Spatial prediction of river channel topography by kriging

Topographic information is fundamental to geomorphic inquiry, and spatial prediction of bed elevation from irregular survey data is an important component of many reach‐scale studies. Kriging is a geostatistical technique for obtaining these predictions along with measures of their reliability, and this paper outlines a specialized framework intended for application to river channels. Our modular approach includes an algorithm for transforming the coordinates of data and prediction locations to a channel‐centered coordinate system, several different methods of representing the trend component of topographic variation and search strategies that incorporate geomorphic information to determine which survey data are used to make a prediction at a specific location. For example, a relationship between curvature and the lateral position of maximum depth can be used to include cross‐sectional asymmetry in a two‐dimensional trend surface model, and topographic breaklines can be used to restrict which data are retained in a local neighborhood around each prediction location. Using survey data from a restored gravel‐bed river, we demonstrate how transformation to the channel‐centered coordinate system facilitates interpretation of the variogram, a statistical model of reach‐scale spatial structure used in kriging, and how the choice of a trend model affects the variogram of the residuals from that trend. Similarly, we show how decomposing kriging predictions into their trend and residual components can yield useful information on channel morphology. Cross‐validation analyses involving different data configurations and kriging variants indicate that kriging is quite robust and that survey density is the primary control on the accuracy of bed elevation predictions. The root mean‐square error of these predictions is directly proportional to the spacing between surveyed cross‐sections, even in a reconfigured channel with a relatively simple morphology; sophisticated methods of spatial prediction are no substitute for field data. Copyright © 2007 John Wiley & Sons, Ltd.     

Spatial prediction on river networks: comparison of top‐kriging with regional regression

Top‐kriging is a method for estimating stream flow‐related variables on a river network. Top‐kriging treats these variables as emerging from a two‐dimensional spatially continuous process in the landscape. The top‐kriging weights are estimated by regularising the point variogram over the catchment area (kriging support), which accounts for the nested nature of the catchments. We test the top‐kriging method for a comprehensive Austrian data set of low stream flows. We compare it with the regional regression approach where linear regression models between low stream flow and catchment characteristics are fitted independently for sub‐regions of the study area that are deemed to be homogeneous in terms of flow processes. Leave‐one‐out cross‐validation results indicate that top‐kriging outperforms the regional regression on average over the entire study domain. The coefficients of determination (cross‐validation) of specific low stream flows are 0.75 and 0.68 for the top‐kriging and regional regression methods, respectively. For locations without upstream data points, the performances of the two methods are similar. For locations with upstream data points, top‐kriging performs much better than regional regression as it exploits the low flow information of the neighbouring locations. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonparametric geostatistical risk mapping

In this work, a fully nonparametric geostatistical approach to estimate threshold exceeding probabilities is proposed. To estimate the large-scale variability (spatial trend) of the process, the nonparametric local linear regression estimator, with the bandwidth selected by a method that takes the spatial dependence into account, is used. A bias-corrected nonparametric estimator of the variogram, obtained from the nonparametric residuals, is proposed to estimate the small-scale variability. Finally, a bootstrap algorithm is designed to estimate the unconditional probabilities of exceeding a threshold value at any location. The behavior of this approach is evaluated through simulation and with an application to a real data set.     

Kriging with external drift for functional data for air quality monitoring

Functional data featured by a spatial dependence structure occur in many environmental sciences when curves are observed, for example, along time or along depth. Recently, some methods allowing for the prediction of a curve at an unmonitored site have been developed. However, the existing methods do not allow to include in a model exogenous variables that, for example, bring meteorology information in modeling air pollutant concentrations. In order to introduce exogenous variables, potentially observed as curves as well, we propose to extend the so-called kriging with external drift—or regression kriging—to the case of functional data by means of a three-step procedure involving functional modeling for the trend and spatial interpolation of functional residuals. A cross-validation analysis allows to choose smoothing parameters and a preferable kriging predictor for the functional residuals. Our case study considers daily PM₁₀ concentrations measured from October 2005 to March 2006 by the monitoring network of Piemonte region (Italy), with the trend defined by meteorological time-varying covariates and orographical constant-in-time variables. The performance of the proposed methodology is evaluated by predicting PM₁₀ concentration curves on 10 validation sites, even with simulated realistic datasets on a larger number of spatial sites. In this application the proposed methodology represents an alternative to spatio-temporal modeling but it can be applied more generally to spatially dependent functional data whose domain is not a time interval.     

Nonparametric bias-corrected variogram estimation under non-constant trend

In geostatistics, the approximation of the spatial dependence structure of a process, through the estimation of the variogram or the covariogram of the variable under consideration, is an important issue. In this work, under a general spatial model, including a mean or trend function, and without assuming any parametric model for this function and for the dependence structure of the process, a general nonparametric estimator of the variogram is proposed. The new approach consists in applying an iterative algorithm, using the residuals obtained from a nonparametric local linear estimation of the trend function, jointly with a correction of the bias due to the use of these residuals. A simulation study checks the validity of the presented approaches in practice. The broad applicability of the procedures is demonstrated on a real data set.     

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.     

Comparison between the indirect approach and kriging with samples of different support for estimation using samples of different length

In this paper we compare two estimation methods to deal with samples of different support: (1) the indirect approach using accumulation and (2) kriging with samples of different support. These two methods were tested in a simple example. The estimates of the two methods were compared against a benchmark scenario. The benchmark consisted of kriging using a complete set of samples on the same support. The effects of the nugget effect, variogram range and type on the weight of long samples, the estimate, and the error variance were assessed. Kriging with samples of different support led to lower error variance and to estimates closer to the estimates of the benchmark scenario. Furthermore, in the case of spatially continuous attributes (low nugget effect), the indirect approach assigns greater weight to long samples than kriging with samples of different support. A cross validation study comparing the two methods with a database from a bauxite deposit was performed. The results of the cross validation study showed that kriging with samples of different support resulted in more precise estimates.     

Comparison of ordinary kriging and artificial neural network for spatial mapping of arsenic contamination of groundwater

In this technical note, we investigate the hypothesis that ‘non-linearity matters in the spatial mapping of complex patterns of groundwater arsenic contamination’. The spatial mapping pertained to data-driven techniques of spatial interpolation based on sampling data at finite locations. Using the well known example of extensive groundwater contamination by arsenic in Bangladesh, we find that the use of a highly non-linear pattern learning technique in the form of an artificial neural network (ANN) can yield more accurate results under the same set of constraints when compared to the ordinary kriging method. One ANN and a variogram model were used to represent the spatial structure of arsenic contamination for the whole country. The probability for successful detection of a well as safe or unsafe was found to be atleast 15% larger than that by kriging under the country-wide scenario. The probability of false hopes, which is a serious issue in public health monitoring was found to be significantly lower (by more than 10%) than that by kriging.     

Moving window kriging with geographically weighted variograms

This study adds to our ability to predict the unknown by empirically assessing the performance of a novel geostatistical-nonparametric hybrid technique to provide accurate predictions of the value of an attribute together with locally-relevant measures of prediction confidence, at point locations for a single realisation spatial process. The nonstationary variogram technique employed generalises a moving window kriging (MWK) model where classic variogram (CV) estimators are replaced with information-rich, geographically weighted variogram (GWV) estimators. The GWVs are constructed using kernel smoothing. The resultant and novel MWK–GWV model is compared with a standard MWK model (MWK–CV), a standard nonlinear model (Box–Cox kriging, BCK) and a standard linear model (simple kriging, SK), using four example datasets. Exploratory local analyses suggest that each dataset may benefit from a MWK application. This expectation was broadly confirmed once the models were applied. Model performance results indicate much promise in the MWK–GWV model. Situations where a MWK model is preferred to a BCK model and where a MWK–GWV model is preferred to a MWK–CV model are discussed with respect to model performance, parameterisation and complexity; and with respect to sample scale, information and heterogeneity.     

Exploring a valid model for the variogram of an isotropic spatial process

The variogram is one of the most important tools in the assessment of spatial variability and a crucial parameter for kriging. It is widely known that an estimator for the variogram cannot be used as its representator in some contexts because of its lack of conditional semi negative definiteness. Consequently, once the variogram is estimated, a valid family must be chosen to fit an appropriate model. Under isotropy, this selection is carried out by eye from the observation of the variogram estimated curve. In this paper, a statistical methodology is proposed to explore a valid model for the variogram. The statistic for this approach is based on quadratic forms depending on smoothed random variables which gather the underlying spatial variation. The distribution of the test statistic is approximated by a shifted chi-square distribution. A simulation study is also carried out to check the power and size of the test. Reference bands, as a complementary graphical tool, are calculated. An example from the literature is used to illustrate the methodologies presented.     

An ArcGIS® Approach to Include Tectonic Structures in Point Data Regionalization

Point data derived from drilling logs must often be regionalized. However, aquifers may show discontinuous surface structures, such as the offset of an aquitard caused by tectonic faults. One main challenge has been to incorporate these structures into the regionalization process of point data. We combined ordinary kriging and inverse distance weighted (IDW) interpolation to account for neotectonic structures in the regionalization process. The study area chosen to test this approach is the largest porous aquifer in Austria. It consists of three basins formed by neotectonic events and delimited by steep faults with a vertical offset of the aquitard up to 70 m within very short distances. First, ordinary kriging was used to incorporate the characteristic spatial variability of the aquitard location by means of a variogram. The tectonic faults could be included into the regionalization process by using breaklines with buffer zones. All data points inside the buffer were deleted. Last, IDW was performed, resulting in an aquitard map representing the discontinuous surface structures. This approach enables one to account for such surfaces using the standard software package ArcGIS^®; therefore, it could be adopted in many practical applications.     

Improving Regional Seismic Event Location in China

—?In an effort to improve our ability to locate seismic events in China using only regional data, we have developed empirical propagation path corrections and applied such corrections using traditional location routines. Thus far, we have concentrated on corrections to observed P arrival times for crustal events using travel-time observations available from the USGS Earthquake Data Reports, the International Seismic Centre Bulletin, the preliminary International Data Center Reviewed Event Bulletin, and our own travel-time picks from regional data. Location ground truth for events used in this study ranges from 25?km for well-located teleseimic events, down to 2?km for nuclear explosions located using satellite imagery. We also use eight events for which depth is constrained using several waveform methods. We relocate events using the EvLoc algorithm from a region encompassing much of China (latitude 20°–55°N; longitude 65°–115°E). We observe that travel-time residuals exhibit a distance-dependent bias using IASPEI91 as our base model. To remedy this bias, we have developed a new 1-D model for China, which removes a significant portion of the distance bias. For individual stations having sufficient P-wave residual data, we produce a map of the regional travel-time residuals from all well-located teleseismic events. Residuals are used only if they are smaller than 10?s in absolute value and if the seismic event is located with accuracy better than 25?km. From the residual data, correction surfaces are constructed using modified Bayesian kriging. Modified Bayesian kriging offers us the advantage of providing well-behaved interpolants and their errors, but requires that we have adequate error estimates associated with the travel-time residuals from which they are constructed. For our P-wave residual error estimate, we use the sum of measurement and modeling errors, where measurement error is based on signal-to-noise ratios when available, and on the published catalog estimate otherwise. Our modeling error originates from the variance of travel-time residuals for our 1-D China model. We calculate propagation path correction surfaces for 74 stations in and around China, including six stations from the International Monitoring System. The statistical significance of each correction surface is evaluated using a cross-validation technique. We show relocation results for nuclear tests from the Balapan and Lop Nor test sites, and for earthquakes located using interferometric synthetic aperture radar. These examples show that the use of propagation path correction surfaces in regional relocations eliminates distance bias in the residual curves and significantly improves the accuracy and precision of seismic event locations.     

Declustering experimental variograms by global estimation with fourth order moments

The variogram is a key parameter for geostatistical estimation and simulation. Preferential sampling may bias the spatial structure and often leads to noisy and unreliable variograms. A novel technique is proposed to weight variogram pairs in order to compensate for preferential or clustered sampling . Weighting the variogram pairs by global kriging of the quadratic differences between the tail and head values gives each pair the appropriate weight, removes noise and minimizes artifacts in the experimental variogram. Moreover, variogram uncertainty could be computed by this technique. The required covariance between the pairs going into variogram calculation, is a fourth order covariance that must be calculated by second order moments. This introduces some circularity in the calculation whereby an initial variogram must be assumed before calculating how the pairs should be weighted for the experimental variogram. The methodology is assessed by synthetic and realistic examples. For synthetic example, a comparison between the traditional and declustered variograms shows that the declustered variograms are better estimates of the true underlying variograms. The realistic example also shows that the declustered sample variogram is closer to the true variogram.     

Compositional Bayesian indicator estimation

Indicator kriging is widely used for mapping spatial binary variables and for estimating the global and local spatial distributions of variables in geosciences. For continuous random variables, indicator kriging gives an estimate of the cumulative distribution function, for a given threshold, which is then the estimate of a probability. Like any other kriging procedure, indicator kriging provides an estimation variance that, although not often used in applications, should be taken into account as it assesses the uncertainty of the estimate. An alternative approach to indicator estimation is proposed in this paper. In this alternative approach the complete probability density function of the indicator estimate is evaluated. The procedure is described in a Bayesian framework, using a multivariate Gaussian likelihood and an a priori distribution which are both combined according to Bayes theorem in order to obtain a posterior distribution for the indicator estimate. From this posterior distribution, point estimates, interval estimates and uncertainty measures can be obtained. Among the point estimates, the median of the posterior distribution is the maximum entropy estimate because there is a fifty-fifty chance of the unknown value of the estimate being larger or smaller than the median; that is, there is maximum uncertainty in the choice between two alternatives. Thus in some sense, the latter is an indicator estimator, alternative to the kriging estimator, that includes its own uncertainty. On the other hand, the mode of the posterior distribution estimator, assuming a uniform prior, is coincidental with the simple kriging estimator. Additionally, because the indicator estimate can be considered as a two-part composition which domain of definition is the simplex, the method is extended to compositional Bayesian indicator estimation. Bayesian indicator estimation and compositional Bayesian indicator estimation are illustrated with an environmental case study in which the probability of the content of a geochemical element in soil being over a particular threshold is of interest. The computer codes and its user guides are public domain and freely available.     

Regional flood frequency analysis using residual kriging in physiographical space

In this article, an approach using residual kriging (RK) in physiographical space is proposed for regional flood frequency analysis. The physiographical space is constructed using physiographical/climatic characteristics of gauging basins by means of canonical correlation analysis (CCA). This approach is a modified version of the original method, based on ordinary kriging (OK). It is intended to handle effectively any possible spatial trends within the hydrological variables over the physiographical space. In this approach, the trend is first quantified and removed from the hydrological variable by a quadratic spatial regression. OK is therefore applied to the regression residual values. The final estimated value of a specific quantile at an ungauged station is the sum of the spatial regression estimate and the kriged residual. To evaluate the performance of the proposed method, a cross‐validation procedure is applied. Results of the proposed method indicate that RK in CCA physiographical space leads to more efficient estimates of regional flood quantiles when compared to the original approach and to a straightforward regression‐based estimator. Copyright © 2010 John Wiley & Sons, Ltd.     

Interpolations of groundwater table elevation in dissected uplands

The variable elevation of the groundwater table in the St. Louis area was estimated using multiple linear regression (MLR), ordinary kriging, and cokriging as part of a regional program seeking to assess liquefaction potential. Surface water features were used to determine the minimum water table for MLR and supplement the principal variables for ordinary kriging and cokriging. By evaluating the known depth to the water and the minimum water table elevation, the MLR analysis approximates the groundwater elevation for a contiguous hydrologic system. Ordinary kriging and cokriging estimate values in unsampled areas by calculating the spatial relationships between the unsampled and sampled locations. In this study, ordinary kriging did not incorporate topographic variations as an independent variable, while cokriging included topography as a supporting covariable. Cross validation suggests that cokriging provides a more reliable estimate at known data points with less uncertainty than the other methods. Profiles extending through the dissected uplands terrain suggest that: (1) the groundwater table generated by MLR mimics the ground surface and elicits a exaggerated interpolation of groundwater elevation; (2) the groundwater table estimated by ordinary kriging tends to ignore local topography and exhibits oversmoothing of the actual undulations in the water table; and (3) cokriging appears to give the realistic water surface, which rises and falls in proportion to the overlying topography. The authors concluded that cokriging provided the most realistic estimate of the groundwater surface, which is the key variable in assessing soil liquefaction potential in unconsolidated sediments.     

Sequential kriging and cokriging: Two powerful geostatistical approaches

A sequential linear estimator is developed in this study to progressively incorporate new or different spatial data sets into the estimation. It begins with a classical linear estimator (i.e., kriging or cokriging) to estimate means conditioned to a given observed data set. When an additional data set becomes available, the sequential estimator improves the previous estimate by using linearly weighted sums of differences between the new data set and previous estimates at sample locations. Like the classical linear estimator, the weights used in the sequential linear estimator are derived from a system of equations that contains covariances and cross-covariances between sample locations and the location where the estimate is to be made. However, the covariances and cross-covariances are conditioned upon the previous data sets. The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. It also provides a new way to incorporate additional information into a previous estimation.     

Mapping gamma radiation and its uncertainty from weathering products in a Tasmanian landscape with a proximal sensor and random forest kriging

The radionuclides of potassium (⁴⁰ K), uranium (²³⁸U) and thorium (²³²Th) emit from the land surface gamma radiation that is characteristic of the underlying rocks and the distribution of their weathering products in the landscape. We measured the radiation along widely separated transects using a mobile proximal sensor over a 10 000‐ha region of Tasmania. We supplemented the transect data with information from soil and geological maps and dense data from LandSat and SPOT imagery, a digital elevation model and terrain attributes on a grid at 30‐m intervals so as to map the radionuclides. We used a sequence of steps, starting with a spatial bootstrap and random forests to predict emissions across the study area and at sampling points excluded from the bootstrap samples. The predictions at the sampling points were compared with the observed values to obtain residuals, which were then used to krige them at all points on the 30 m grid. We combined the random forest and kriging predictions on the 30‐m grid to obtain our random forest kriging predictions. Repeating the procedure 100 times provided confidence limits on our results and predictions. The resulting maps of the radionuclides accord well with what we know of the soil, lithology and topography of the region from other sources. Alluvial deposits with large amounts of potassium extend from the foot slopes of the Great Western Tiers and along the flood plains of the Meander River, providing evidence of previous widespread weathering and deposition of the material. The fertile Red Ferrosols (roughly equivalent to Ferralic Nitisols in the World reference base (WRB) classification) on the extensive Tertiary basalt plateau emit little gamma radiation as a result of deep weathering and of potassium movement and accumulation down its talus slopes. The maps show the complexity of the region in terms of soil, lithology and terrain, and they show the merits of gamma radiometry for mapping and understanding the distribution of materials in such regions. Copyright © 2013 John Wiley & Sons, Ltd.     

Geostatistical analysis of ground‐survey elevation data to elucidate spatial and temporal river channel change

A digital elevation model (DEM) of a fluvial environment represented landform surface variability well and provided a medium for monitoring morphological change over time. Elevation was measured above an arbitrary datum using a ground‐based three‐dimensional tacheometric survey in two reaches of the River Nent, UK, in July 1998, October 1998 (after flood conditions) and June 1999. A detailed geostatistical analysis of the elevation data was used to model the spatial variation of elevation and to produce DEMs in each reach and for each survey period. Maps of the difference in elevation were produced and volumetric change was calculated for each reach and each survey period. The parameters of variogram models were used to describe the morphological character of each reach and to elucidate the linkages between process and the form of channel change operating at different spatial and temporal scales. The analysis of channel change on the River Nent shows the potential of geostatistics for investigating the magnitude and frequency of geomorphic work in other rivers. A flood modified the channel features, but low magnitude and high frequency flows rationalized the morphology. In spite of relatively small amounts of net flux the channel features changed as a consequence of the reworking of existing material. The blocking of chute entrances and redirection of the channel had a considerable effect on the behaviour of the channel. Such small changes suggested that the distributary system was sensitive to variation in sediment regime. Plots of the kriging variances against sampling intervals were used to quantify the temporal variation in sampling redundancy (ranging between ?11 per cent and +93 per cent). These curves illustrated the importance of bespoke sampling designs to reduce sampling effort by incorporating anisotropic variation in space and geomorphic information on flow regime. Variation in the nugget parameter of the variogram models was interpreted as sampling inaccuracy caused by variability in particle size and is believed to be important for future work on surface roughness. Copyright © 2003 John Wiley & Sons, Ltd.     

Data‐Driven Approach for Analyzing Hydrogeology and Groundwater Quality Across Multiple Scales

Recent trends of assimilating water well records into statewide databases provide a new opportunity for evaluating spatial dynamics of groundwater quality and quantity. However, these datasets are scarcely rigorously analyzed to address larger scientific problems because they are of lower quality and massive. We develop an approach for utilizing well databases to analyze physical and geochemical aspects of groundwater systems, and apply it to a multiscale investigation of the sources and dynamics of chloride (Cl^?) in the near‐surface groundwater of the Lower Peninsula of Michigan. Nearly 500,000 static water levels (SWLs) were critically evaluated, extracted, and analyzed to delineate long‐term, average groundwater flow patterns using a nonstationary kriging technique at the basin‐scale (i.e., across the entire peninsula). Two regions identified as major basin‐scale discharge zones—the Michigan and Saginaw Lowlands—were further analyzed with regional‐ and local‐scale SWL models. Groundwater valleys (“discharge” zones) and mounds (“recharge” zones) were identified for all models, and the proportions of wells with elevated Cl^? concentrations in each zone were calculated, visualized, and compared. Concentrations in discharge zones, where groundwater is expected to flow primarily upwards, are consistently and significantly higher than those in recharge zones. A synoptic sampling campaign in the Michigan Lowlands revealed concentrations generally increase with depth, a trend noted in previous studies of the Saginaw Lowlands. These strong, consistent SWL and Cl^? distribution patterns across multiple scales suggest that a deep source (i.e., Michigan brines) is the primary cause for the elevated chloride concentrations observed in discharge areas across the peninsula.