of 10, 25, 50 and 100 mm rainfall amounts are achieved by this probability function.     

Analysis of Turkish precipitation data: homogeneity and the Southern Oscillation forcings on frequency distributions

The homogeneity of newly compiled 212 precipitation records in Turkey for the period 1973‐2002 was checked by the Standard Normal Homogeneity Test (SNHT) and Pettitt Test. Stations were considered inhomogeneous if at least one of the tests rejects the homogeneity. As a result, 43 out of 212 stations were found to be inhomogeneous. In addition, the previously detected Southern Oscillation (SO)‐related precipitation anomalies by the authors were quantified at each station using the gamma distribution. The observed SO‐related shifts in the median precipitation amounts expressed as gamma percentiles may be considered as a typical SO response of that station. The results of this study confirm the wet responses of Turkish precipitations to El Nino events, whereas those for La Nina events seem to be masked by sampling variations within the study period. Copyright © 2007 John Wiley & Sons, Ltd.     

Distributions of microcanonical cascade weights of rainfall at small timescales

Empirical frequency distributions of multiplicative cascade weights, or breakdown coefficients, at small timescales are analyzed for 5-min precipitation time series from four gauges in Germany. It is shown that histograms of the weights, W, are strongly deformed by the recording precision of rainfall amounts. A randomization procedure is proposed to statistically remove the artifacts due to precision errors in the original series. Evolution of the probability distributions of W from beta-like for large timescales to combined beta-normal distribution with a pronounced peak at W ≈ 0.5 for small timescales is observed. A new 3N-B distribution built from 3 separate normal, N, distributions and one beta, B, distribution is proposed for reproduction of the empirical histograms of W at small timescales. Parameters of the 3N-B distributions are fitted for all gauges and analyzed timescales. Microcanonical cascades models with a generator based on 3N-B distributions are developed and their performance at disaggregating precipitation at 1280-min intervals down to 5-min intervals is evaluated.     

Downscaling two versions of a general circulation model to estimate local hydroclimatic factors under climate change

Abstract

The regional hydroclimatological effect of global climate change has been estimated and compared using a semi-empirical downscaling method with two versions (T21 and T42) of the general circulation model (GCM) developed at the Max Planck Institute for Meteorology, Germany. The comparisons were performed with daily mean temperature and daily precipitation amounts for the continental climate of the state of Nebraska, USA. Both the T21 and the T42 versions resulted in an increase of daily mean temperature under a 2 x C02 climatess. The magnitude of warming was substantially greater for T21 than for T42, except for February and June and at some stations in July where the T42 model suggested greater warming. Both GCMs resulted in a slight decrease in precipitation frequency and an increase in the amount of precipitation on wet days. Here, the T42 model again led to smaller changes. Different locations within Nebraska exhibited somewhat different temperature and precipitation responses with both GCM versions.     

Field experiments constraining the probability distribution of particle travel distances during natural rainstorms on different slope gradients

Rain splash erosion is an important soil transport mechanism on steep hillslopes. The rain splash process is highly stochastic; here we seek to constrain the probability distribution of splash transport distances on natural hillslopes as a function of hillslope gradient and total precipitation depth. Field experiments were conducted under natural precipitation events to observe splash travel on varying slope gradients. The downslope fraction of splash transport on 15°, 25° and 33° gradients were 85%, 96% and 96%, respectively. Maximum splash transport (L_max) was related to the rain splash detachment of soil particles and slope gradient. An empirical relationship of L_max to the precipitation depth and gradient was obtained; it is linearly proportional to hillslope gradient and logarithmically related to precipitation depth. Measured splash distances were calibrated to the fully two‐dimensional (2D) model of splash transport of Furbish et al. (Journal of Geophysical Research 112 : F01001, 2007) that is based on the assumption that radial splash distances are exponentially distributed; calibrated values of mean splash transport distances are an order of magnitude greater than those previously determined in a controlled laboratory setting. We also compared measured data with several one‐dimensional (1D) probability distributions to asses if splash transport distances could be better explained by a heavy‐tailed probability distribution rather than an exponential probability distribution. We find that for hillslopes of 15° and 25°, although a log‐normal probability distribution best describes the data, we find its likelihood is nearly indistinguishable from an exponential distribution based on computing maximum likelihood estimators for all 1D distributions (exponential, log‐normal and Weibull). At 33°, however, we find stronger evidence that measured travel distances are heavy‐tailed. Copyright © 2011 John Wiley & Sons, Ltd.     

Spatial variability and interpolation of daily precipitation amount

Daily precipitation amounts show spatial variation over sub-continential regions. Point measurements, representative for regions of land, have to be interpolated towards unobserved locations. In this study four days in 1984 were selected to investigate the spatial variability of daily precipitation amount in North-western Europe in relation to the meteorological conditions. Data were interpolated using Kriging. Crossvalidation was used to compare interpolated values with measured values. Large differences in the spatial structure of daily precipitation amount are obsered as a result of different meterological conditions. Stratification of the study area into a coastal, a mountainous and an interior stratum proved to be successful, reducing the Mean Squared Error of Prediction with up to 55%.     

Spatial variability and interpolation of daily precipitation amount

Daily precipitation amounts show spatial variation over sub-continential regions. Point measurements, represntative for regions of land, have to be interpolated towards unobserved locations. In this study four days in 1984 were selected to investigate the spatial variability of daily precipitation amount in north-western Europe in relation to the meteorological conditions. Data were interpolated using kriging. Crossvalidation was used to compare interpolated values with measured values. Large differences in the spatial structure of daily precipitation amount are observed as a result of different meteorological conditions. Stratification of the study area into a coast, a mountain and an interior stratum proved to be successful, reducing the Mean Squared Error of Prediction with up to 55%.This article was inadvertently printed in SHH 6(3) 1992 without figures and figure legends. The article is being reprinted in this issue in complete form. The editor apologizes for this error in publication.     

Determination of probability distributions for Strahler stream lengths based on Poisson process and DEM

Abstract

One of the basic tasks in geomorphologic analysis is to know the probability distributions of the stream lengths of different orders. In practical applications, this information is useful for basin rainfall-runoff modelling. The objective of this study is to determine the length distributions of the Strahler streams. A Poisson process was used to derive the theoretical distributions. The result showed that the length distribution of the first-order stream is an exponential distribution and the second-order or higher order stream length is a gamma distribution. In order to verify the theoretical distributions, a digital elevation model (DEM) was adopted to calculate the stream lengths of four basins in Taiwan. Kolmogorov-Smirnov and chi-square tests were used to test the goodness-of-fit of the data. Results showed that the length distributions of the first- and second-order streams analysed by using DEM correspond with those from the derived distribution method.