Effect of elevation resolution on evapotranspiration simulations using MODFLOW

Surface elevations represented in MODFLOW head-dependent packages are usually derived from digital elevation models (DEMs) that are available at much high resolution. Conventional grid refinement techniques to simulate the model at DEM resolution increases computational time, input file size, and in many cases are not feasible for regional applications. This research aims at utilizing the increasingly available high resolution DEMs for effective simulation of evapotranspiration (ET) in MODFLOW as an alternative to grid refinement techniques. The source code of the evapotranspiration package is modified by considering for a fixed MODFLOW grid resolution and for different DEM resolutions, the effect of variability in elevation data on ET estimates. Piezometric head at each DEM cell location is corrected by considering the gradient along row and column directions. Applicability of the research is tested for the lower Rio Grande (LRG) Basin in southern New Mexico. The DEM at 10 m resolution is aggregated to resampled DEM grid resolutions which are integer multiples of MODFLOW grid resolution. Cumulative outflows and ET rates are compared at different coarse resolution grids. Results of the analysis conclude that variability in depth-to-groundwater within the MODFLOW cell is a major contributing parameter to ET outflows in shallow groundwater regions. DEM aggregation methods for the LRG Basin have resulted in decreased volumetric outflow due to the formation of a smoothing error, which lowered the position of water table to a level below the extinction depth.     

Hydrodynamic modelling of mesoscale eddies in the Black Sea

A three dimensional structure of mesoscale circulation in the Black Sea is simulated using the Proudman Oceanographic Laboratory Coastal Ocean Modelling System. A number of sensitivity tests reveal the response of the model to changes in the horizontal resolution, time steps, and diffusion coefficients. Three numerical grids are examined with x-fine (3.2 km), fine (6.7 km) and coarse (25 km) resolution. It is found that the coarse grid significantly overestimates the energy of the currents and is not adequate even for the study of basin-scale circulation. The x-fine grid, on the other hand, does not give significant advantages compared to the fine grid, and the latter is used for the bulk of simulations. The most adequate parameters are chosen from the sensitivity study and used to model both the basin-scale circulation and day-to-day variability of mesoscale currents for the months of May and June of 2000. The model is forced with actual wind data every 6 h and monthly climatic data for evaporation, precipitation, heat fluxes and river run-off. The results of the fine grid model are compared favourably against the satellite imagery. The model adequately reproduces the general circulation and many mesoscale features including cyclonic and anticyclonic eddies, jets and filaments in different parts of the Black Sea. The model gives a realistic geographical distribution and parameters of mesoscale currents, such as size, shape and evolution of the eddies.     

On the effects of triangulated terrain resolution on distributed hydrologic model response

Distributed hydrologic models based on triangulated irregular networks (TIN) provide a means for computational efficiency in small to large‐scale watershed modelling through an adaptive, multiple resolution representation of complex basin topography. Despite previous research with TIN‐based hydrology models, the effect of triangulated terrain resolution on basin hydrologic response has received surprisingly little attention. Evaluating the impact of adaptive gridding on hydrologic response is important for determining the level of detail required in a terrain model. In this study, we address the spatial sensitivity of the TIN‐based Real‐time Integrated Basin Simulator (tRIBS) in order to assess the variability in the basin‐averaged and distributed hydrologic response (water balance, runoff mechanisms, surface saturation, groundwater dynamics) with respect to changes in topographic resolution. Prior to hydrologic simulations, we describe the generation of TIN models that effectively capture topographic and hydrographic variability from grid digital elevation models. In addition, we discuss the sampling methods and performance metrics utilized in the spatial aggregation of triangulated terrain models. For a 64 km² catchment in northeastern Oklahoma, we conduct a multiple resolution validation experiment by utilizing the tRIBS model over a wide range of spatial aggregation levels. Hydrologic performance is assessed as a function of the terrain resolution, with the variability in basin response attributed to variations in the coupled surface–subsurface dynamics. In particular, resolving the near‐stream, variable source area is found to be a key determinant of model behaviour as it controls the dynamic saturation pattern and its effect on rainfall partitioning. A relationship between the hydrologic sensitivity to resolution and the spatial aggregation of terrain attributes is presented as an effective means for selecting the model resolution. Finally, the study highlights the important effects of terrain resolution on distributed hydrologic model response and provides insight into the multiple resolution calibration and validation of TIN‐based hydrology models. Copyright © 2005 John Wiley & Sons, Ltd.     

Impact of model development,calibration and validation decisions on hydrological simulations in West Lake Erie Basin

Watershed simulation models are used extensively to investigate hydrologic processes, landuse and climate change impacts, pollutant load assessments and best management practices (BMPs). Developing, calibrating and validating these models require a number of critical decisions that will influence the ability of the model to represent real world conditions. Understanding how these decisions influence model performance is crucial, especially when making science‐based policy decisions. This study used the Soil and Water Assessment Tool (SWAT) model in West Lake Erie Basin (WLEB) to examine the influence of several of these decisions on hydrological processes and streamflow simulations. Specifically, this study addressed the following objectives (1) demonstrate the importance of considering intra‐watershed processes during model development, (2) compare and evaluated spatial calibration versus calibration at outlet and (3) evaluate parameter transfers across temporal and spatial scales. A coarser resolution (HUC‐12) model and a finer resolution model (NHDPlus model) were used to support the objectives. Results showed that knowledge of watershed characteristics and intra‐watershed processes are critical to produced accurate and realistic hydrologic simulations. The spatial calibration strategy produced better results compared to outlet calibration strategy and provided more confidence. Transferring parameter values across spatial scales (i.e. from coarser resolution model to finer resolution model) needs additional fine tuning to produce realistic results. Transferring parameters across temporal scales (i.e. from monthly to yearly and daily time‐steps) performed well with a similar spatial resolution model. Furthermore, this study shows that relying solely on quantitative statistics without considering additional information can produce good but unrealistic simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

An investigation of numerical grid effects in parameter estimation

Modern ground water characterization and remediation projects routinely require calibration and inverse analysis of large three-dimensional numerical models of complex hydrogeological systems. Hydrogeologic complexity can be prompted by various aquifer characteristics including complicated spatial hydrostratigraphy and aquifer recharge from infiltration through an unsaturated zone. To keep the numerical models computationally efficient, compromises are frequently made in the model development, particularly, about resolution of the computational grid and numerical representation of the governing flow equation. The compromise is required so that the model can be used in calibration, parameter estimation, performance assessment, and analysis of sensitivity and uncertainty in model predictions. However, grid properties and resolution as well as applied computational schemes can have large effects on forward-model predictions and on inverse parameter estimates. We investigate these effects for a series of one- and two-dimensional synthetic cases representing saturated and variably saturated flow problems. We show that "conformable" grids, despite neglecting terms in the numerical formulation, can lead to accurate solutions of problems with complex hydrostratigraphy. Our analysis also demonstrates that, despite slower computer run times and higher memory requirements for a given problem size, the control volume finite-element method showed an advantage over finite-difference techniques in accuracy of parameter estimation for a given grid resolution for most of the test problems.     

Eddy energetics in the Faroe-Shetland channel: a model resolution study

Observations indicate that, perhaps as a result of intense shears produced by the topographic steering of the slope current and merging of the warm North Atlantic water with cold Nordic waters, the Faroe-Shetland Channel is a fertile ground for energetic eddies and meanders. How these meso-scale ( 10 km) features are resolved in a three-dimensional numerical model is studied here with grid-nesting, in which the convergence of the simulated energetics is examined as a function of grid sizes Δ =10, 10/3, 10/5 and 10/7 km. Large-scale forcing to the fine-grid nest consists of steady slope-current transport (2 Sv) and north/south climatological density difference only. The mean fields were found to converge at Δ = 10/3 km, but the eddy fields required Δ = 2 km or less. At the finest grids, meso-scale features more in conformity with those observed were found: meanders with amplitudes of O(10 km) and along-slope propagation speeds of about 0.14 m s- 1 cross-channel vacillation of the slope current, with amplitude of about 10 km and periods of 5–10 days, and vertical excursions of isotherms of O(100 m). The meanders were highly ageostrophic (ζ/f> 1, ζ=relative vorticity, f = Coriolis parameter), and developed as a result of conditions favorable to flow instability originated from south of the channel. When adequately resolved, the meander energetics are comparable to those produced by (large-scale) winds, which suggests that a high-resolution grid (Δ < 2 km) will be required for hindcast and forecast studies.     

Model structure and uncertainty for stochastic non-point source modelling applications

Abstract

The uncertainty associated with a rainfall–runoff and non-point source loading (NPS) model can be attributed to both the parameterization and model structure. An interesting implication of the areal nature of NPS models is the direct relationship between model structure (i.e. sub-watershed size) and sample size for the parameterization of spatial data. The approach of this research is to find structural limitations in scale for the use of the conceptual NPS model, then examine the scales at which suitable stochastic depictions of key parameter sets can be generated. The overlapping regions are optimal (and possibly the only suitable regions) for conducting meaningful stochastic analysis with a given NPS model. Previous work has sought to find optimal scales for deterministic analysis (where, in fact, calibration can be adjusted to compensate for sub-optimal scale selection); however, analysis of stochastic suitability and uncertainty associated with both the conceptual model and the parameter set, as presented here, is novel; as is the strategy of delineating a watershed based on the uncertainty distribution. The results of this paper demonstrate a narrow range of acceptable model structure for stochastic analysis in the chosen NPS model. In the case examined, the uncertainties associated with parameterization and parameter sensitivity are shown to be outweighed in significance by those resulting from structural and conceptual decisions.

Citation Parker, G. T. Rennie, C. D. & Droste, R. L. (2011) Model structure and uncertainty for stochastic non-point source modelling applications. Hydrol. Sci. J. 56(5), 870–882.     

A two-stage sensitivity analysis for parameter identification and calibration of a physically-based distributed model in a river basin

Hydrological models demand large numbers of input parameters, which are to be optimally identified for better simulation of various hydrological processes. Identifying the most relevant parameters and their values using efficient sensitivity analysis methods helps to better understand model performance. In this study, the physically-based distributed model SHETRAN is used for hydrological simulation on the Netravathi River Basin in south India and the most important parameters are identified using the Morris screening method. Further, the influence of a particular model parameter on streamflow is quantified using local sensitivity analysis and optimal parameters are obtained for calibration of the SHETRAN model. The results demonstrate the capability of two-stage sensitivity analysis, combining qualitative and quantitative methods in the initial screening-out of insignificant model parameters, identifying parameter interactions and quantifying the contribution of each model parameter to the streamflow. The results of the sensitivity analysis simplified the calibration procedure of SHETRAN for the study area.     

Spatial Quantification of Groundwater Abstraction in the Irrigated Indus Basin

Groundwater abstraction and depletion were assessed at a 1‐km resolution in the irrigated areas of the Indus Basin using remotely sensed evapotranspiration (ET) and precipitation; a process‐based hydrological model and spatial information on canal water supplies. A calibrated Soil and Water Assessment Tool (SWAT) model was used to derive total annual irrigation applied in the irrigated areas of the basin during the year 2007. The SWAT model was parameterized by station corrected precipitation data (R) from the Tropical Rainfall Monitoring Mission, land use, soil type, and outlet locations. The model was calibrated using a new approach based on spatially distributed ET fields derived from different satellite sensors. The calibration results were satisfactory and strong improvements were obtained in the Nash‐Sutcliffe criterion (0.52 to 0.93), bias (?17.3% to ?0.4%), and the Pearson correlation coefficient (0.78 to 0.93). Satellite information on R and ET was then combined with model results of surface runoff, drainage, and percolation to derive groundwater abstraction and depletion at a nominal resolution of 1 km. It was estimated that in 2007, 68 km³ (262 mm) of groundwater was abstracted in the Indus Basin while 31 km³ (121 mm) was depleted. The mean error was 41 mm/year and 62 mm/year at 50% and 70% probability of exceedance, respectively. Pakistani and Indian Punjab and Haryana were the most vulnerable areas to groundwater depletion and strong measures are required to maintain aquifer sustainability.     

Evaluating the effect of scale in flood inundation modelling in urban environments

Cellular‐based approaches for flood inundation modelling have been extensively calibrated and evaluated for the prediction of flood flows on rural river reaches. However, there has only been limited application of these approaches to urban environments, where the need for flood management is greatest. Practical application of two‐dimensional (2D) flood inundation models is often limited by computation time and processing power on standard desktop PCs when attempting to resolve flows on the high‐resolution grids necessary to replicate urban features. Consequently, it is necessary to evaluate the effectiveness of coarse grids to represent flood flows through urban environments. To examine these effects, LISFLOOD‐FP, a 2D storage cell model, is applied to hypothetical flooding scenarios in Greenfields, Glasgow. Grid resampling techniques in GIS software packages are evaluated and a bilinear gridding technique appears to provide the most accurate and physically intuitive results. A gridding method maintaining sharp elevation changes at building interfaces and neighbouring land is presented and estimates of the discretization noise associated with the coarse resolution grids suggest little improvement over current gridding methods. The variation in model results from the friction sensitivity analysis suggests a non‐stationary response to Manning's n with changing model resolution. Model results suggests that a coarse resolution model for urban applications is limited by the representation of urban media in coarse model grids. Furthermore, critical length scales related to building dimensions and building separation distances exist in urban areas that determine maximum possible grid resolutions for hydraulic models of urban flooding. Copyright ©, 2008 John Wiley & Sons, Ltd.     

Prediction uncertainty of conceptual rainfall-runoff models caused by problems in identifying model parameters and structure

Abstract

The uncertainties arising from the problem of identifying a representative model structure and model parameters in a conceptual rainfall-runoff model were investigated. A conceptual model, the HBV model, was applied to the mountainous Brugga basin (39.9 km”) in the Black Forest, southwestern Germany. In a first step, a Monte Carlo procedure with randomly generated parameter sets was used for calibration. For a ten-year calibration period, different parameter sets resulted in an equally good correspondence between observed and simulated runoff. A few parameters were well defined (i.e. best parameter values were within small ranges), but for most parameters good simulations were found with values varying over wide ranges. In a second step, model variants with different numbers of elevation and landuse zones and various runoff generation conceptualizations were tested. In some cases, representation of more spatial variability gave better simulations in terms of discharge. However, good results could be obtained with different and even unrealistic concepts. The computation of design floods and low flow predictions illustrated that the parameter uncertainty and the uncertainty of identifying a unique best model variant have implications for model predictions. The flow predictions varied considerably. The peak discharge of a flood with a probability of 0.01 year^?1, for instance, varied from 40 to almost 60 mm day^?1. It was concluded that model predictions, particularly in applied studies, should be given as ranges rather than as single values.     

The NAFE’06 data set: Towards soil moisture retrieval at intermediate resolution

The National Airborne Field Experiment 2006 (NAFE’06) was conducted during a three week period of November 2006 in the Murrumbidgee River catchment, located in southeastern Australia. One objective of NAFE’06 was to explore the suitability of the area for SMOS (Soil Moisture and Ocean Salinity) calibration/validation and develop downscaling and assimilation techniques for when SMOS does come on line. Airborne L-band brightness temperature was mapped at 1 km resolution 11 times (every 1–3 days) over a 40 by 55 km area in the Yanco region and 3 times over a 40 by 50 km area that includes Kyeamba Creek catchment. Moreover, multi-resolution, multi-angle and multi-spectral airborne data including surface temperature, surface reflectance (green, read and near infrared), lidar data and aerial photos were acquired over selected areas to develop downscaling algorithms and test multi-angle and multi-spectral retrieval approaches. The near-surface soil moisture was measured extensively on the ground in eight sampling areas concurrently with aircraft flights, and the soil moisture profile was continuously monitored at 41 sites. Preliminary analyses indicate that (i) the uncertainty of a single ground measurement was typically less than 5% vol. (ii) the spatial variability of ground measurements at 1 km resolution was up to 10% vol. and (iii) the validation of 1 km resolution L-band data is facilitated by selecting pixels with a spatial soil moisture variability lower than the point-scale uncertainty. The sensitivity of passive microwave and thermal data is also compared at 1 km resolution to illustrate the multi-spectral synergy for soil moisture monitoring at improved accuracy and resolution. The data described in this paper are available at www.nafe.unimelb.edu.au.     

一种基于温度历史和瞬态地温场模拟的剥露历史反演方法

根据低温热年代学数据,提取岩石从深部剥露到地表的信息,对理解诸多地质问题(如造山带演化、地表过程及其相互作用等)具有重要意义.本文提出一种基于岩石温度历史(可利用古温标、热年代计等方法制约),并考虑剥露过程对地温场扰动的剥露历史反演计算方法.基于假定的与真实数据的正反演模拟和参数敏感性分析表明:热扩散率的变化对剥蚀量计算影响不大,在常规岩石热扩散率变化范围内(20～35km2/Ma),总剥蚀量变化小于10%;传统计算方法低估了剥蚀总量,对于现今地温梯度小于20℃/km、冷却速率大于2～3℃/Ma,或现今地温梯度大于30℃/km、冷却速率大于5～10℃/Ma的地区,需要考虑热平流对剥蚀量计算的影响;匀速冷却的热历史指示剥蚀速率持续减小,而非匀速剥蚀.本文将该方法应用到龙门山南段和四川盆地,反演计算显示龙门山南段15Ma以来的剥蚀总量为8km,四川盆地中部80Ma以来剥蚀总量约为3km、东部约5km.     

华北及周边地区上地幔P波速度结构

本文基于层析成像方法研究了华北及其邻区(30°N~43°N,100°E~130°E)的上地幔P波速度结构.该区域是我国地震活动强烈的地区之一,曾多次发生破坏性地震,地质构造十分复杂.本研究采用中国数字化地震台网1996~2002年和中国科学院地质与地球物理所（IGGCAS）流动地震台阵2001~2003年地震记录中的P波走时,挑选了6870个地震事件,其中包括1382个区域地震事件,5488个远震事件.在对实际资料反演之前,进行了分辨率测试以确定反演结果的可靠性,对反演中不同参数对结果的影响做了分析,以确定最佳的的模型和参数.根据上述分析,以水平方向1°×1°,垂直方向以5 km、15 km、30 km为间隔划分网格,并建立初始速度模型,采用伪弯曲射线追踪方法计算走时和射线路径,通过基于LSQR算法的反演重建了研究区域40 km以下、480 km以上的三维P波速度结构.成像结果显示：（A）研究区域西南部的四川盆地北部在40 km以上为明显的低速异常,40 km以下一直到360 km均为高速异常;（B）华北盆地在90 km之上高速异常比较明显,而在120 km~360 km的深度,低速异常为区域主要特征,它与渤海湾地区的低速区连接,形成地幔楔,随着深度的增加,由于太平洋俯冲板块的影响,华北盆地和渤海湾地区下的高速异常突出.     

Enhanced identification of a hydrologic model using streamflow and satellite water storage data: A multicriteria sensitivity analysis and optimization approach

Hydrologic model development and calibration have continued in most cases to focus only on accurately reproducing streamflows. However, complex models, for example, the so‐called physically based models, possess large degrees of freedom that, if not constrained properly, may lead to poor model performance when used for prediction. We argue that constraining a model to represent streamflow, which is an integrated resultant of many factors across the watershed, is necessary but by no means sufficient to develop a high‐fidelity model. To address this problem, we develop a framework to utilize the Gravity Recovery and Climate Experiment's (GRACE) total water storage anomaly data as a supplement to streamflows for model calibration, in a multiobjective setting. The VARS method (Variogram Analysis of Response Surfaces) for global sensitivity analysis is used to understand the model behaviour with respect to streamflow and GRACE data, and the BORG multiobjective optimization method is applied for model calibration. Two subbasins of the Saskatchewan River Basin in Western Canada are used as a case study. Results show that the developed framework is superior to the conventional approach of calibration only to streamflows, even when multiple streamflow‐based error functions are simultaneously minimized. It is shown that a range of (possibly false) system trajectories in state variable space can lead to similar (acceptable) model responses. This observation has significant implications for land‐surface and hydrologic model development and, if not addressed properly, may undermine the credibility of the model in prediction. The framework effectively constrains the model behaviour (by constraining posterior parameter space) and results in more credible representation of hydrology across the watershed.     

Transient Recharge Estimability Through Field‐Scale Groundwater Model Calibration

The estimation of recharge through groundwater model calibration is hampered by the nonuniqueness of recharge and aquifer parameter values. It has been shown recently that the estimability of spatially distributed recharge through calibration of steady‐state models for practical situations (i.e., real‐world, field‐scale aquifer settings) is limited by the need for excessive amounts of hydraulic‐parameter and groundwater‐level data. However, the extent to which temporal recharge variability can be informed through transient model calibration, which involves larger water‐level datasets, but requires the additional consideration of storage parameters, is presently unknown for practical situations. In this study, time‐varying recharge estimates, inferred through calibration of a field‐scale highly parameterized groundwater model, are systematically investigated subject to changes in (1) the degree to which hydraulic parameters including hydraulic conductivity (K) and specific yield (S_y) are constrained, (2) the number of water‐level calibration targets, and (3) the temporal resolution (up to monthly time steps) at which recharge is estimated. The analysis involves the use of a synthetic reality (a reference model) based on a groundwater model of Uley South Basin, South Australia. Identifiability statistics are used to evaluate the ability of recharge and hydraulic parameters to be estimated uniquely. Results show that reasonable estimates of monthly recharge (<30% recharge root‐mean‐squared error) require a considerable amount of transient water‐level data, and that the spatial distribution of K is known. Joint estimation of recharge, S_y and K, however, precludes reasonable inference of recharge and hydraulic parameter values. We conclude that the estimation of temporal recharge variability through calibration may be impractical for real‐world settings.     

A two-step sensitivity analysis for hydrological signatures in Jinhua River Basin,East China

Parameter calibration and sensitivity analysis (SA) are usually not straightforward tasks for distributed hydrological models, owing to the complexity of models and the large number of parameters. A two-step SA approach is proposed for analysing hydrological signatures based on the distributed hydrology–soil–vegetation model (DHSVM) in the Jinhua River Basin, East China. A preliminary SA is conducted to obtain influential parameters via analysis of variance. These parameters are further analysed through a variance-based global sensitivity analysis method to achieve robust rankings and parameter contributions. Parallel computing is designed to reduce the computational burden. The results reveal that only a few parameters are significantly sensitive and that interactions between parameters cannot be ignored. When analysing hydrological signatures, it is found that water yield is simulated very well for most samples. Small and medium floods are simulated very well, while slight underestimations happen for large floods.     

Sensitivity of regionalized transfer-function noise models to the input and parameter transfer method / Sensibilité de modèles de type fonction de transfert bruit régionalisée (FTBR) aux données d'entrée et aux méthodes de transfert de paramètres

Abstract

Monthly shallow water-table depths can be predicted from climate forcings on a large scale by the combination of a transfer-function noise (TFN) model with a parameter regionalization scheme. To investigate the sensitivity of regionalized transfer-function noise (RTFN) models, simulations with different input series and parameter regionalization methods, combined with topographic external drifts, are presented. The input series include the precipitation, precipitation surplus or infiltration calculated by the variable infiltration capacity (VIC) land surface model, and the regionalization methods include those based on a combination of the principal component analysis (PCA) classification or Gaussian mixture model (GMM) clustering, with the external drifts, such as the elevation or terrain slope, generated from digital elevation models (DEM). Verification and cross-validation show that the infiltration series reduces the errors for regionalization by 12.5–18.8%, but cannot improve the calibration. The PCA method outperforms its alternative GMM method for the input series and the external drifts. The combination of the infiltration series and the regionalization method based on PCA classification and elevation produces the best results with respect to the mean absolute error and root mean squared error. The spatial and temporal variations of water-table depths at a macro-scale in continental China are predicted by the combination.     

Grid resolution study of ground water flow and transport

Three-dimensional grids representing a heterogeneous, ground water system are generated at 10 different resolutions in support of a site-scale flow and transport modeling effort. These grids represent hydrostratigraphy near Yucca Mountain, Nevada, consisting of 18 stratigraphic units with contrasting fluid flow and transport properties. The grid generation method allows the stratigraphy to be modeled by numerical grids of different resolution so that comparison studies can be performed to test for grid quality and determine the resolution required to resolve geologic structure and physical processes such as fluid flow and solute transport. The process of generating numerical grids with appropriate property distributions from geologic conceptual models is automated, thus making the entire process easy to implement with fewer user-induced errors. The series of grids of various resolutions are used to assess the level at which increasing resolution no longer influences the flow and solute transport results. Grid resolution is found to be a critical issue for ground water flow and solute transport. The resolution required in a particular instance is a function of the feature size of the model, the intrinsic properties of materials, the specific physics of the problem, and boundary conditions. The asymptotic nature of results related to flow and transport indicate that for a hydrologic model of the heterogeneous hydrostratigraphy under Yucca Mountain, a horizontal grid spacing of 600 m and vertical grid spacing of 40 m resolve the hydrostratigraphic model with sufficient precision to accurately model the hypothetical flow and solute transport to within 5% of the value that would be obtained with much higher resolution.