A Linking Test that investigates the feasibility of inverse modelling: application to a simple rainfall interception model for Mt Gambier,southeast South Australia

Interception loss has an important influence on the water yield of forested areas. Nevertheless, in most studies stemflow is not measured, therefore the question of how to determine the feasibility of optimizing interception and stemflow parameters simultaneously by matching daily simulated throughfall to fortnightly measurements of cumulative throughfall is an important one. By applying a daily empirical interception model, a goodness fit of 2·2 mm/day is obtained between observed and simulated cumulative throughfall. However, by applying the simple but robust Linking Test, it was shown that the parameters are non‐unique and falsely linked, i.e. inter‐relationships between different vegetation parameter sets give similar throughfall but non‐unique net precipitation. The Linking Test investigates the causes of obtaining falsely linked parameters and shows that objective equifinality is not the source of the problem. Objective equifinality occurs when an inappropriate objective function is used. The Linking Test also shows that falsely linked parameters are not caused by measuring throughfall on a non‐daily basis (termed frequency sampling equifinality). By expanding the interception model to the second degree, it was found that the non‐uniqueness is due to the inherent nature of interception and stemflow functions that behave similarly and therefore can easily compensate each other (termed similarity equifinality). It is also shown that a simple daily empirical exponential interception model developed for conifers in the uplands of the United Kingdom is suitable to model interception in Pinus radiata plantations in the Mediterranean climate of southern Australia by using only daily gross precipitation data and two parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

Inverse modelling for freshwater lens in small islands: Kish Island,Persian Gulf

A number of challenges including instability, nonconvergence, nonuniqueness, nonoptimality, and lack of a general guideline for inverse modelling have limited the application of automatic calibration by generic inversion codes in solving the saltwater intrusion problem in real‐world cases. A systematic parameter selection procedure for the selection of a small number of independent parameters is applied to a real case of saltwater intrusion in a small island aquifer system in the semiarid region of the Persian Gulf. The methodology aims at reducing parameter nonuniqueness and uncertainty and the time spent on inverse modelling computations. Subsequent to the automatic calibration of the numerical model, uncertainty is analysed by constrained nonlinear optimization of the inverse model. The results define the percentage of uncertainty in the parameter estimation that will maintain the model inside a user‐defined neighbourhood of the best possible calibrated model. Sensitivity maps of both pressure and concentration for the small island aquifer system are also developed. These sensitivity maps indicate higher sensitivity of pressure to model parameters compared with concentration. These sensitivity maps serve as a benchmark for correlation analysis and also assist in the selection of observations points of pressure and concentration in the calibration process. Copyright © 2012 John Wiley & Sons, Ltd.     

Impact of the numbers of observations and calibration parameters on equifinality,model performance,and output and parameter uncertainty

The level of model complexity that can be effectively supported by available information has long been a subject of many studies in hydrologic modelling. In particular, distributed parameter models tend to be regarded as overparameterized because of numerous parameters used to describe spatially heterogeneous hydrologic processes. However, it is not clear how parameters and observations influence the degree of overparameterization, equifinality of parameter values, and uncertainty. This study investigated the impact of the numbers of observations and parameters on calibration quality including equifinality among calibrated parameter values, model performance, and output/parameter uncertainty using the Soil and Water Assessment Tool model. In the experiments, the number of observations was increased by expanding the calibration period or by including measurements made at inner points of a watershed. Similarly, additional calibration parameters were included in the order of their sensitivity. Then, unique sets of parameters were calibrated with the same objective function, optimization algorithm, and stopping criteria but different numbers of observations. The calibration quality was quantified with statistics calculated based on the ‘behavioural’ parameter sets, identified using 1% and 5% cut‐off thresholds in a generalized likelihood uncertainty estimation framework. The study demonstrated that equifinality, model performance, and output/parameter uncertainty were responsive to the numbers of observations and calibration parameters; however, the relationship between the numbers, equifinality, and uncertainty was not always conclusive. Model performance improved with increased numbers of calibration parameters and observations, and substantial equifinality did neither necessarily mean bad model performance nor large uncertainty in the model outputs and parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Glacial isostatic adjustment at the Laurentide ice sheet margin: Models and observations in the Great Lakes region

Glacial Isostatic Adjustment (GIA) modelling in North America relies on relative sea level information which is primarily obtained from areas far away from the uplift region. The lack of accurate geodetic observations in the Great Lakes region, which is located in the transition zone between uplift and subsidence due to the deglaciation of the Laurentide ice sheet, has prevented more detailed studies of this former margin of the ice sheet. Recently, observations of vertical crustal motion from improved GPS network solutions and combined tide gauge and satellite altimetry solutions have become available. This study compares these vertical motion observations with predictions obtained from 70 different GIA models. The ice sheet margin is distinct from the centre and far field of the uplift because the sensitivity of the GIA process towards Earth parameters such as mantle viscosity is very different. Specifically, the margin area is most sensitive to the uppermost mantle viscosity and allows for better constraints of this parameter. The 70 GIA models compared herein have different ice loading histories (ICE-3/4/5G) and Earth parameters including lateral heterogeneities. The root-mean-square differences between the 6 best models and the two sets of observations (tide gauge/altimetry and GPS) are 0.66 and 1.57 mm/yr, respectively. Both sets of independent observations are highly correlated and show a very similar fit to the models, which indicates their consistent quality. Therefore, both data sets can be considered as a means for constraining and assessing the quality of GIA models in the Great Lakes region and the former margin of the Laurentide ice sheet.     

Glacial isostatic adjustment at the Laurentide ice sheet margin: Models and observations in the Great Lakes region

Glacial Isostatic Adjustment (GIA) modelling in North America relies on relative sea level information which is primarily obtained from areas far away from the uplift region. The lack of accurate geodetic observations in the Great Lakes region, which is located in the transition zone between uplift and subsidence due to the deglaciation of the Laurentide ice sheet, has prevented more detailed studies of this former margin of the ice sheet. Recently, observations of vertical crustal motion from improved GPS network solutions and combined tide gauge and satellite altimetry solutions have become available. This study compares these vertical motion observations with predictions obtained from 70 different GIA models. The ice sheet margin is distinct from the centre and far field of the uplift because the sensitivity of the GIA process towards Earth parameters such as mantle viscosity is very different. Specifically, the margin area is most sensitive to the uppermost mantle viscosity and allows for better constraints of this parameter. The 70 GIA models compared herein have different ice loading histories (ICE-3/4/5G) and Earth parameters including lateral heterogeneities. The root-mean-square differences between the 6 best models and the two sets of observations (tide gauge/altimetry and GPS) are 0.66 and 1.57 mm/yr, respectively. Both sets of independent observations are highly correlated and show a very similar fit to the models, which indicates their consistent quality. Therefore, both data sets can be considered as a means for constraining and assessing the quality of GIA models in the Great Lakes region and the former margin of the Laurentide ice sheet.     

The scaling characteristics of soil parameters: From plot scale heterogeneity to subgrid parameterization

The variation in soil texture, surface moisture or vertical soil moisture gradient in larger scale atmospheric models may lead to significant variations in simulated surface fluxes of water and heat. The parameterization of soil moisture fluxes at spatial scales compatible with the grid size of distributed hydrological models and mesoscale atmospheric models ( 100 km²) faces principal problems which relate to the underlying microscopic or field scale heterogeneity in soil characteristics.

The most widely used parameterization in soil hydrology, the Darcy-Richards (DR) equation, is gaining increasing importance in mesoscale and climate modelling. This is mainly due to the need to introduce plant-interactive soil water depletion and stomatal conductance parameterizations and to improve the calculation of deep percolation and runoff. Covering a grid of several hundreds of square kilometres, the DR parameterization in soil-vegetation-atmosphere-transfer schemes (SVATs) is assumed to be scale-invariant. The parameters describing the non-linear, area-average soil hydraulic functions in this scale-invariant DR-equation should be treated as calibration-parameters, which do not necessarily have a physical meaning. The saturated hydraulic conductivity is one of the soil parameters to which the models show very high sensitivity. It is shown that saturated hydraulic conductivity can be scaled in both vertical and horizontal directions for large flow domains.

In this paper, a distinction is made between effective and aggregated soil parameters. Effective parameters are defined as area-average values or distributions over a domain with a single, distinct textural soil type. They can be obtained by scaling or inverse modelling. Aggregated soil parameters represent grid-domains with several textural soil types. In soil science dimensional methods have been developed to scale up soil hydraulic characteristics. With some specific assumptions, these techniques can be extrapolated from classical field-scale problems in soil heterogeneity to larger domains, compatible with the grid-size of large scale models. Particularly promising is the estimation of effective soil hydraulic parameters from area averaging measurements through inverse modelling of the unsaturated flow.

Techniques to scale and aggregate the soil characteristics presented in this paper qualify for direct or indirect use in large scale meteorological models. One of the interesting results is the effective behaviour of the reference curve, which can be obtained from similar media scaling. If the conclusions of this paper survive further studies, a relatively simple method will become available to parameterize soil variability at large scales. The inverse technique is found to provide effective soil parameters which perform well in predicting both the area-average evaporation and the area-average soil moisture fluxes, such as subsurface runoff. This is not the case for aggregated soil parameters. Obtained from regression relationships between soil textural composition and hydraulic characteristics, these aggregated parameters predict evaporation fluxes well, but fail to predict water balance terms such as percolation and runoff. This is a serious drawback which could eventually hamper the improvement of the representation of the hydrological cycle in mesoscale atmospheric models and in GCMs.     

Sensitivity analysis and uncertainty assessment in water budgets simulated by the variable infiltration capacity model for Canadian subarctic watersheds

In this study, we evaluate uncertainties propagated through different climate data sets in seasonal and annual hydrological simulations over 10 subarctic watersheds of northern Manitoba, Canada, using the variable infiltration capacity (VIC) model. Further, we perform a comprehensive sensitivity and uncertainty analysis of the VIC model using a robust and state-of-the-art approach. The VIC model simulations utilize the recently developed variogram analysis of response surfaces (VARS) technique that requires in this application more than 6,000 model simulations for a 30-year (1981–2010) study period. The method seeks parameter sensitivity, identifies influential parameters, and showcases streamflow sensitivity to parameter uncertainty at seasonal and annual timescales. Results suggest that the Ensemble VIC simulations match observed streamflow closest, whereas global reanalysis products yield high flows (0.5–3.0 mm day⁻¹) against observations and an overestimation (10–60%) in seasonal and annual water balance terms. VIC parameters exhibit seasonal importance in VARS, and the choice of input data and performance metrics substantially affect sensitivity analysis. Uncertainty propagation due to input forcing selection in each water balance term (i.e., total runoff, soil moisture, and evapotranspiration) is examined separately to show both time and space dimensionality in available forcing data at seasonal and annual timescales. Reliable input forcing, the most influential model parameters, and the uncertainty envelope in streamflow prediction are presented for the VIC model. These results, along with some specific recommendations, are expected to assist the broader VIC modelling community and other users of VARS and land surface schemes, to enhance their modelling applications.     

Towards a comprehensive approach to parameter estimation in land surface parameterization schemes

In climate models, the land–atmosphere interactions are described numerically by land surface parameterization (LSP) schemes. The continuing improvement in realism in these schemes comes at the expense of the need to specify a large number of parameters that are either directly measured or estimated. Also, an emerging problem is whether the relationships used in LSPs are universal and globally applicable. One plausible approach to evaluate this is to first minimize uncertainty in model parameters by calibration. In this paper, we conduct a comprehensive analysis of some model diagnostics using a slightly modified version of the Simple Biosphere 3 model for a variety of biomes located mainly in the Amazon. First, the degree of influence of each individual parameter in simulating surface fluxes is identified. Next, we estimate parameters using a multi‐operator genetic algorithm applied in a multi‐objective context and evaluate simulations of energy and carbon fluxes against observations. Compared with the default parameter sets, these parameter estimates improve the partitioning of energy fluxes in forest and cropland sites and provide better simulations of daytime increases in assimilation of net carbon during the dry season at forest sites. Finally, a detailed assessment of the parameter estimation problem was performed by accounting for the decomposition of the mean squared error to the total model uncertainty. Analysis of the total prediction uncertainty reveals that the parameter adjustments significantly improve reproduction of the mean and variability of the flux time series at all sites and generally remove seasonality of the errors but do not improve dynamical properties. Our results demonstrate that error decomposition provides a meaningful and intuitive way to understand differences in model performance. To make further advancements in the knowledge of these models, we encourage the LSP community to adopt similar approaches in the future. Copyright © 2012 John Wiley & Sons, Ltd.     

Constraining water table depth simulations in a land surface model using estimated baseflow

Several recent studies have shown the significance of representing groundwater in land surface hydrologic simulations. However, optimal methods for model parameter calibration in order to realistically simulate baseflow and groundwater depth have received little attention. Most studies still use globally constant groundwater parameters due to the lack of available datasets for calibration. Moreover, when models are calibrated, various parameter combinations are found to exhibit equifinality in simulated total runoff due to model parameter interactions. In this study, a simple lumped groundwater model is incorporated into the Community Land Model (CLM), in which the water table is interactively coupled to soil moisture through the groundwater recharge fluxes. The coupled model (CLMGW) is successfully validated in Illinois using a 22-year (1984–2005) monthly observational dataset. Baseflow estimates from the digital recursive filter technique are used to calibrate the CLMGW parameters. The advantage obtained from incorporating baseflow calibration in addition to traditional calibration based on measured streamflow alone is demonstrated by a Monte Carlo-type simulation analysis. Using the optimal parameter sets identified from baseflow calibration, flow partitioning and water table depth simulations using CLMGW are improved, and the equifinality problem is alleviated. For other regions that lack observations of water table depth, the baseflow calibration approach can be used to enhance parameter estimation and constrain water table depth simulations.     

Simultaneous determination of aquifer parameters and zone structures with fuzzy c-means clustering and meta-heuristic harmony search algorithm

This study proposes an inverse solution algorithm through which both the aquifer parameters and the zone structure of these parameters can be determined based on a given set of observations on piezometric heads. In the zone structure identification problem fuzzy c-means (FCM) clustering method is used. The association of the zone structure with the transmissivity distribution is accomplished through an optimization model. The meta-heuristic harmony search (HS) algorithm, which is conceptualized using the musical process of searching for a perfect state of harmony, is used as an optimization technique. The optimum parameter zone structure is identified based on three criteria which are the residual error, parameter uncertainty, and structure discrimination. A numerical example given in the literature is solved to demonstrate the performance of the proposed algorithm. Also, a sensitivity analysis is performed to test the performance of the HS algorithm for different sets of solution parameters. Results indicate that the proposed solution algorithm is an effective way in the simultaneous identification of aquifer parameters and their corresponding zone structures.     

Simultaneous identification of parameter,initial condition,and boundary condition in groundwater modelling

In this paper, we perform an inverse method to simultaneously estimate aquifer parameters, initial condition, and boundary conditions in groundwater modelling. The parameter estimation is extended to a complete inverse problem that makes the calibrated groundwater flow model more realistic. The adjoint state method, the gradient search method, and the least square error algorithm are combined to build the optimization procedure. Horizontal two‐dimensional groundwater flow in a confined aquifer is exemplified to demonstrate the correlation between unknowns, the contribution of observation, as well as the suitability of applying the inverse method. The correlation analysis shows the connection between storage coefficient and initial condition. Besides, transmissivity and boundary conditions are also highly correlated. More observations at different location and time are necessary to provide sufficient information. A time series of unsteady head is requested for estimation of storage coefficient and initial condition. Observation near boundary is very effective for boundary condition estimation. The observation at pumping well mostly contributes to the estimation of transmissivity. According to all observations, it is possible to identify parameters, initial condition, and boundary condition simultaneously. Furthermore, the results not only illustrate the traditional assumption of known boundary condition but also initial condition, which may cause an incorrect estimation. Copyright © 2009 John Wiley & Sons, Ltd.     

Sensitivity analysis of hydraulic conductivity and Manning's n parameters lead to new method to scale effective hydraulic conductivity across model resolutions

Hydrological modelling is an important tool for research, policy, and management, but uncertainty remains about parameters transferability from field observations made at small scale to models at the catchment scale and larger. This uncertainty compels the need to develop parameter relationships that are translatable across scale. In this study, we compare the changes to modelled processes as resolution is coarsened from 100‐m to 1‐km in a topographically complex, 255‐km² Colorado River headwater catchment. We conducted a sensitivity analysis for hydraulic conductivity (K) and Manning's n parameters across four orders of magnitude. Results showed that K acts as a moderator between surface and subsurface contributions to streamflow, whereas n moderates the duration of high intensity, infiltration‐excess flow. The parametric sensitivity analysis informed development of a new method to scale effective hydraulic conductivity across modelling resolutions in order to compensate for the loss of topographic gradients as resolution is coarsened. A similar mathematical relationship between n and lateral resolution changes was not found, possibly because n is also sensitive to time discretization. This research provides an approach to translate hydraulic conductivity parameters from a calibrated coarse model to higher resolutions where the number of simulations are limited by computational demand.     

Tidal and wind-driven surface currents in the German Bight: HFR observations versus model simulations

Tidal and wind-driven surface currents in the German Bight between shallow mudflats of the North Frisian islands and the island of Helgoland are studied using coastal high-frequency radar (HFR) observations and hindcasts from a primitive equation numerical model. The setup of the observational system is described, and estimates of expected measurement errors are given. A quantitative comparison of numerical model results and observations is performed. The dominant tidal components are extracted from the two data sources using tidal harmonic analysis and the corresponding tidal ellipses are defined. Results show that the spatial patterns of different tidal ellipse parameters are consistent in the two data sets. Model sensitivity studies with constant and variable salinity and temperature distributions are used to study density-related mechanisms of circulation. Furthermore, the role of the surface wind field in driving the German Bight circulation is investigated using the complex correlation between wind and surface current vectors. The observed change of the respective correlation patterns from the coastal to open ocean is shown to be due to a combination of density effects, the coastline and topography. The overall conclusion is that HFR observations resolve the small-scale and rapidly evolving characteristics of coastal currents well in the studied area and could present an important component for regional operational oceanography when combined with numerical modelling. Some unresolved issues associated with the complex circulation and large instability of circulation in front of the Elbe River Estuary justify further considerations of this area using dedicated surveys and modelling efforts.     

Coupling basin- and site-scale inverse models of the Española aquifer

Large-scale models are frequently used to estimate fluxes to small-scale models. The uncertainty associated with these flux estimates, however, is rarely addressed. We present a case study from the Espa?ola Basin, northern New Mexico, where we use a basin-scale model coupled with a high-resolution, nested site-scale model. Both models are three-dimensional and are analyzed by codes FEHM and PEST. Using constrained nonlinear optimization, we examine the effect of parameter uncertainty in the basin-scale model on the nonlinear confidence limits of predicted fluxes to the site-scale model. We find that some of the fluxes are very well constrained, while for others there is fairly large uncertainty. Site-scale transport simulation results, however, are relatively insensitive to the estimated uncertainty in the fluxes. We also compare parameter estimates obtained by the basin- and site-scale inverse models. Differences in the model grid resolution (scale of parameter estimation) result in differing delineation of hydrostratigraphic units, so the two models produce different estimates for some units. The effect is similar to the observed scale effect in medium properties owing to differences in tested volume. More important, estimation uncertainty of model parameters is quite different at the two scales. Overall, the basin inverse model resulted in significantly lower estimates of uncertainty, because of the larger calibration dataset available. This suggests that the basin-scale model contributes not only important boundary condition information but also improved parameter identification for some units. Our results demonstrate that caution is warranted when applying parameter estimates inferred from a large-scale model to small-scale simulations, and vice versa.     

A finite element sea ice model of the Canadian Arctic Archipelago

The Canadian Arctic Archipelago (CAA) is a complex area formed by narrow straits and islands in the Arctic. It is an important pathway for freshwater and sea-ice transport from the Arctic Ocean to the Labrador Sea and ultimately to the Atlantic Ocean. The narrow straits are often crudely represented in coupled sea-ice–ocean models, leading to a misrepresentation of transports through these straits. Unstructured meshes are an alternative in modelling this complex region, since they are able to capture the complex geometry of the CAA. This provides higher resolution in the flow field and allows for more accurate transports (but not necessarily better modelling). In this paper, a finite element sea-ice model of the Arctic region is described and used to estimate the sea-ice fluxes through the CAA. The model is a dynamic–thermodynamic sea-ice model with elastic–viscous–plastic rheology and is coupled to a slab ocean, where the temperature and salinity are restored to climatology, with no velocities and surface elevation. The model is spun-up from 1973 to 1978 with NCEP/NARR reanalysis data. From 1979 to 2007, the model is forced by NCEP/DoE reanalysis data. The large scale sea-ice characteristics show good agreement with observations. The total sea-ice area agrees very well with observations and shows a sensitivity to the Arctic oscillation (AO). For 1998–2002, we find estimates for the sea-ice volume and area fluxes through Admunsen Gulf, McClure Strait and the Queen Elizabeth Islands that compare well with observation and are slightly better than estimates from other models. For Nares Strait, we find that the fluxes are much lower than observed, due to the missing effect of topographic steering on the atmospheric forcing fields. The 1979–2007 fluxes show large seasonal and interannual variability driven primarily by variability in the ice velocity field and a sensitivity to the AO and other large-scale atmospheric variability, which suggests that accurate atmospheric forcing might be crucial to modelling the CAA.     

Inversion for seismic anisotropy using genetic algorithms1

A general inversion scheme based on a genetic algorithm is developed to invert seismic observations for anisotropic parameters. The technique is applied to the inversion of shear-wave observations from two azimuthal VSP data sets from the Conoco test site in Oklahoma. Horizontal polarizations and time-delays are inverted for hexagonal and orthorhombic symmetries. The model solutions are consistent with previous studies using trial and error matching of full waveform synthetics. The shear-wave splitting observations suggest the presence of a shear-wave line singularity and are consistent with a dipping fracture system which is known to exist at the test site. Application of the inversion scheme prior to full waveform modelling demonstrates that a considerable saving in time is possible whilst retaining the same degree of accuracy.     

Modern trends in the solution of forward and inverse 3D electromagnetic induction problems

The main results obtained during the last 5–8 yr in the solution of forward and inverse problems of 3D induction studies are summarized. The up-to-date status of 3D modelling is presented and prospective improvements in the formulation and numerical solution of forward problems are discussed. Approximate techniques and practical aspects of 3D modelling are specially considered. The general scheme of 3D interpretation of electromagnetic geophysical observations is outlined and realistic formalized approaches to solving 3D inverse problems, namely direct inversion and formalized model fitting, are studied.     

Multiple objective evaluation of a simple phosphorus transfer model

A simple phosphorus (P) transfer model of the Welland catchment, UK, is evaluated against multiple objective functions using a Monte Carlo approach that combines calibration, identifiability, sensitivity and uncertainty analysis. The model is based on simple conceptual rainfall‐runoff and river routing components, combined with estimates of the daily non‐point source load derived from annual landuse‐based export coefficients, disaggregated as a function of the runoff. The model has limited data requirements, consistent with data availability, and is parsimoneous with respect to the number of parameters identified through inverse modelling. The best performing parameter sets capture the main aspects of the observed flow and total P (TP) concentrations and provide a suitable basis for a decision‐support tool. However, a trade‐off is evident between matching the observed flow peaks, flow recessions and TP concentrations simultaneously, highlighting some limitations of the model structure and/or calibration data. Model analysis indicates that daily non‐point source load cannot be described as a function of near‐surface runoff and land use alone, but that other influences, including seasonality, are important. However, further model development to improve performance is likely to introduce additional complexity (in terms of parameter numbers), and hence additional problems of parameter identifiability and output uncertainty, which in turn raises issues of the information content of the available data. Copyright © 2004 John Wiley & Sons, Ltd.     

An iterative scheme for use in MT 2D forward modelling — to speed up production of the Jacobian and to improve accuracy of the inverse matrix

An important stage in two-dimensional magnetotelluric modelling is the calculation of the Earth's response functions for an assumed conductivity model and the calculation of the associated Jacobian relating those response functions to the model parameters. The efficiency of the calculation of the Jacobian will affect the efficiency of the inversion modelling. Rodi (1976) produced all the Jacobian elements by inverting a single matrix and using an approximate first-order algorithm. Since only one inverse matrix required calculation the procedure speeded up the inversion. An iterative scheme to improve the approximation to the Jacobian information is presented in this paper. While this scheme takes a little longer than Rodi's algorithm, it enables a more accurate determination of the Jacobian information. It is found that the Jacobian elements can be produced in 10% of the time required to calculate an inverse matrix or to calculate a 2D starting model. A modification of the algorithm can further be used to improve the accuracy of the original inverse matrix calculated in a 2D finite difference program and hence the solution this program produces. The convergence of the iteration scheme is found to be related both to the originally calculated inverse matrix and to the change in the newly formed matrix arising from perturbation of the model parameter. A ridge regression inverse algorithm is used in conjunction with the iterative scheme for forward modelling described in this paper to produce a 2D conductivity section from field data.     

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.