Cokriging for enhanced spatial interpolation of rainfall in two Australian catchments

Rainfall data in continuous space provide an essential input for most hydrological and water resources planning studies. Spatial distribution of rainfall is usually estimated using ground‐based point rainfall data from sparsely positioned rain‐gauge stations in a rain‐gauge network. Kriging has become a widely used interpolation method to estimate the spatial distribution of climate variables including rainfall. The objective of this study is to evaluate three geostatistical (ordinary kriging [OK], ordinary cokriging [OCK], kriging with an external drift [KED]), and two deterministic (inverse distance weighting, radial basis function) interpolation methods for enhanced spatial interpolation of monthly rainfall in the Middle Yarra River catchment and the Ovens River catchment in Victoria, Australia. Historical rainfall records from existing rain‐gauge stations of the catchments during 1980–2012 period are used for the analysis. A digital elevation model of each catchment is used as the supplementary information in addition to rainfall for the OCK and kriging with an external drift methods. The prediction performance of the adopted interpolation methods is assessed through cross‐validation. Results indicate that the geostatistical methods outperform the deterministic methods for spatial interpolation of rainfall. Results also indicate that among the geostatistical methods, the OCK method is found to be the best interpolator for estimating spatial rainfall distribution in both the catchments with the lowest prediction error between the observed and estimated monthly rainfall. Thus, this study demonstrates that the use of elevation as an auxiliary variable in addition to rainfall data in the geostatistical framework can significantly enhance the estimation of rainfall over a catchment.     

On the geostatistical approach to the inverse problem

The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity.     

Performance of geostatistical interpolation methods for modeling sampled data with non-stationary mean

The measured ozone pollution peak in the atmosphere of Mexico City region was considered in order to study the effect of a non-stationary mean of the sampled data in geostatistics interpolation methods. With this objective the local mean value of the sampled data was estimated through a linear regression analysis of their values on the monitoring station’s coordinates. The residuals obtained by removing the data trend are considered as a set of stationary random variables. Several interpolation methods used in geostatistics, such as inverse distance weighted, kriging, and artificial neural networks techniques were considered. In an effort to optimize and evaluate its performance, we fit interpolated values to sampled data, obtaining optimal values for the parameters defining the used model, that means, the values of the parameters that give the lowest mean RMSE between the interpolated value and measured data at 20 stations at 1500 hours for a set of 21 days of December 2001, which was chosen as the training set. The training set is conformed by all the days in December 2001 excepting the days (3,6,9,12,...,27,30) which were considered as the testing set. Once the optimal model is obtained, it is used to interpolate the values at the stations at 1500 hours for the testing days. The RMSE between interpolated and measured values at monitoring stations was also evaluated for these testing values and is shown as a percentage in Table 2. These values and the defined generalization parameter G, can be used to evaluate the performance and the ability of the models to predict and reproduce the peak of ozone concentrations. Scatter plots for testing data are presented for each interpolation method. An interpretation of the ozone pollution levels obtained at 1500 hours at December 21 was given using the wind field that prevailed in the region 1 h before the same day.     

A nonhomogeneous Poisson process geostatistical model

This paper introduces a new geostatistical model for counting data under a space-time approach using nonhomogeneous Poisson processes, where the random intensity process has an additive formulation with two components: a Gaussian spatial component and a component accounting for the temporal effect. Inferences of interest for the proposed model are obtained under the Bayesian paradigm. To illustrate the usefulness of the proposed model, we first develop a simulation study to test the efficacy of the Markov Chain Monte Carlo (MCMC) method to generate samples for the joint posterior distribution of the model’s parameters. This study shows that the convergence of the MCMC algorithm used to simulate samples for the joint posterior distribution of interest is easily obtained for different scenarios. As a second illustration, the proposed model is applied to a real data set related to ozone air pollution collected in 22 monitoring stations in Mexico City in the 2010 year. The proposed geostatistical model has good performance in the data analysis, in terms of fit to the data and in the identification of the regions with the highest pollution levels, that is, the southwest, the central and the northwest regions of Mexico City.     

Multiscale adaptive estimation of the conductivity field using pump test data

An inverse problem is posed in terms of log-conductivities which are decomposed into macroscale deterministic and microscale stochastic components. The macroscale and microscale conductivities conceptualize hierarchical, scale-dependent aquifer parameters. A deterministic parameter estimation scheme divides a flow domain into a limited number of macroscale constant conductivity zones. A stochastic microscale parameter estimation scheme is used to obtain fluctuations about the macroscale averages in terms of geostatistical models. Both the macroscale and the microscale conductivities are estimated via maximum likelihood, adjoint-state methodologies. Monte Carlo-type approaches are used to examine the distribution of macroscale and microscale conductivity estimates.     

A method for the interpolation of nonnegative functions with an application to contaminant load estimation

The objective of this work is to extend kriging, a geostatistical interpolation method, to honor parameter nonnegativity. The new method uses a prior probability distribution based on reflected Brownian motion that enforces this constraint. The work presented in this paper focuses on interpolation problems where the unknown is a function of a single variable (e.g. time), and is developed both for the case with and without measurement error in the available data. The algorithms presented for conditional simulations are computationally efficient, particularly in the case with no measurement error. We present an application to the interpolation of dissolved arsenic concentration data from the North Fork of the Humboldt River, Nevada.     

Copula-based geostatistical modeling of continuous and discrete data including covariates

It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.     

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.     

Approximate copula-based estimation and prediction of discrete spatial data

The present paper reports on the use of copula functions to describe the distribution of discrete spatial data, e.g. count data from environmental mapping or areal data analysis. In particular, we consider approaches to parameter point estimation and propose a fast method to perform approximate spatial prediction in copula-based spatial models with discrete marginal distributions. We assess the goodness of the resulting parameter estimates and predictors under different spatial settings and guide the analyst on which approach to apply for the data at hand. Finally, we illustrate the methodology by analyzing the well-known Lansing Woods data set. Software that implements the methods proposed in this paper is freely available in Matlab language on the author’s website.     

A stability analysis of the geostatistical approach to aquifer transmissivity identification

In recent years, geostatistical concepts have been applied to the inverse problem of transmissivity estimation from piezometric head data. It has been claimed that such methods overcome various difficulties encountered in other approaches. However, the reconstruction of transmissivity from head measurements is ill-posed as it depends on derivatives of the head field. Consequently, any accurate method for its solution is likely to encounter numerically ill-conditioned systems. This paper reviews the geostatistical approach, and uses the stability analyses of linear algebra to show that, as the amount of available data increases and the discretization of the system is refined, both a numerically ill-conditioned parameter estimation problem and ill-conditioned cokriging equations may appear. Therefore, while the geostatistical approach does have conceptual appeal, it does not avoid the fundamental difficulties arising out of the ill-posed nature of transmissivity identification. Instead, the method is likely to be quite sensitive to these difficulties, so care must be taken in its formulation to minimize their effects. A means to stabilize the geostatistical method is suggested and numerical experiments that highlight key points of our analysis are given.     

Evaluation of a water equivalent of snow cover map in a small catchment area using a geostatistical approach

We describe an objective method for evaluating the spatial distribution of water equivalents of the snow cover within a small catchment. Regression analysis is used to quantify the relationship between elevation, presence or absence of forest, and potential direct solar radiation as independent variables and water equivalent as measured at a number of sites. First, this regression relationship is used to interpolate water equivalent data all over the basin area. Then we interpolate the residuals of the regression using a geostatistical approach. Superimposing the results obtained by interpolating the regression relationship and the interpolated residuals eventually yields the water equivalent distribution over the test area. The advantages of the interpolation method used lie in the optimal (effective, unbiased) estimation of the interpolated values as well as in the possibility to quantify the associated estimation variances.     

Compared performances of different algorithms for estimating annual nutrient loads discharged by the eutrophic River Loire

Good estimates of pollutant fluxes are required for Earth systems sciences and water quality management. The gradual accumulation of water quality data records over the past few decades has increased the value of these data for examining long‐term trends. On many major rivers, however, infrequent sampling of most pollutants makes flux estimates and their analysis difficult. This paper explores the performance of different methods for estimating nutrient fluxes. The objective is to assess the accuracy (bias) and precision (dispersion) of annual nutrient fluxes based on monthly sampling, which is the frequency with which 80% of French water quality surveys have been carried out since 1971. The study is based on a data set of nutrient concentrations surveyed at high frequency during a 5 year pilot study (1981–85) at the Orléans station in the middle reaches of the River Loire, France. The mean specific fluxes were 641 (nitrate‐N), 96 (total‐P) and 37 kg year⁻¹ km⁻² (orthophosphate‐P). For each year, the data set was then ‘resampled’ by randomly simulating 12 sampling dates. 100 simulated monthly samplings were generated, upon which seven estimation methods were tested. The evaluations indicate that, when concentrations of specific substances in large rivers exhibit seasonal variation, a simple method based on linear interpolation between samples taken at approximately monthly intervals is advocated. With the monthly sampling interval, the precision (confidence level of 95%) of annual nutrient fluxes obtained by the appropriate methods was 13% for nitrates, 20% for total‐P, 26% for orthophosphates, and 34% for particulate‐P. The frequency of water quality surveys required to obtain an annual nutrient flux with 10% precision was around 15 days for nitrate, 10 days for orthophosphate‐P and total‐P, and about 5 days in the case of particulate‐P. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Estimating threshold-exceeding probability maps of environmental variables with Markov chain random fields

Estimating and mapping spatial uncertainty of environmental variables is crucial for environmental evaluation and decision making. For a continuous spatial variable, estimation of spatial uncertainty may be conducted in the form of estimating the probability of (not) exceeding a threshold value. In this paper, we introduced a Markov chain geostatistical approach for estimating threshold-exceeding probabilities. The differences of this approach compared to the conventional indicator approach lie with its nonlinear estimators—Markov chain random field models and its incorporation of interclass dependencies through transiograms. We estimated threshold-exceeding probability maps of clay layer thickness through simulation (i.e., using a number of realizations simulated by Markov chain sequential simulation) and interpolation (i.e., direct conditional probability estimation using only the indicator values of sample data), respectively. To evaluate the approach, we also estimated those probability maps using sequential indicator simulation and indicator kriging interpolation. Our results show that (i) the Markov chain approach provides an effective alternative for spatial uncertainty assessment of environmental spatial variables and the probability maps from this approach are more reasonable than those from conventional indicator geostatistics, and (ii) the probability maps estimated through sequential simulation are more realistic than those through interpolation because the latter display some uneven transitions caused by spatial structures of the sample data.     

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.     

Improving the assessment of uncertainty and risk in the spatial prediction of environmental impacts: a new approach for fitting geostatistical model parameters based on dual Kriging in the presence of a trend

Modern methods of geostatistics deliver an essential contribution to Environmental Impact Assessment (EIA). These methods allow for spatial interpolation, forecast and risk assessment of expected impact during and after mining projects by integrating different sources of data and information. Geostatistical estimation and simulation algorithms are designed to provide both, a most likely forecast as well as information about the accuracy of the prediction. The representativeness of these measures depends strongly on the quality of the inferred model parameters, which are mainly defined by the parameters of the variogram or the covariance function. Available data may be sparse, trend affected and of different data type making the inference of representative geostatistical model parameters difficult. This contribution introduces a new method for best fitting of the geostatistical model parameters in the presence of a trend, which utilizes the empirical and theoretical differences between Universal Kriging and trend-predictions. The method extends well known approaches of cross validation in two aspects. Firstly, the model evaluation is not only limited to sample data locations but is performed on any prediction locations of the attribute in the domain. Secondly, it extends the measure used in cross validation, based on a single point replacement by using error curves. These allow defining rings of influence representing errors resulting from separate variogram lags. By analyzing the different variogram lags the fit of the complete covariance can be assessed and the influence of the several model parameters separated. The use of the proposed method in an EIA context is illustrated in a case study related on the prediction of mining-induced ground movements.     

Improving the reliability of groundwater monitoring networks using combined numerical,geostatistical and neural network-based simulation models

ABSTRACT

A new methodology is proposed for improving the accuracy of groundwater-level estimations and increasing the efficiency of groundwater-level monitoring networks. Three spatio-temporal (S-T) simulation models, numerical groundwater flow, artificial neural network and S-T kriging, are implemented to simulate water-table level variations. Individual models are combined using model fusion techniques and the more accurate of the individual and combined simulation models is selected for the estimation. Leave-one-out cross-validation shows that the estimation error of the best fusion model is significantly less than that of the three individual models. The selected fusion model is then considered for optimal S-T redesign of the groundwater monitoring network of the Dehgolan Plain (Iran). Using a Bayesian maximum entropy interpolation technique, soft data are included in the geostatistical analyses. Different scenarios are defined to incorporate economic considerations and different levels of precision in selecting the best monitoring network; a network of 37 wells is proposed as the best configuration. The mean variance estimation errors of all scenarios decrease significantly compared to that of the existing monitoring network. A reduction in equivalent uniform annual costs of different scenarios is achieved.     

基于贝叶斯原理的PP波和PS波AVO联合反演方法研究(英文)

基于Aki-Richards公式和贝叶斯原理,本文发展了利用叠前PP波和PS波资料联合反演P波速度比、S波速度比和密度比的方法。该方法假设参数之间满足正态分布,引入参数协方差矩阵来描述反演参数之间的相关性以提高反演过程的稳定性,并同时使反演的参数序列服从Cauchy分布,引入矩阵Q来描述参数序列的稀疏性以提高反演结果的分辨率。采用本文提出的方法对模型数据和实际多波资料进行反演,结果表明:本文方法正确有效;与传统的单一PP波反演相比,PP波和PS波AVO联合反演具有稳定性更好和反演精度更高等优点。     

Parameter estimation for 2-parameter generalized pareto distribution by POME

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 2-parameter generalized Pareto (GP2) distribution. Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). The parameter estimates yielded by POME were comparable or better within certain ranges of sample size and coefficient of variation.     

Remote sensing of the global distribution of total ozone and the inferred upper-tropospheric circulation from Nimbus IRIS experiments

The global distribution of total ozone is derived for the period April, May, June and July of 1969 from Nimbus-3 Infrared Interferometer Spectrometer (IRIS) experiment. Preliminary estimates of ozone amounts from Nimbus-4 IRIS for the same period of 1970 show similar results. The standard error of estimation of total ozone from both IRIS experiments is 6% with respect to Dobson Spectrophotometer measurements. A systematic variation in the ozone distribution from April to July in the tropical, middle and polar latitudes is observed indicating the changes in the lower stratospheric circulation.The total ozone measurements show a strong correlation with the upper tropospheric geopotential height in the extratropical latitudes. From this relationship total ozone is used as a quasi-stream function to deduce geostrophic winds at the 200 mb level over extratropical regions of the northern and southern hemispheres. These winds reveal the subtropical and polar jet streams over the globe.Allied research associates.