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.     

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.     

Assessing the relationships between elevation and extreme precipitation with various durations in southern Taiwan using spatial regression models

A spatially autocorrelated effect exists in precipitation of a mountainous basin. This study examines the relationship between maximum annual rainfall and elevation in the Kaoping River Basin of southern Taiwan using spatial regression models (i.e. geographically weighted regression (GWR), simultaneous autoregression (SAR), and conditional autoregression (CAR)). Results show that the GWR, SAR, and CAR models can improve spatial data fitting and provide an enhanced estimation for the rainfall–elevation relationship than the ordinary least squares approach. In particular, GWR achieves the most accurate estimation, and SAR and CAR achieve similar performance in terms of the Akaike information criterion. The relationship between extreme rainfall and elevation for longer duration is more concise than that for short durations. Results show that the spatial distribution of precipitation depends on elevation and that rainfall patterns in study area are heterogeneous between the southwestern plain and the eastern mountain area. Copyright © 2011 John Wiley & Sons, Ltd.     

A comparison of gauge and radar precipitation data for simulating an extreme hydrological event in the Severn Uplands,UK

This paper provides a comparison of gauge and radar precipitation data sources during an extreme hydrological event. November–December 2006 was selected as a time period of intense rainfall and large river flows for the Severn Uplands, an upland catchment in the United Kingdom. A comparison between gauge and radar precipitation time‐series records for the event indicated discrepancies between data sources, particularly in areas of higher elevation. The HEC‐HMS rainfall‐runoff model was selected to assess the accuracy of the precipitation to simulate river flows for the extreme event. Gauge, radar and gauge‐corrected radar rainfall were used as model inputs. Universal cokriging was used to geostatistically interpolate gauge data with radar and elevation data as covariates. This interpolated layer was used to calculate the mean‐field bias and correct the radar composites. Results indicated that gauge‐ and gauge‐corrected radar‐driven models replicated flows adequately for the extreme event. Gauge‐corrected flow predictions produced an increase in flow prediction accuracy when compared with the raw radar, yet predictions were comparative in accuracy to those using the interpolated gauge network. Subsequent investigations suggested this was due to an adequate spatial and temporal resolution of the precipitation gauge network within the Severn Uplands. Results suggested that the six rain gauges could adequately represent precipitation variability of the Severn Uplands to predict flows at an approximately equal accuracy to that obtained by radar. Temporally, radar produced an increase in flow prediction accuracy in mountainous reaches once the gauge time step was in excessive of an hourly interval. Copyright © 2010 John Wiley & Sons, Ltd.     

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

ABSTRACT

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

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

Suitability of a lumped rainfall–runoff model for flashy tropical watersheds in New Caledonia

ABSTRACT

The GR4H lumped hourly rainfall–runoff model was assessed for its integration in a ridge-to-reef modelling framework. Particular attention was paid to rainfall representation, robustness of parameter estimates and ability to reproduce the main runoff features. The study was conducted in four tropical mountainous watersheds in New Caledonia, which are exposed to intense rainfall events, large annual climatic variations triggered by El Niño oscillation, and wildfires. The inverse distance and elevation weighting algorithm outperformed other classical rainfall interpolation methods under data-limited conditions. The time span of data needed for robust calibration was site specific and varied from 6–7 years to 10 years, which may be linked to El Niño events and to wildfires. With sufficient data, simulation quality was equivalent during the calibration and validation periods. The GR4H model was generally able to simulate both flash floods and large annual variations. The model was more reliable when simulating wet years and watersheds not subject to land-cover changes.     

Nonlinear estimation of spatial distribution of rainfall — An indicator cokriging approach

Indicator cokriging (Journel 1983) is examined as a tool for real-time estimation of rainfall from rain gage measurements. The approach proposed in this work obviates real-time estimation of real-time statistics of rainfall by using ensemble or climatological statistics exclusively, and reduces computational requirements attendant to indicator cokriging by employing only a few auxiliary cutoffs in estimation of conditional probabilities. Due to unavailability of suitable rain gage measurements, hourly radar rain fall data were used for both indicator covariance estimation and a comparative evaluation. Preliminary results suggest that the indicator cokriging approach is clearly superior to its ordinary kriging counterpart, whereas the indicator kriging approach is not. The improvement is most significant in estimation of light rainfall, but drops off significantly for heavy rainfall. The lack of predictability in spatial estimation of heavy rainfall is borne out in the integral scale of indicator correlation: peaking to its maximum for cutoffs near the median, indicator correlation scale becomes increasingly smaller for larger cutoffs of rainfall depth. A derived-distribution analysis, based on the assumption that radar rainfall is a linear sum of ground-truth and a random error, suggests that, at low cutoffs, indicator correlation scale of ground-truth can significantly differ from that of radar rainfall, and points toward inclusion of rainfall intermittency, for example, within the framework proposed in this work.     

The relationship between altitude of meteorological stations and average monthly and annual precipitation

The aim of this study was to prove that altitudinal variability of average monthly and annual precipitation is better summarised when the altitude observed within a radius of several kilometres around a meteorological station is taken into consideration, instead of the altitude of the station itself. The use of the variable Z′, which combines the altitude of the closest mountain with its distance from the station, is compared against the use of altitude alone in simple linear and multiple quadratic regression equations for the altitudinal interpolation of precipitation over Greece. The data-set comprised precipitation observations from 516 meteorological stations. The comparison between the two variables is discussed on the basis of the resulting determination coefficients (R²) and standard errors of estimate (S). For all seasons, except summer, it was found that the variable Z′ improves the predictive ability of the regression equations, thus showing its potential for further use in interpolation procedures.     

A comparision of interpolation methods for producing digital elevation models at the field scale

Digital elevation models have been used in many applications since they came into use in the late 1950s. It is an essential tool for applications that are concerned with the Earth's surface such as hydrology, geology, cartography, geomorphology, engineering applications, landscape architecture and so on. However, there are some differences in assessing the accuracy of digital elevation models for specific applications. Different applications require different levels of accuracy from digital elevation models. In this study, the magnitudes and spatial patterning of elevation errors were therefore examined, using different interpolation methods. Measurements were performed with theodolite and levelling. Previous research has demonstrated the effects of interpolation methods and the nature of errors in digital elevation models obtained with indirect survey methods for small‐scale areas. The purpose of this study was therefore to investigate the size and spatial patterning of errors in digital elevation models obtained with direct survey methods for large‐scale areas, comparing Inverse Distance Weighting, Radial Basis Functions and Kriging interpolation methods to generate digital elevation models. The study is important because it shows how the accuracy of the digital elevation model is related to data density and the interpolation algorithm used. Cross validation, split‐sample and jack‐knifing validation methods were used to evaluate the errors. Global and local spatial auto‐correlation indices were then used to examine the error clustering. Finally, slope and curvature parameters of the area were modelled depending on the error residuals using ordinary least regression analyses. In this case, the best results were obtained using the thin plate spline algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Bilinear surface smoothing for spatial interpolation with optional incorporation of an explanatory variable. Part 2: Application to synthesized and rainfall data

ABSTRACT

The non-parametric mathematical framework of bilinear surface smoothing (BSS) methodology provides flexible means for spatial (two dimensional) interpolation of variables. As presented in a companion paper, interpolation is accomplished by means of fitting consecutive bilinear surface into a regression model with known break points and adjustable smoothing terms defined by means of angles formed by those bilinear surface. Additionally, the second version of the methodology (BSSE) incorporates, in an objective manner, the influence of an explanatory variable available at a considerably denser dataset. In the present study, both versions are explored and illustrated using both synthesized and real world (hydrological) data, and practical aspects of their application are discussed. Also, comparison and validation against the results of commonly used spatial interpolation methods (inverse distance weighted, spline, ordinary kriging and ordinary cokriging) are performed in the context of the real world application. In every case, the method’s efficiency to perform interpolation between data points that are interrelated in a complicated manner was confirmed. Especially during the validation procedure presented in the real world case study, BSSE yielded very good results, outperforming those of the other interpolation methods. Given the simplicity of the approach, the proposed mathematical framework’s overall performance is quite satisfactory, indicating its applicability for diverse tasks of scientific and engineering hydrology and beyond.

Editor Z. W. Kundzewicz; Associate editor A. Carsteanu     

Improving interpolation of daily precipitation for hydrologic modelling: spatial patterns of preferred interpolators

Detailed hydrologic models require high‐resolution spatial and temporal data. This study aims at improving the spatial interpolation of daily precipitation for hydrologic models. Different parameterizations of (1) inverse distance weighted (IDW) interpolation and (2) A local weighted regression (LWR) method in which elevation is the explanatory variable and distance, elevation difference and aspect difference are weighting factors, were tested at a hilly setting in the eastern Mediterranean, using 16 years of daily data. The preferred IDW interpolation was better than the preferred LWR scheme in 27 out of 31 validation gauges (VGs) according to a criteria aimed at minimizing the absolute bias and the mean absolute error (MAE) of estimations. The choice of the IDW exponent was found to be more important than the choice of whether or not to use elevation as explanatory data in most cases. The rank of preferred interpolators in a specific VG was found to be a stable local characteristic if a sufficient number of rainy days are averaged. A spatial pattern of the preferred IDW exponents was revealed. Large exponents (3) were more effective closer to the coast line whereas small exponents (1) were more effective closer to the mountain crest. This spatial variability is consistent with previous studies that showed smaller correlation distances of daily precipitation closer to the Mediterranean coast than at the hills, attributed mainly to relatively warm sea‐surface temperature resulting in more cellular convection coastward. These results suggest that spatially variable, physically based parameterization of the distance weighting function can improve the spatial interpolation of daily precipitation. Copyright © 2009 John Wiley & Sons, Ltd.     

Topographic accuracy assessment of bare earth lidar-derived unstructured meshes

This study is focused on the integration of bare earth lidar (Light Detection and Ranging) data into unstructured (triangular) finite element meshes and the implications on simulating storm surge inundation using a shallow water equations model. A methodology is developed to compute root mean square error (RMSE) and the 95th percentile of vertical elevation errors using four different interpolation methods (linear, inverse distance weighted, natural neighbor, and cell averaging) to resample bare earth lidar and lidar-derived digital elevation models (DEMs) onto unstructured meshes at different resolutions. The results are consolidated into a table of optimal interpolation methods that minimize the vertical elevation error of an unstructured mesh for a given mesh node density. The cell area averaging method performed most accurate when DEM grid cells within 0.25 times the ratio of local element size and DEM cell size were averaged. The methodology is applied to simulate inundation extent and maximum water levels in southern Mississippi due to Hurricane Katrina, which illustrates that local changes in topography such as adjusting element size and interpolation method drastically alter simulated storm surge locally and non-locally. The methods and results presented have utility and implications to any modeling application that uses bare earth lidar.     

Spatial interpolation of snow water equivalency using surface observations and remotely sensed images of snow‐covered area

As demand for water continues to escalate in the western Unites States, so does the need for accurate monitoring of the snowpack in mountainous areas. In this study, we describe a simple methodology for generating gridded‐estimates of snow water equivalency (SWE) using both surface observations of SWE and remotely sensed estimates of snow‐covered area (SCA). Multiple regression was used to quantify the relationship between physiographic variables (elevation, slope, aspect, clear‐sky solar radiation, etc.) and SWE as measured at a number of sites in a mountainous basin in south‐central Idaho (Big Wood River Basin). The elevation of the snowline, obtained from the SCA estimates, was used to constrain the predicted SWE values. The results from the analysis are encouraging and compare well to those found in previous studies, which often utilized more sophisticated spatial interpolation techniques. Cross‐validation results indicate that the spatial interpolation method produces accurate SWE estimates [mean R² = 0·82, mean mean absolute error (MAE) = 4·34 cm, mean root mean squared error (RMSE) = 5·29 cm]. The basin examined in this study is typical of many mid‐elevation mountainous basins throughout the western United States, in terms of the distribution of topographic variables, as well as the number and characteristics of sites at which the necessary ground data are available. Thus, there is high potential for this methodology to be successfully applied to other mountainous basins. Copyright © 2010 John Wiley & Sons, Ltd.     

Incorporating elevation in rainfall interpolation in Tunisia using geostatistical methods

Abstract

This paper compares the performance of three geostatistical algorithms, which integrate elevation as an auxiliary variable: kriging with external drift (KED); kriging combined with regression, called regression kriging (RK) or kriging after detrending; and co-kriging (CK). These three methods differ by the way by in which the secondary information is introduced into the prediction procedure. They are applied to improve the prediction of the monthly average rainfall observations measured at 106 climatic stations in Tunisia over an area of 164 150 km² using the elevation as the auxiliary variable. The experimental sample semivariograms, residual semivariograms and cross-variograms are constructed and fitted to estimate the rainfall levels and the estimation variance at the nodes of a square grid of 20 km?×?20 km resolution and to develop corresponding contour maps. Contour diagrams for KED and RK were similar and exhibited a pattern corresponding more closely to local topographic features when (a) the network is sparse and (b) the rainfall–elevation correlation is poor, while CK showed a smooth zonal pattern. Smaller prediction variances are obtained for the RK algorithm. The cross-validation showed that the RMSE obtained for CK gave better results than for KED or RK.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Feki, H., Slimani, M., and Cudennec, C., 2012. Incorporating elevation in rainfall interpolation in Tunisia using geostatistical methods. Hydrological Sciences Journal, 57 (7), 1294–1314.     

Correlation functions of rainfall field and their application in network design in the tropics

The study examines the correlation function of tropical monsoon rainfall on monthly, seasonal and annual time scales and obtains the relationship between this function and the distance. The area selected for study is Vidarbha with a fairly dense network of rain gauges. Vidarbha is a meteorological sub-division of the state of Maharashtra in India. Utilizing the relationship between the correlation function of the rainfall field and the distance, the errors of optimum interpolation of rainfall at a point have been computed by applying the method of optimum interpolation byGandin (1970). Relationships between the errors of interpolation and distance have been evaluated and from this the maximum spacing allowed between rain gauges for a specified tolerable error in interpolation has been estimated for each of the periods.     

Radar for hydrology: Unfulfilled promise or unrecognized potential?

Hydrology requires accurate and reliable rainfall input. Because of the strong spatial and temporal variability of precipitation, estimation of spatially distributed rain rates is challenging. Despite the fact that weather radars provide high-resolution (but indirect) observations of precipitation, they are not used in hydrological applications as extensively as one could expect. The goal of the present review paper is to investigate this question and to provide a clear view of the opportunities (e.g., for flash floods, urban hydrology, rainfall spatial extremes) the limitations (e.g., complicated error structure, need for adjustment) and the challenges for the use of weather radar in hydrology (i.e., validation studies, precipitation forecasting, mountainous precipitation, error propagation in hydrological models).     

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.     

Spatial prediction and modeling of soil salinity using simple cokriging,artificial neural networks,and support vector machines in El Outaya plain,Biskra, southeastern Algeria

Soil salinization is one of the most predominant environmental hazards responsible for agricultural land degradation, especially in the arid and semi-arid regions.An accurate spatial prediction and modeling of soil salinity in agricultural land are so important for farmers and decision-makers to develop the appropriate mechanisms to prevent the loss of fertile soil and increase crop production.El Outaya plain is marked by soil salinity increases due to the excessive use of poor groundwater quality for irrigation. This study aims to compare the performance of simple kriging, cokriging(SCOK), multilayer perceptron neural networks(MLP-NN), and support vector machines(SVM)in the prediction of topsoil and subsoil salinity. The field covariates including geochemical properties of irrigation groundwater and physical properties of soil and environmental covariates including digital elevation model and remote sensing derivatives were used as input candidates to SCOK, MLP-NN, and SVM. The optimal input combination was determined using multiple linear stepwise regression(MLSR). The results revealed that the SCOK using field covariates including water electrical conductivity(ECw) and sand percentage(sand %), and environmental covariates including land surface temperature(LST), topographic wetness index(TWI), and elevation could significantly increase the accuracy of soil salinity spatial prediction. The comparison of the prediction accuracy of the different modeling techniques using the Taylor diagram indicated that MLP-NN using LST, TWI, and elevation as inputs were more accurate in predicting the topsoil salinity [ECs(TS)] with a mean absolute error(MAE) of 0.43, root mean square error(RMSE) of 0.6 and correlation coefficient of 0.946. MLP-NN using ECw and sand % as inputs were more accurate in predicting the subsoil salinity [ECs(SS)] with MAE of 0.38, RMSE of0.6, and R of 0.968.