Kriging in the hydrosciences

Most of the methods currently used in hydrosciences for interpolation and spatial averaging fail to quantify the accuracy of the estimates.The theory of regionalized variables enables one to point out the relationship between the spatial correlation of hydrometeorological or hydrogeological fields and the precision of interpolation, or determination of average values, over these fields.A new estimation method called kriging has proven to be quite well adapted to solving water resources problems. The author presents a series of case-studies in automatic contouring, data input for numerical models, estimation of average precipitation over a given catchment area, and measurement network design.     

Analysis of precipitation extremes with the assessment of regional climate models over the Willamette River Basin,USA

An appropriate, rapid and effective response to extreme precipitation and any potential flood disaster is essential. Providing an accurate estimate of future changes to such extreme events due to climate change are crucial for responsible decision making in flood risk management given the predictive uncertainties. The objective of this article is to provide a comparison of dynamically downscaled climate models simulations from multiple model including 12 different combinations of General Circulation Model (GCM)–regional climate model (RCM), which offers an abundance of additional data sets. The three major aspects of this study include the bias correction of RCM scenarios, the application of a newly developed performance metric and the extreme value analysis of future precipitation. The dynamically downscaled data sets reveal a positive overall bias that is removed through quantile mapping bias correction method. The added value index was calculated to evaluate the models' simulations. Results from this metric reveal that not all of the RCMs outperform their host GCMs in terms of correlation skill. Extreme value theory was applied to both historic, 1980–1998, and future, 2038–2069, daily data sets to provide estimates of changes to 2‐ and 25‐year return level precipitation events. The generalized Pareto distribution was used for this purpose. The Willamette River basin was selected as the study region for analysis because of its topographical variability and tendency for significant precipitation. The extreme value analysis results showed significant differences between model runs for both historical and future periods with considerable spatial variability in precipitation extremes. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved bias correction scheme based on comparative precipitation characteristics

Bias correction is a necessary post‐processing procedure in order to use regional climate model (RCM) simulated local climate variables as the input data for hydrological models owing to systematic errors of RCMs. Most of present bias correction methods adjust statistical properties between observed and simulated data on the basis of calendar periods, e.g. month or season. However, this matching statistic is only a necessary condition, not a sufficient condition, as temporal distribution of the precipitation between observed and simulated data is ignored. This study suggests an improved bias correction scheme that considers not only statistical properties but also the temporal distribution between the time series of observed and modelled data. The ratio of the observed precipitation to simulated precipitation is used to compare the behaviour between the observed and modelled precipitation data, and three criteria are proposed when dividing bias correction periods: (1) underestimation of precipitation, (2) stability of /underestimation of precipitation, (2) stability of precipitation ratio and (3) oscillation of precipitation ratio. The results show that the output of the proposed bias correction method follows the trend of the observed precipitation better than that of the conventional bias correction method. This study indicates that temporal distribution should not be ignored when choosing a comparison period for bias correction. However, the study is only a preliminary attempt to address this important issue, and we hope it will stimulate more research activities to improve the methodology. Future efforts on several unsolved problems have been suggested such as how to find out the optimal group number to avoid the overfitting and underfitting conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation of the annual maximum distribution from samples of maxima in separate seasons

Quantile estimates of the annual maximum distribution can be obtained by fitting theoretical distributions to the maxima in separate seasons, e.g. to the monthly maxima. In this paper, asymptotic expressions for the bias and the variance of such estimates are derived for the case that the seasonal maxima follow a Gumbel distribution. Results from these expressions are presented for a situation with no seasonal variation and for maximum precipitation depths at Uccle/Ukkel (Belgium). It is shown that the bias is often negligible and that the variance reduction by using seasonal maxima instead of just the annual maxima strongly depends on the seasonal variation in the data. A comparison is made between the asymptotic standard error of quantile estimates from monthlymaxima with those from a partial duration series. Much attention is paid to the effect of model misspecification on the resulting quantile estimates of the annual maximum distribution. The use of seasonal maxima should be viewed with caution when the upper tail of this distribution is of interest.     

Missing precipitation data estimation using optimal proximity metric-based imputation,nearest-neighbour classification and cluster-based interpolation methods

Abstract

New optimal proximity-based imputation, K-nearest neighbour (K-NN) classification and K-means clustering methods are proposed and developed for estimation of missing daily precipitation records. Mathematical programming formulations are developed to optimize the weighting, classification and clustering schemes used in these methods. Ten different binary and real-valued distance metrics are used as proximity measures. Two climatic regions, Kentucky and Florida, (temperate and tropical) in the USA, with different gauge density and network structure, are used as case studies to evaluate the new methods. A comprehensive exercise is undertaken to compare the performances of the new methods with those of several deterministic and stochastic spatial interpolation methods. The results from these comparisons indicate that the proposed methods performed better than existing methods. Use of optimal proximity metrics as weights, spatial clustering of observation sites and classification of precipitation data resulted in improvement of missing data estimates.

Editor D. Koutsoyiannis; Associate editor C. Onof     

Impact of the spatial variability of daily precipitation on hydrological projections: A comparison of GCM‐ and RCM‐driven cases in the Han River basin,Korea

In this study, we investigate the impact of the spatial variability of daily precipitation on hydrological projections based on a comparative assessment of streamflow simulations driven by a global climate model (GCM) and two regional climate models (RCMs). A total of 12 different climate input datasets, that is, the raw and bias‐corrected GCM and raw and bias‐corrected two RCMs for the reference and future periods, are fed to a semidistributed hydrological model to assess whether the bias correction using quantile mapping and dynamical downscaling using RCMs can improve streamflow simulation in the Han River basin, Korea. A statistical analysis of the daily precipitation demonstrates that the precipitation simulated by the GCM fails to capture the large variability of the observed daily precipitation, in which the spatial autocorrelation decreases sharply within a relatively short distance. However, the spatial variability of precipitation simulated by the two RCMs shows better agreement with the observations. After applying bias correction to the raw GCM and raw RCMs outputs, only a slight change is observed in the spatial variability, whereas an improvement is observed in the precipitation intensity. Intensified precipitation but with the same spatial variability of the raw output from the bias‐corrected GCM does not improve the heterogeneous runoff distributions, which in turn regulate unrealistically high peak downstream streamflow. GCM‐simulated precipitation with a large bias correction that is necessary to compensate for the poor performance in present climate simulation appears to distort streamflow patterns in the future projection, which leads to misleading projections of climate change impacts on hydrological extremes.     

On robustness of large quantile estimates to largest elements of the observation series

The robustness of large quantile estimates of largest elements in a small sample by the methods of moments (MOM), L‐moments (LMM) and maximum likelihood (MLM) was evaluated and compared. Bias (B) and mean square error (MSE) were used to measure the estimation methods performance. Quantiles were estimated by eight two‐parameter probability distributions with the variation coefficient being the shape parameter. The effect of dropping largest elements of the series on large quantile values was assessed for various variation coefficient (C_V)/sample size (n) ‘combinations’ with n = 30 as the basic value. To that end, both the Monte Carlo sampling experiments and an asymptotic approach consisting in distribution truncation were applied. In general, both sampling and asymptotic approaches point to MLM as the most robust method of the three considered, with respect to bias of large quantiles. Comparing the performance of two other methods, the MOM estimates were found to be more robust for small and moderate hydrological samples drawn from distributions with zero lower‐bound than were the LMM estimates. Extending the evaluation to outliers, it was shown that all the above findings remain valid. However, using the MSE variation as a measure of performance, the LMM was found to be the best for most distribution/variation coefficient combinations, whereas MOM was found to be the worst. Moreover, removal of the largest sample element need not result in a loss of estimation efficiency. The gain in accuracy is observed for the heavy‐tailed and log‐normal distributions, being particularly distinctive for LMM. In practice, while dealing with a single sample deprived of its largest element, one should choose the estimation method giving the lowest MSE of large quantiles. For n = 30 and several distribution/variation coefficient combinations, the MLM outperformed the two other methods in this respect and its supremacy grew with sample size, while MOM was usually the worst. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparative evaluation of Poisson partial duration models for mixed flood populations

Simple homogeneous formulations of two extreme value partial duration flood models are compared to more sophisticated compound formulations in terms of asymptotic performance of quantile estimates. The compound model formulations were developed to model flood series resulting from mixed climatological processes. It was found that only in the case of marked nonhomogeneity in the data samples did the compound formulation of the models offer significant advantages in terms of variance of quantile estimates. However, the estimates from the homogeneous model were significantly biased in the negative direction. This negative bias of quantile estimates from the simple model was even more pronounced when the more sophisticated Weibull model was used as the base.     

Comparative evaluation of Poisson partial duration models for mixed flood populations

Simple homogeneous formulations of two extreme value partial duration flood models are compared to more sophisticated compound formulations in terms of asymptotic performance of quantile estimates. The compound model formulations were developed to model flood series resulting from mixed climatological processes. It was found that only in the case of marked nonhomogeneity in the data samples did the compound formulation of the models offer significant advantages in terms of variance of quantile estimates. However, the estimates from the homogeneous model were significantly biased in the negative direction. This negative bias of quantile estimates from the simple model was even more pronounced when the more sophisticated Weibull model was used as the base.     

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.     

Spatial interpolation schemes of daily precipitation for hydrologic modeling

Distributed hydrologic models typically require spatial estimates of precipitation interpolated from sparsely located observational points to the specific grid points. We compare and contrast the performance of regression-based statistical methods for the spatial estimation of precipitation in two hydrologically different basins and confirmed that widely used regression-based estimation schemes fail to describe the realistic spatial variability of daily precipitation field. The methods assessed are: (1) inverse distance weighted average; (2) multiple linear regression (MLR); (3) climatological MLR; and (4) locally weighted polynomial regression (LWP). In order to improve the performance of the interpolations, the authors propose a two-step regression technique for effective daily precipitation estimation. In this simple two-step estimation process, precipitation occurrence is first generated via a logistic regression model before estimate the amount of precipitation separately on wet days. This process generated the precipitation occurrence, amount, and spatial correlation effectively. A distributed hydrologic model (PRMS) was used for the impact analysis in daily time step simulation. Multiple simulations suggested noticeable differences between the input alternatives generated by three different interpolation schemes. Differences are shown in overall simulation error against the observations, degree of explained variability, and seasonal volumes. Simulated streamflows also showed different characteristics in mean, maximum, minimum, and peak flows. Given the same parameter optimization technique, LWP input showed least streamflow error in Alapaha basin and CMLR input showed least error (still very close to LWP) in Animas basin. All of the two-step interpolation inputs resulted in lower streamflow error compared to the directly interpolated inputs.     

The implications of bias correction methods and climate model ensembles on soil erosion projections under climate change

Climate change will most likely cause an increase in extreme precipitation and consequently an increase in soil erosion in many locations worldwide. In most cases, climate model output is used to assess the impact of climate change on soil erosion; however, there is little knowledge of the implications of bias correction methods and climate model ensembles on projected soil erosion rates. Using a soil erosion model, we evaluated the implications of three bias correction methods (delta change, quantile mapping and scaled distribution mapping) and climate model selection on regional soil erosion projections in two contrasting Mediterranean catchments. Depending on the bias correction method, soil erosion is projected to decrease or increase. Scaled distribution mapping best projects the changes in extreme precipitation. While an increase in extreme precipitation does not always result in increased soil loss, it is an important soil erosion indicator. We suggest first establishing the deviation of the bias-corrected climate signal with respect to the raw climate signal, in particular for extreme precipitation. Furthermore, individual climate models may project opposite changes with respect to the ensemble average; hence climate model ensembles are essential in soil erosion impact assessments to account for climate model uncertainty. We conclude that the impact of climate change on soil erosion can only accurately be assessed with a bias correction method that best reproduces the projected climate change signal, in combination with a representative ensemble of climate models. © 2018 John Wiley & Sons, Ltd.     

Spatial interpolation using nonlinear mathematical programming models for estimation of missing precipitation records

Abstract

New mathematical programming models are proposed, developed and evaluated in this study for estimating missing precipitation data. These models use nonlinear and mixed integer nonlinear mathematical programming (MINLP) formulations with binary variables. They overcome the limitations associated with spatial interpolation methods relevant to the arbitrary selection of weighting parameters, the number of control points within a neighbourhood, and the size of the neighbourhood itself. The formulations are solved using genetic algorithms. Daily precipitation data obtained from 15 rain gauging stations in a temperate climatic region are used to test and derive conclusions about the efficacy of these methods. The developed methods are compared with some naïve approaches, multiple linear regression, nonlinear least-square optimization, kriging, and global and local trend surface and thin-plate spline models. The results suggest that the proposed new mathematical programming formulations are superior to those obtained from all the other spatial interpolation methods tested in this study.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi

Citation Teegavarapu, R.S.V., 2012. Spatial interpolation using nonlinear mathematical programming models for estimation of missing precipitation records. Hydrological Sciences Journal, 57 (3), 383–406.     

Homogeneity analysis of Turkish meteorological data set

The missing value interpolation and homogeneity analysis were performed on the meteorological data of Turkey. The data set has the observations of six variables: the maximum air temperature, the minimum air temperature, the mean air temperature, the total precipitation, the relative humidity and the local pressure of 232 stations for the period 1974–2002. The missing values on the monthly data set were estimated using two methods: the linear regression (LR) and the expectation maximization (EM) algorithm. Because of higher correlations between test and reference series, EM algorithm results were preferred. The homogeneity analysis was performed on the annual data using a relative test and four absolute homogeneity tests were used for the stations where non‐testable series were found due to the low correlation coefficients between the test and the reference series. A comparison was accomplished by the graphics where relative and absolute tests provided different outcomes. Absolute tests failed to detect the inhomogeneities in the precipitation series at the significance level 1%. Interestingly, most of the inhomogeneities detected on the temperature variables existed in the Aegean region of Turkey. It is considered that theseinhomogeneities were mostly caused by non‐natural effects such as relocation. Because of changes at topography at short distance in this region intensify non‐random characteristics of the temperature series when relocation occurs even in small distances. The marine effect, which causes artifical cooling effect due to sea breezes has important impact on temperature series and the orograhpy allows this impact go through the inner parts in this region. Copyright © 2010 John Wiley & Sons, Ltd.     

Exploratory analysis of statistical post‐processing methods for hydrological ensemble forecasts

Despite many recent improvements, ensemble forecast systems for streamflow often produce under‐dispersed predictive distributions. This situation is problematic for their operational use in water resources management. Many options exist for post‐processing of raw forecasts. However, most of these have been developed using meteorological variables such as temperature, which displays characteristics very different from streamflow. In addition, streamflow data series are often very short or contain numerous gaps, thus compromising the estimation of post‐processing statistical parameters. For operational use, a post‐processing method has to be effective while remaining as simple as possible. We compared existing post‐processing methods using normally distributed and gamma‐distributed synthetic datasets. To reflect situations encountered with ensemble forecasts of daily streamflow, four normal distribution parameterizations and six gamma distribution parameterizations were used. Three kernel‐based approaches were tested, namely, the ‘best member’ method and two improvements thereof, and one regression‐based approach. Additional tests were performed to assess the ability of post‐processing methods to cope with short calibration series, missing values or small numbers of ensemble members. We thus found that over‐dispersion is best corrected by the regression method, while under‐dispersion is best corrected by kernel‐based methods. This work also shows key limitations associated with short data series, missing values, asymmetry and bias. One of the improved best member methods required longer series for the estimation of post‐processing parameters, but if provided with adequate information, yielded the best improvement of the continuous ranked probability score. These results suggest guidelines for future studies involving real operational datasets. Copyright © 2014 John Wiley & Sons, Ltd.     

Seismic data interpolation using frequency‐domain complex nonstationary autoregression

We have developed a novel method for missing seismic data interpolation using f‐x‐domain regularised nonstationary autoregression. f‐x regularised nonstationary autoregression interpolation can deal with the events that have space‐varying dips. We assume that the coefficients of f‐x regularised nonstationary autoregression are smoothly varying along the space axis. This method includes two steps: the estimation of the coefficients and the interpolation of missing traces using estimated coefficients. We estimate the f‐x regularised nonstationary autoregression coefficients for the completed data using weighted nonstationary autoregression equations with smoothing constraints. For regularly missing data, similar to Spitz f‐x interpolation, we use autoregression coefficients estimated from low‐frequency components without aliasing to obtain autoregression coefficients of high‐frequency components with aliasing. For irregularly missing or gapped data, we use known traces to establish nonstationary autoregression equations with regularisation to estimate the f‐x autoregression coefficients of the complete data. We implement the algorithm by iterated scheme using a frequency‐domain conjugate gradient method with shaping regularisation. The proposed method improves the calculation efficiency by applying shaping regularisation and implementation in the frequency domain. The applicability and effectiveness of the proposed method are examined by synthetic and field data examples.     

Frequency based imputation of precipitation

Changing climate and precipitation patterns make the estimation of precipitation, which exhibits two-dimensional and sometimes chaotic behavior, more challenging. In recent decades, numerous data-driven methods have been developed and applied to estimate precipitation; however, these methods suffer from the use of one-dimensional approaches, lack generality, require the use of neighboring stations and have low sensitivity. This paper aims to implement the first generally applicable, highly sensitive two-dimensional data-driven model of precipitation. This model, named frequency based imputation (FBI), relies on non-continuous monthly precipitation time series data. It requires no determination of input parameters and no data preprocessing, and it provides multiple estimations (from the most to the least probable) of each missing data unit utilizing the series itself. A total of 34,330 monthly total precipitation observations from 70 stations in 21 basins within Turkey were used to assess the success of the method by removing and estimating observation series in annual increments. Comparisons with the expectation maximization and multiple linear regression models illustrate that the FBI method is superior in its estimation of monthly precipitation. This paper also provides a link to the software code for the FBI method.     

Exploration of optimal time steps for daily precipitation bias correction: a case study using a single grid of RCM on the River Exe in southwest England

ABSTRACT

Bias correction is a necessary post-processing procedure in order to use regional climate model (RCM)-simulated local climate variables as the input data for hydrological models due to systematic errors of RCMs. Most of the present bias-correction methods adjust statistical properties between observed and simulated data based on a predefined duration (e.g. a month or a season). However, there is a lack of analysis of the optimal period for bias correction. This study attempted to address the question whether there is an optimal number for bias-correction groups (i.e. optimal bias-correction period). To explore this we used a catchment in southwest England with the regional climate model HadRM3 precipitation data. The proposed methodology used only one grid of RCM in the Exe catchment, one emissions scenario (A1B) and one member (Q0) among 11 members of HadRM3. We tried 13 different bias-correction periods from 3-day to 360-day (i.e. the whole of one year) correction using the quantile mapping method. After the bias correction a low pass filter was used to remove the high frequencies (i.e. noise) followed by estimating Akaike’s information criterion. For the case study catchment with the regional climate model HadRM3 precipitation, the results showed that a bias-correction period of about 8 days is the best. We hope this preliminary study on the optimum number bias-correction period for daily RCM precipitation will stimulate more research to improve the methodology with different climatic conditions. Future efforts on several unsolved problems have been suggested, such as how strong the filter should be and the impact of the number of bias correction groups on river flow simulations.

Editor M.C. Acreman Associate editor S. Kanae