Kriging in estuaries: as the crow flies, or as the fish swims?

doi:10.1016/S0022-0981(97)00006-3

Journal of Experimental Marine Biology and Ecology

Volume 213, Issue 1, 1 June 1997, Pages 1-11

https://doi.org/10.1016/S0022-0981(97)00006-3 Get rights and content

Abstract

Geostatistical methods are becoming an essential tool for understanding the spatial distribution of biological and chemical species in estuaries. At the heart of these methods are the spatial prediction/mapping methods known as “kriging”; these can construct statistically optimal predictions for data at unobserved locations using a relatively small, spatially explicit sample. The prediction at any given location is a weighted average of the sample values, where the weights depend on the distances between the sample sites and the target location. For most geostatistical settings, distances are computed “as the crow flies”, i.e. Euclidean distance. For measurements made in estuarine streams, however, intuition suggests that distances between sites should be measured “as the fish swims”, i.e. the length of the shortest in-water path between two sites. Our study evaluated the relative accuracy of eight kriging methods for predicting contaminant and water quality variables measured in an urbanized estuary in South Carolina. The eight methods were defined by all combinations of three factors, each at two levels: (a) Distance metric (Euclidean vs. in-water); (b) semivariogram type (spherical vs. linear) and (c) model trend component (distance to the inlet mouth; without vs. with). For four of the eight variables studied, the in-water distances provided prediction accuracy improvement on the order of 10–30% of prediction error variance. In two of these cases, the improvement only occurred when in-water distances were used together with a model trend component. Choice of semivariogram did not have much effect on prediction accuracy. Although the overall improvement in prediction accuracy was unpredictable and modest, considering the additional difficulties associated with in-water distances, the results suggest that the integration of Geographic Information System (GIS)-based network analysis with kriging using in-water distances merits further research.

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Classical kriging models use Euclidean distance when modeling spatial autocorrelation. However for regions with irregular boundaries and holes, such as estuaries and coastlines, a measure of within-domain distance may capture a system’s proximity dependencies more accurately. Standard kriging techniques are not guaranteed to yield a valid covariance structure when defined in terms of non-Euclidean distances. In this paper, we develop a new kriging model for irregularly shaped domains. Our model uses an approximation to a diffusion process to define a valid covariance structure that reflects the domain topology. A covariance matrix is defined through the use of random walks on a lattice, process convolutions, and the kriging equations. A simulation study demonstrates that for commonly encountered topologies, our diffusion kriging estimator is superior to a kriging estimator based on shortest within-domain distance. We also illustrate our method using water quality data from Puget Sound and Lake Peipsi to map chlorophyll concentration.
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However, these distances were directly used in Kriging to perform prediction at unsampled locations, which can result in invalid covariance functions. Euclidean distance was also replaced by shortest-path distance in Little et al. (1997), Rathbun (1998), and Jensen et al. (2006) which can also lead to invalid models. This issue has been discussed in detail by Curriero (2006) and Ver Hoef (2018).
Crash data observed on a road network often exhibit spatial correlation due to unobserved effects with inherent spatial correlation following the structure of the road network. It is important to model this spatial correlation while accounting for the road network structure. In this study, we introduce the network process convolution (NPC) model. In this model, the spatial correlation among crash data is captured by a Gaussian Process (GP) approximated through a kernel convolution approach. The GP’s covariance function is based on path distance computed between a limited set of knots and crash data points on the road network. The proposed model offers a straightforward approach for predicting crash frequency at unobserved locations where covariates are available, and for interpolating the GP values anywhere on the network. Inference procedure is performed following the Bayesian paradigm and is implemented in R-INLA, which offers an estimation procedure that is very efficient compared to Markov Chain Monte Carlo sampling algorithms. We fitted our model to synthetic data and to crash data from Ottawa, Canada. We compared the proposed approach with a proper Conditional Autoregressive (pCAR) model, and with Poisson Regression (PR) and Negative Binomial (NB) models without latent effects. The results of the study indicated that although the pCAR model has comparable fitting performance, the NPC model outperforms pCAR when the main goal is to predict unobserved locations of interest. The proposed model also offers lower mean absolute error rates for cross validated crash counts, latent variable values, fixed-effect coefficients, as well as shorter interval scores for singletons. The NPC provides a natural way to account for the road network structure when considering the inclusion of spatially structured latent random effects in the modelling of crash data. It also offers an improved predictive capability for crash data on a road network.
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Numerous studies have indeed reported fish movements and variations of their habitat distribution in relation to tidal cycles, particularly for pelagic juveniles (Laffaille et al., 2001; Alp and Pichon, 2020; Martinho et al., 2008). Moreover most estuaries are also irregularly-shaped non-convex domains, and the use of euclidean distance may not be appropriate for the analysis of their spatial structures’ dependence (Rathbun, 1998; Little et al., 1997). The observation of spatial auto-correlation in estuarine domains yields therefore to several challenges and it is crucial to understand the physical nature of estuaries, and the environmental preferendum of the studied juvenile fishes to produce relevant abundance indices (Walmsley et al., 2018).
Estuaries play a fundamental role in the renewal of fisheries resources, as they hold nurseries for many juvenile fish species. Estimating juveniles’ abundance in estuaries is therefore key to improve stock assessment models, anticipate future recruitment and prevent crises related to biomass collapse. While geostatistical methods have been widely used in fisheries science to estimate species’ abundance during offshore scientific surveys, difficulties arise when using these methods in estuaries. Indeed, these ecosystems are characterized by their irregular and often non-convex morphology, their environmental gradients (salinity, depth), and their tidal dynamics which question the validity of the hypothesis of second-order stationarity, fundamental to the theory of intrinsic geostatistics. Therefore, we tested the performance of different geostatistical methods to account for the complexity of these ecosystems and quantify robust indices of abundance adapted to estuaries. We used density data of juvenile sea bass (Dicentrarchus labrax) sampled with demersal trawls in the Loire River collected over three consecutive years and tested a metric space for which the distance along the estuary is considered. We took into account the non-stationarity of densities with either a transitive approach or an intrinsic approach with spatio-temporal external drifts, which takes into account the effects of tides and environmental gradients. These geostatistical methods allowed us to produce densities distribution maps and had substantially greater predictive capabilities than the stratified random estimator (classical reference estimator). However, geostatistical methods consistently had larger CVs than the stratified random estimator because the latter ignores the spatio-temporal distribution of sampling points leading to uncertainties underestimates and hence overly optimistic confidence intervals. The use of geostatistically computed abundance indices in an assessment model appears to be a conservative approach, whose uncertainties would allow a more robust adjustment trade-off between different indices when estimating recruitment in estuaries.
Introducing a Kriging-based Gaussian Process approach in pedotransfer functions: Evaluation for the prediction of soil water retention with temperate and tropical datasets
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Citation Excerpt :
Kriging is a geostatistical interpolation method, originally developed for improved ore content prediction in mining (Krige, 1951; Matheron, 1963). Nowadays, it is an essential tool for spatial interpolation in domains such as geology, geotechnics (Soulie et al., 1990), precision agriculture (Wollenhaupt et al., 1994), and fisheries (Little et al., 1997). It has become an successful interpolation method for non-spatial variables as well, in particular as a metamodelling tool (Sacks et al., 1989; Simpson et al., 2001; Wang and Shan, 2007; Kleijnen, 2009).
Data mining algorithms such as Artificial Neural Networks (ANN) and k-Nearest Neighbour (kNN) have proven their merits in pedotransfer function modelling. Kriging is a well-known algorithm for spatial interpolation, but in this study it is proposed as an alternative data mining technique. It was compared to kNN as a benchmark pedotransfer function to predict soil water retention for a wide range of datasets, containing soil data from both temperate and tropical regions. The performance of both methods was compared through Monte Carlo cross-validation and the precision of the predictions was assessed with an ensemble procedure. Across all datasets, a significant improvement in prediction bias, accuracy and precision was found with Kriging, compared to kNN. Moreover, it was demonstrated how predictions with Kriging are more robust and insensitive to non-correlated predictor variables, and how the optimized hyperparameters provide additional insight in the training dataset properties. Kriging was found to be a accurate, precise and robust data mining solution for pedotransfer function modelling.
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The spatial variability evaluation of the water table of an aquifer provides useful information in water resources management plans. Geostatistical methods are often employed to map the free surface of an aquifer. In geostatistical analysis using Kriging techniques the selection of the optimal variogram is very important for the optimal method performance. This work compares three different criteria to assess the theoretical variogram that fits to the experimental one: the Least Squares Sum method, the Akaike Information Criterion and the Cressie’s Indicator. Moreover, variable distance metrics such as the Euclidean, Minkowski, Manhattan, Canberra and Bray-Curtis are applied to calculate the distance between the observation and the prediction points, that affects both the variogram calculation and the Kriging estimator. A Fuzzy Logic System is then applied to define the appropriate neighbors for each estimation point used in the Kriging algorithm. The two criteria used during the Fuzzy Logic process are the distance between observation and estimation points and the groundwater level value at each observation point. The proposed techniques are applied to a data set of 250 hydraulic head measurements distributed over an alluvial aquifer. The analysis showed that the Power-law variogram model and Manhattan distance metric within ordinary kriging provide the best results when the comprehensive geostatistical analysis process is applied. On the other hand, the Fuzzy Logic approach leads to a Gaussian variogram model and significantly improves the estimation performance. The two different variogram models can be explained in terms of a fractional Brownian motion approach and of aquifer behavior at local scale. Finally, maps of hydraulic head spatial variability and of predictions uncertainty are constructed for the area with the two different approaches comparing their advantages and drawbacks.
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Citation Excerpt :
In temporally dynamic and topographically fragmented coastal waters irregular changes hinder the use of temporal gap-filling methods based on temporally adjacent values, as they cannot follow irregular changes over long intervals. Spatial gap-filling such as interpolation techniques, in turn, are limited because of the anisotropic characteristics of coastal waters, due to barriers, currents and tides (Dunn and Ridgway, 2002; Little et al., 1997; Suominen et al., 2010). All the above-mentioned challenges are present in the northern Baltic Sea, where the coasts are in many cases bordered by archipelagos.
A topographically fragmental archipelago with dynamic waters set the preconditions for assessing coherent remotely sensed information. We generated a turbidity dataset for an archipelago coast in the Baltic Sea from MERIS data (FSG L1b), using CoastColour L1P, L2R and L2W processors. We excluded land and mixed pixels by masking the imagery with accurate (1:10 000) shoreline data. Using temporal linear averaging (TLA), we produced satellite-imagery datasets applicable to temporal composites for the summer seasons of three years. The turbidity assessments and temporally averaged data were compared to in situ observations obtained with coastal monitoring programs. The ability of TLA to estimate missing pixel values was further assessed by cross-validation with the leave-one-out method. The correspondence between L2W turbidity and in situ observations was good (r = 0.89), and even after applying TLA the correspondence remained acceptable (r = 0.78). The datasets revealed spatially divergent temporal water characteristics, which may be relevant to the management, design of monitoring and habitat models. Monitoring observations may be spatially biased if the temporal succession of water properties is not taken into account in coastal areas with anisotropic dispersion of waters and asynchronous annual cycles. Accordingly, areas of varying turbidity may offer a different habitat for aquatic biota than areas of static turbidity, even though they may appear similar if water properties are measured for short annual periods.

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Kriging in estuaries: as the crow flies, or as the fish swims?

Abstract

Spatial aspects of ecological data

Fitting variogram models by weighted least squares

J. Int. Assoc. Math. Geol.

On the stability of the geostatistical method

Math. Geol.

Use of geographic information processing for the identification of ‘indirect’ impacts associated with regulatory permitting programs: For now, a conceptual model

A Florida GIS for estuarine management

The role of Geographic Information Systems in managing Florida's coastal wetland resource

Basin-wide management: A remote sensing/GIS approach

Design and implementation of a coastal resource Geographic Information System: administrative considerations

Spatial models of ecological systems and processes: the role of GIS

Developing GIS data layers for estuarine resource management