Interpolation processes using multivariate geostatistics for mapping of climatological precipitation mean in the Sannio Mountains (southern Italy)

The spatial variability of precipitation has often been a topic of research, since accurate modelling of precipitation is a crucial condition for obtaining reliable results in hydrology and geomorphology. In mountainous areas, the sparsity of the measurement networks makes an accurate and reliable spatialization of rainfall amounts at the local scale difficult. The purpose of this paper is to show how the use of a digital elevation model can improve interpolation processes at the subregional scale for mapping the mean annual and monthly precipitation from rainfall observations (40 years) recorded in a region of 1400 km² in southern Italy. Besides linear regression of precipitation against elevation, two methods of interpolation are applied: inverse squared distance and ordinary cokriging. Cross‐validation indicates that the inverse distance interpolation, which ignores the information on elevation, yields the largest prediction errors. Smaller prediction errors are produced by linear regression and ordinary cokriging. However, the results seem to favour the multivariate geostatistical method including auxiliary information (related to elevation). We conclude that ordinary cokriging is a very flexible and robust interpolation method because it can take into account several properties of the landscape; it should therefore be applicable in other mountainous regions, especially where precipitation is an important geomorphological factor. Copyright © 2005 John Wiley & Sons, Ltd.     

Statistical corrections of spatially interpolated missing precipitation data estimates

Spatial interpolation methods used for estimation of missing precipitation data generally under and overestimate the high and low extremes, respectively. This is a major limitation that plagues all spatial interpolation methods as observations from different sites are used in local or global variants of these methods for estimation of missing data. This study proposes bias‐correction methods similar to those used in climate change studies for correcting missing precipitation estimates provided by an optimal spatial interpolation method. The methods are applied to post‐interpolation estimates using quantile mapping, a variant of equi‐distant quantile matching and a new optimal single best estimator (SBE) scheme. The SBE is developed using a mixed‐integer nonlinear programming formulation. K‐fold cross validation of estimation and correction methods is carried out using 15 rain gauges in a temperate climatic region of the U.S. Exhaustive evaluation of bias‐corrected estimates is carried out using several statistical, error, performance and skill score measures. The differences among the bias‐correction methods, the effectiveness of the methods and their limitations are examined. The bias‐correction method based on a variant of equi‐distant quantile matching is recommended. Post‐interpolation bias corrections have preserved the site‐specific summary statistics with minor changes in the magnitudes of error and performance measures. The changes were found to be statistically insignificant based on parametric and nonparametric hypothesis tests. The correction methods provided improved skill scores with minimal changes in magnitudes of several extreme precipitation indices. The bias corrections of estimated data also brought site‐specific serial autocorrelations at different lags and transition states (dry‐to‐dry, dry‐to‐wet, wet‐to‐wet and wet‐to‐dry) close to those from the observed series. Bias corrections of missing data estimates provide better serially complete precipitation time series useful for climate change and variability studies in comparison to uncorrected filled data series. Copyright © 2013 John Wiley & Sons, Ltd.     

Homotopy perturbation method for solving the space–time fractional advection–dispersion equation

In this paper we present a reliable algorithm, the homotopy perturbation method, to construct numerical solutions of the space–time fractional advection–dispersion equation in the form of a rapidly convergent series with easily computable components. Fractional advection–dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to space–time fractional advection–dispersion equations.     

The comparison of sensitivity analysis of hydrological uncertainty estimates by GLUE and Bayesian method under the impact of precipitation errors

The input uncertainty is as significant as model error, which affects the parameter estimation, yields bias and misleading results. This study performed a comprehensive comparison and evaluation of uncertainty estimates according to the impact of precipitation errors by GLUE and Bayesian methods using the Metropolis Hasting algorithm in a validated conceptual hydrological model (WASMOD). It aims to explain the sensitivity and differences between the GLUE and Bayesian method applied to hydrological model under precipitation errors with constant multiplier parameter and random multiplier parameter. The 95 % confidence interval of monthly discharge in low flow, medium flow and high flow were selected for comparison. Four indices, i.e. the average relative interval length, the percentage of observations bracketed by the confidence interval, the percentage of observations bracketed by the unit confidence interval and the continuous rank probability score (CRPS) were used in this study for sensitivity analysis under model input error via GLUE and Bayesian methods. It was found that (1) the posterior distributions derived by the Bayesian method are narrower and sharper than those obtained by the GLUE under precipitation errors, but the differences are quite small; (2) Bayesian method performs more sensitive in uncertainty estimates of discharge than GLUE according to the impact of precipitation errors; (3) GLUE and Bayesian methods are more sensitive in uncertainty estimate of high flow than the other flows by the impact of precipitation errors; and (4) under the impact of precipitation, the results of CRPS for low and medium flows are quite stable from both GLUE and Bayesian method while it is sensitive for high flow by Bayesian method.     

Long-term time-dependent stochastic modelling of extreme waves

This paper presents a literature survey on time-dependent statistical modelling of extreme waves and sea states. The focus is twofold: on statistical modelling of extreme waves and space- and time-dependent statistical modelling. The first part will consist of a literature review of statistical modelling of extreme waves and wave parameters, most notably on the modelling of extreme significant wave height. The second part will focus on statistical modelling of time- and space-dependent variables in a more general sense, and will focus on the methodology and models used also in other relevant application areas. It was found that limited effort has been put on developing statistical models for waves incorporating spatial and long-term temporal variability and it is suggested that model improvements could be achieved by adopting approaches from other application areas. In particular, Bayesian hierarchical space–time models were identified as promising tools for spatio-temporal modelling of extreme waves. Finally, a review of projections of future extreme wave climate is presented.     

A spatio-temporal climate-based model of early dengue fever warning in southern Taiwan

Dengue Fever (DF) has been identified by the World Health organization (WHO) as one of the most serious vector-borne infectious diseases in tropical and sub-tropical areas. During 2007, in particular, there were over 2,000 DF cases in Taiwan, which was the highest number of cases in the recorded history of Taiwan epidemics. Most DF studies have focused mainly on temporal DF patterns and its close association with climatic covariates, whereas they have understated spatial DF patterns (spatial dependence and clustering) and composite space–time effects. The present study proposes a spatio-temporal DF prediction approach based on stochastic Bayesian Maximum Entropy (BME) analysis. Core and site-specific knowledge bases are considered, including climate and health datasets under conditions of uncertainty, space–time dependence functions, and a Poisson regression model of climatic variables contributing to DF occurrences in southern Taiwan during 2007. The results show that the DF outbreaks in the study area are highly influenced by climatic conditions. Furthermore, the analysis can provide the required “one-week-ahead” outbreak warnings based on spatio-temporal predictions of DF distributions. Therefore, the proposed approach can provide the Taiwan Disease Control Agency with a valuable tool to timely identify, control, and even efficiently prevent DF spreading across space–time.     

Interpolation of runoff applying objective methods

The paper treats the problem of interpolating annual runoff from regular streamflow measurements in a regional scale applying objective methods. These methods are adapted to point processes like temperature and precipitation. Modifications are needed to account for the fact that streamflow is an integrated process following the hierarchical structure of river systems. The most straightforward method is therefore to relate the interpolation to the existing river network. For theoretical reasons it is preferable to interpolate the lateral inflow rather than the flow in the river itself. Procedures for the interpolation with the different approaches are developed and discussed. Special attention is put on the question how the equation of continuity can be satisfied. The Laagen drainage basin in southern Norway is used as a test area. The data consist of annual observations of streamflow and digital map information on river networks and drainage basin boundaries.     

Modelling ocean wave climate with a Bayesian hierarchical space–time model and a log-transform of the data

Long-term trends in the ocean wave climate because of global warming are of major concern to many stakeholders within the maritime industries, and there is a need to take severe sea state conditions into account in design of marine structures and in marine operations. Various stochastic models of significant wave height are reported in the literature, but most are based on point measurements without exploiting the flexible framework of Bayesian hierarchical space–time models. This framework allows modelling of complex dependence structures in space and time and incorporation of physical features and prior knowledge, yet remains intuitive and easily interpreted. This paper presents a Bayesian hierarchical space–time model with a log-transform for significant wave height data for an area in the North Atlantic ocean. The different components of the model will be outlined, and the results from applying the model to data of different temporal resolutions will be discussed. Different model alternatives have been tried and long-term trends in the data have been identified for all model alternatives. Overall, these trends are in reasonable agreement and also agree fairly well with previous studies. The log-transform was included in order to account for observed heteroscedasticity in the data, and results are compared to previous results where a similar model was employed without a log-transform. Furthermore, a discussion of possible extensions to the model, e.g. incorporating regression terms with relevant meteorological data, will be presented.     

Interpolation of runoff applying objective methods

The paper treats the problem of interpolating annual runoff from regular streamflow measurements in a regional scale applying objective methods. These methods are adapted to point processes like temperature and precipitation. Modifications are needed to account for the fact that streamflow is an integrated process following the hierarchical structure of river systems. The most straightforward method is therefore to relate the interpolation to the existing river network. For theoretical reasons it is preferable to interpolate the lateral inflow rather than the flow in the river itself. Procedures for the interpolation with the different approaches are developed and discussed. Special attention is put on the question how the equation of continuity can be satisfied. The Laagen drainage basin in southern Norway is used as a test area. The data consist of annual observations of streamflow and digital map information on river networks and drainage basin boundaries.     

非线性二次规划贝叶斯叠前反演

叠前反演的目的是基于弹性波理论从地震数据中获得地层参数的可靠估计,进而用于描述地层的流体和岩性特征.然而叠前反演问题都是高维的和非适定的,并且容易受各种噪声和采集过程中不确定因素的影响,因此,为了获得稳定可靠的解必需对反演过程加以合理的约束.本文提出了一种基于非线性二次规划的叠前三参数反演方法.首先基于贝叶斯参数估计理论,假设似然函数服从高斯分布,并使待反演的参数服从于改进的Cauchy分布,从而提高了反演结果的分辨率;其次用协方差矩阵来描述参数间的相关程度,进一步提高了反演结果的稳定性;最后将问题转化为一个非线性二次规划的求解问题,并在多种约束下得到问题的解.仿真实验和实际应用皆已表明,本文提出的反演方法运算速度快捷,既使在信噪比很低的情况下也可获得较好的反演结果,为储层的进一步识别提供更多的物性参数.     

Geospatial–temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America

A quantification of the spatio-temporal dependence among precipitation extremes is important for investigating the properties of intense storms as well as flood or flash-flood related hazards. Extreme value theory has been widely applied to the hydrologic sciences and hydraulic engineering. However, rigorous approaches to quantify dependence structures among extreme values in space and time have not been reported in the literature. Previous researchers have quantified the dependence among extreme values through the concept of (pairwise bivariate) tail dependence coefficients. For estimation of the tail dependence coefficients, we apply a recently developed method [Kuhn G. On dependence and extremes. PhD thesis (Advisor: C. Klüppelberg), Munich University of Technology, 2006] which utilized the multivariate tail dependence function of a subclass of elliptical copulas. This study extends the previous approach in the context of space and time by considering pairs of spatial grids in South America and quantifying the dependence among precipitation extremes based on the time series at each spatial grid. In addition, Kendall’s τ is used to estimate the pairwise copula correlation (for an elliptical copula) of precipitation between all grids in South America. The geospatial–temporal dependence measures are applied to precipitation observations from 1940 to 2005 as well as simulations from the Community Climate System Model version 3 (CCSM3) for 1940–2099. New insights are obtained regarding the spatio-temporal dependence structures for precipitation over South America both with regard to correlation as well as tail dependence.     

Accuracy assessment of optimum interpolation in multichannel precipitation analysis

The accuracy of an optimum interpolation technique in filling missing values in multichannel (or multisite) hydrologic series containing time-coincident data gaps is examined. The applied methodology is based on the maximum entropy method (MEM) of spectral estimation or multivariate autoregressive modeling and heavily depends upon the properties of multichannel prediction error filter (PEF). Six precipitation time series spatially located within a hydrologic basin are used and time-coincident artificial gaps are created in all six series. The performance of the technique is assessed by comparing the filled-in series to the observed and by employing spectral analysis. The results reveal the usefulness of the method in multichannel hydrologic analysis.     

Accuracy assessment of optimum interpolation in multichannel precipitation analysis

The accuracy of an optimum interpolation technique in filling missing values in multichannel (or multisite) hydrologic series containing time-coincident data gaps is examined. The applied methodology is based on the maximum entropy method (MEM) of spectral estimation or multivariate autoregressive modeling and heavily depends upon the properties of multichannel prediction error filter (PEF). Six precipitation time series spatially located within a hydrologic basin are used and time-coincident artificial gaps are created in all six series. The performance of the technique is assessed by comparing the filled-in series to the observed and by employing spectral analysis. The results reveal the usefulness of the method in multichannel hydrologic analysis.     

Estimation of regional meteorological and hydrological drought characteristics: a case study for Denmark

Information on regional drought characteristics provides critical information for adequate water resource management. This study introduces a method to calculate the probability of a specific area to be affected by a drought of a given severity and demonstrates its potential for calculating both meteorological and hydrological drought characteristics. The method is demonstrated using Denmark as a case study. The calculation procedure was applied to monthly precipitation and streamflow series separately, which were linearly transformed by the Empirical Orthogonal Functions (EOF) method. Denmark was divided into 260 grid-cells of 14×17 km, and the monthly mean and the EOF-weight coefficients were interpolated by kriging. The frequency distributions of the first two (streamflow) or three (precipitation) amplitude functions were then derived. By performing Monte Carlo simulations, amplitude functions corresponding to 1000 years of data were generated. Based on these simulated functions as well as interpolated mean and weight coefficients, long time series of precipitation and streamflow were simulated for each grid-cell. The probability distribution functions of the area covered by a drought and the drought deficit volumes were then derived and combined to produce drought severity-area-frequency curves. These curves allowed an estimation of the probability of an area of a certain extent to have a drought of a given severity, and thereby return periods could be assigned to historical drought events. A comparison of drought characteristics showed that streamflow droughts are less homogeneous over the region, less frequent and last for longer time periods than precipitation droughts.     

Observations of submesoscale eddies using high-frequency radar-derived kinematic and dynamic quantities

The spatio-temporal variability of submesoscale eddies off southern San Diego is investigated with two-year observations of subinertial surface currents [O(1) m depth] derived from shore-based high-frequency radars. The kinematic and dynamic quantities — velocity potential, stream function, divergence, vorticity, and deformation rates — are directly estimated from radial velocity maps using optimal interpolation. For eddy detection, the winding-angle approach based on flow geometry is applied to the calculated stream function. A cluster of nearly enclosed streamlines with persistent vorticity in time is identified as an eddy. About 700 eddies were detected for each rotation (clockwise and counter-clockwise). The two rotations show similar statistics with diameters in the range of 5–25 km and Rossby number of 0.2–2. They persist for 1–7 days with weak seasonality and migrate with a translation speed of 4–15 cm s⁻¹ advected by background currents. The horizontal structure of eddies exhibits nearly symmetric tangential velocity with a maximum at the defined radius of the eddy, non-zero radial velocity due to background flows, and Gaussian vorticity with the highest value at the center. In contrast divergence has no consistent spatial shape. Two episodic events are presented with other in situ data (subsurface current and temperature profiles, and local winds) as an example of frontal-scale secondary circulation associated with drifting submesoscale eddies.     

Detection of ENSO‐influence on the monthly precipitation in South Korea

During the latest several decades, there has been considerable interest in revealing the relationship between El Niño–southern oscillation (ENSO) and hydro‐meteorological variables. The oscillation is characterized by a simple index, the southern oscillation index (SOI). However, thus far, there is little evidence for the influence of ENSO in Korea and Japan. The influence of ENSO has also been studied in South Korea, but the estimated results are still qualitative and show an indirect relationship between ENSO and hydro‐meteorological variables. In this study we use simple approaches to reveal the quantitative and direct correlation between SOI and the monthly precipitation at five stations distributed over South Korea. The monthly precipitation data are transformed into nonexceedance probability time series because the data cannot be normally distributed by applying the usual transformations. The SOI is classified into five categories according to their values. Additionally, to detect the nonlinear relationship between categorized SOI and nonexceedance probability of the monthly precipitation, we use Kendall's τ, a nonparametric test. Significant correlations between the categorized SOI and the transformed precipitation are detected. Generally, the monthly precipitation is influenced by a La Niña event with a lag time of 4 months for southern coastal areas and a lag time of 5 months for middle to high regions in South Korea. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of satellite-based rainfall products for hydrological modelling in Morocco

ABSTRACT

Several satellite-based precipitation estimates are becoming available at a global scale, providing new possibilities for water resources modelling, particularly in data-sparse regions and developing countries. This work provides a first validation of five different satellite-based precipitation products (TRMM-3B42 v6 and v7, RFE 2.0, PERSIANN-CDR, CMORPH1.0 version 0.x) in the 1785 km² Makhazine catchment (Morocco). Precipitation products are first compared against ground observations. Ten raingauges and four different interpolation methods (inverse distance, nearest neighbour, ordinary kriging and residual kriging with altitude) were used to compute a set of interpolated precipitation reference fields. Second, a parsimonious conceptual hydrological model is considered, with a simulation approach based on the random generation of model parameters drawn from existing parameter set libraries, to compare the different precipitation inputs. The results indicate that (1) all four interpolation methods, except the nearest neighbour approach, give similar and valid precipitation estimates at the catchment scale; (2) among the different satellite-based precipitation estimates verified, the TRMM-3B42 v7 product is the closest to observed precipitation, and (3) despite poor performance at the daily time step when used in the hydrological model, TRMM-3B42 v7 estimates are found adequate to reproduce monthly dynamics of discharge in the catchment. The results provide valuable perspectives for water resources modelling of data-scarce catchments with satellite-based rainfall data in this region.

Editor M.C. Acreman; Associate editor N. Verhoest     

A Bayesian hierarchical spatio-temporal model for extreme rainfall in Extremadura (Spain)

A statistical study was made of the temporal trend in extreme rainfall in the region of Extremadura (Spain) during the period 1961–2009. A hierarchical spatio-temporal Bayesian model with a GEV parameterization of the extreme data was employed. The Bayesian model was implemented in a Markov chain Monte Carlo framework that allows the posterior distribution of the parameters that intervene in the model to be estimated. The results show a decrease of extreme rainfall in winter and spring and a slight increase in autumn. The uncertainty in the trend parameters obtained with the hierarchical approach is much smaller than the uncertainties obtained from the GEV model applied locally. Also found was a negative relationship between the NAO index and the extreme rainfall in Extremadura during winter. An increase was observed in the intensity of the NAO index in winter and spring, and a slight decrease in autumn.     

Model-based geostatistical interpolation of the annual number of ozone exceedance days in the Netherlands

This paper discusses two model-based geostatistical methods for spatial interpolation of the number of days that ground level ozone exceeds a threshold level. The first method assumes counts to approximately follow a Poisson distribution, while the second method assumes a log-Normal distribution. First, these methods were compared using an extensive data set covering the Netherlands, Belgium and Germany. Second, the focus was placed on only the Netherlands, where only a small data set was used. Bayesian techniques were used for parameter estimation and interpolation. Parameter estimates are comparable due to the log-link in both models. Incorporating data from adjacent countries improves parameter estimation. The Poisson model predicts more accurately (maximum kriging standard deviation of 2.16 compared to 2.69) but shows smoother surfaces than the log-Normal model. The log-Normal approach ensures a better representation of the observations and gives more realistic patterns (an RMSE of 2.26 compared to 2.44). Model-based geostatistical procedures are useful to interpolate limited data sets of counts of ozone exceedance days. Spatial risk estimates using existing prior information can be made relating health effects to environmental thresholds.     

Effects of temporal scales and space mismatches on the TRMM 3B42 v7 precipitation product in a remote mountainous area

The accurate measurement of precipitation is essential to understanding regional hydrological processes and hydrological cycling. Quantification of precipitation over remote regions such as the Tibetan Plateau is highly unreliable because of the scarcity of rain gauges. The objective of this study is to evaluate the performance of the satellite precipitation product of tropical rainfall measuring mission (TRMM) 3B42 v7 at daily, weekly, monthly, and seasonal scales. Comparison between TRMM grid precipitation and point‐based rain gauge precipitation was conducted using nearest neighbour and bilinear weighted interpolation methods. The results showed that the TRMM product could not capture daily precipitation well due to some rainfall events being missed at short time scales but provided reasonably good precipitation data at weekly, monthly, and seasonal scales. TRMM tended to underestimate the precipitation of small rainfall events (less than 1 mm/day), while it overestimated the precipitation of large rainfall events (greater than 20 mm/day). Consequently, TRMM showed better performance in the summer monsoon season than in the winter season. Through comparison, it was also found that the bilinear weighted interpolation method performs better than the nearest neighbour method in TRMM precipitation extraction.