Performance of Gradient and Gradient-Free Optimizers in Transient Hydraulic Tomography

Sub-surface characterization in fractured aquifers is challenging due to the co-existence of contrasting materials namely matrix and fractures. Transient hydraulic tomography (THT) is proved to be an efficient and robust technique to estimate hydraulic (K_m, K_f) and storage (S_m, S_f) properties in such complex hydrogeologic settings. However, performance of THT is governed by data quality and optimization technique used in inversion. We assessed the performance of gradient and gradient-free optimizers with THT inversion. Laboratory experiments were performed on a two-dimensional, granite rock (80 cm × 45 cm × 5 cm) with known fracture pattern. Cross-hole pumping experiments were conducted at 10 ports (located on fractures), and time-drawdown responses were monitored at 25 ports (located on matrix and fractures). Pumping ports were ranked based on weighted signal-to-noise ratio (SNR) computed at each observation port. Noise-free, good quality (SNR > 100) datasets were inverted using Levenberg–Marquardt: LM (gradient) and Nelder–Mead: NM (gradient-free) methods. All simulations were performed using a coupled simulation-optimization model. Performance of the two optimizers is evaluated by comparing model predictions with observations made at two validation ports that were not used in simulation. Both LM and NM algorithms have broadly captured the preferential flow paths (fracture network) via K and S tomograms, however LM has outperformed NM during validation ( $$). Our results conclude that, while method of optimization has a trivial effect on model predictions, exclusion of low quality (SNR ≤ 100) datasets can significantly improve the model performance.     

Redundant and Nonredundant Information for Model Calibration or Hydraulic Tomography

Drawdown data from independent pumping tests have widely been used to validate the estimated hydraulic parameters from inverse modeling or hydraulic tomography (HT). Yet, the independent pumping test has not been clearly defined. Therefore, the goal of this paper is to define this independent pumping test concept, based on the redundant or nonredundant information about aquifer heterogeneity embedded in the observed heads during cross-hole pumping tests. The definition of complete, moderate redundancy and high nonredundancy of information are stipulated using cross-correlation analysis of the relationship between the head and heterogeneity. Afterward, data from numerical experiments and field sequential pumping test campaigns reinforce the concept and the definition.     

Inherent Limitations of Hydraulic Tomography

We offer a cautionary note in response to an increasing level of enthusiasm regarding high-resolution aquifer characterization with hydraulic tomography. We use synthetic examples based on two recent field experiments to demonstrate that a high degree of nonuniqueness remains in estimates of hydraulic parameter fields even when those estimates are based on simultaneous analysis of a number of carefully controlled hydraulic tests. We must, therefore, be careful not to oversell the technique to the community of practicing hydrogeologists, promising a degree of accuracy and resolution that, in many settings, will remain unattainable, regardless of the amount of effort invested in the field investigation. No practically feasible amount of hydraulic tomography data will ever remove the need to regularize or bias the inverse problem in some fashion in order to obtain a unique solution. Thus, along with improving the resolution of hydraulic tomography techniques, we must also strive to couple those techniques with procedures for experimental design and uncertainty assessment and with other more cost-effective field methods, such as geophysical surveying and, in unconsolidated formations, direct-push profiling, in order to develop methods for subsurface characterization with the resolution and accuracy needed for practical field applications.     

Comparison of Hydraulic Tomography with Traditional Methods at a Highly Heterogeneous Site

Over the past several decades, different groundwater modeling approaches of various complexities and data use have been developed. A recently developed approach for mapping hydraulic conductivity (K) and specific storage (S_s) heterogeneity is hydraulic tomography, the performance of which has not been compared to other more “traditional” methods that have been utilized over the past several decades. In this study, we compare seven methods of modeling heterogeneity which are (1) kriging, (2) effective parameter models, (3) transition probability/Markov Chain geostatistics models, (4) geological models, (5) stochastic inverse models conditioned to local K data, (6) hydraulic tomography, and (7) hydraulic tomography conditioned to local K data using data collected in five boreholes at a field site on the University of Waterloo (UW) campus, in Waterloo, Ontario, Canada. The performance of each heterogeneity model is first assessed during model calibration. In particular, the correspondence between simulated and observed drawdowns is assessed using the mean absolute error norm, (L₁), mean square error norm (L₂), and correlation coefficient (R) as well as through scatterplots. We also assess the various models on their ability to predict drawdown data not used in the calibration effort from nine pumping tests. Results reveal that hydraulic tomography is best able to reproduce these tests in terms of the smallest discrepancy and highest correlation between simulated and observed drawdowns. However, conditioning of hydraulic tomography results with permeameter K data caused a slight deterioration in accuracy of drawdown predictions which suggests that data integration may need to be conducted carefully.     

A New Randomized Binary Prior Model for Hydraulic Tomography in Fractured Aquifers

We present a novel pilot-point-based hydraulic tomography (HT) inversion procedure to delineate preferential flow paths and estimate hydraulic properties in a fractured aquifer. Our procedure considers a binary prior model developed using a randomized algorithm. The randomized algorithm involves discretizing the domain into grid cells, assigning a binary label to each cell, traversing the grid randomly, and choosing the optimal grid configuration cell-by-cell. This binary prior model is used to guide the placement of pilot points and to constrain aquifer parameters during pilot-point-based HT inversion. A two-dimensional fractured granite rock block was considered to test our methodology under controlled laboratory conditions. Multiple pumping tests were conducted at selected ports and the pressure responses were monitored. The pumping datasets thus obtained were preprocessed using median filters to remove random noise, and then analyzed using the proposed procedure. The proposed binary prior algorithm was implemented in C++ by supplying the forward groundwater model, HydroGeoSphere (HGS). Pilot-point-assisted HT inversion was performed using the parameter-estimation tool, coupled to HGS. The resulting parameter distributions were assessed by: (1) a visual comparison of the K- and S_s-tomograms with the known topology of the fractures and (2) comparing model predictions with measurements made at two validation ports that were not used in calibration. The performance assessment revealed that HT with the proposed randomized binary prior could be used to recover fracture-connectivity and to predict drawdowns in fractured aquifers with reasonable accuracy, when compared to a conventional pilot-point inversion scheme.     

A Hidden Markov Model Based Tool for Geophysical Data Exploration

--Unsupervised learning techniques provide a way of investigating scientific data based on automated generation of statistical models. Because these techniques are not dependent on a priori information, they provide an unbiased method for separating data into distinct types. Thus they can be used as an objective method by which to identify data as belonging to previously known classes or to find previously unknown or rare classes and subclasses of data. Hidden Markov model based unsupervised learning methods are particularly applicable to geophysical systems because time relationships between classes, or states of the system, are included in the model. We have applied a modified version of hidden Markov models which employ a deterministic annealing technique to scientific analysis of seismicity and GPS data from the southern California region. Preliminary results indicate that the technique can isolate distinct classes of earthquakes from seismicity data.     

Aquifer Imaging with Oscillatory Hydraulic Tomography: Application at the Field Scale

Modeling and laboratory experiments have demonstrated the ability of oscillatory hydraulic tomography (OHT) to characterize heterogeneity in aquifer hydraulic properties. In OHT, a location is stressed via periodic pumping/injection at a set frequency, and the resulting head signal is measured at a number of monitoring locations. The source of oscillations is repeatedly moved, allowing tomographic imaging of aquifer properties. Changing the period of oscillation also results in observations with additional information. In theory, OHT is comparable to other hydraulic tomography methods in that distributed pressure change measurements provide characterization information. In practice, OHT has several benefits including: (1) little to no water injected into or extracted from the aquifer; and (2) an observational signal at a set period that can be easily extracted in the presence of noise. We report the first field application of OHT, carried out at the Boise Hydrogeophysical Research Site (BHRS) using an oscillating signal generator with a very small cycling volume of <2 L, and a period range of 5 to 70 s. For these tests, signals were detected at distances of over 15 m. After processing to extract periodic signal properties, we perform tomography using a frequency-domain numerical model for groundwater flow. In comparing results against prior characterization results from the BHRS, we find moderate to strong positive correlations between K profiles estimated via different methods at multiple wells, with moderate overall correlation between estimated three-dimensional (3D) K volumes.     

Field Study of Subsurface Heterogeneity with Steady‐State Hydraulic Tomography

Remediation of subsurface contamination requires an understanding of the contaminant (history, source location, plume extent and concentration, etc.), and, knowledge of the spatial distribution of hydraulic conductivity (K) that governs groundwater flow and solute transport. Many methods exist for characterizing K heterogeneity, but most if not all methods require the collection of a large number of small‐scale data and its interpolation. In this study, we conduct a hydraulic tomography survey at a highly heterogeneous glaciofluvial deposit at the North Campus Research Site (NCRS) located at the University of Waterloo, Waterloo, Ontario, Canada to sequentially interpret four pumping tests using the steady‐state form of the Sequential Successive Linear Estimator (SSLE) ( Yeh and Liu 2000 ). The resulting three‐dimensional (3D) K distribution (or K‐tomogram) is compared against: ( 1 ) K distributions obtained through the inverse modeling of individual pumping tests using SSLE, and ( 2 ) effective hydraulic conductivity (K_eff) estimates obtained by automatically calibrating a groundwater flow model while treating the medium to be homogeneous. Such a K_eff is often used for designing remediation operations, and thus is used as the basis for comparison with the K‐tomogram. Our results clearly show that hydraulic tomography is superior to the inversions of single pumping tests or K_eff estimates. This is particularly significant for contaminated sites where an accurate representation of the flow field is critical for simulating contaminant transport and injection of chemical and biological agents used for active remediation of contaminant source zones and plumes.     

A Potential-Based Inversion of Unconfined Steady-State Hydraulic Tomography

The importance of estimating spatially variable aquifer parameters such as transmissivity is widely recognized for studies in resource evaluation and contaminant transport. A useful approach for mapping such parameters is inverse modeling of data from series of pumping tests, that is, via hydraulic tomography. This inversion of field hydraulic tomographic data requires development of numerical forward models that can accurately represent test conditions while maintaining computational efficiency. One issue this presents is specification of boundary and initial conditions, whose location, type, and value may be poorly constrained. To circumvent this issue when modeling unconfined steady-state pumping tests, we present a strategy that analyzes field data using a potential difference method and that uses dipole pumping tests as the aquifer stimulation. By using our potential difference approach, which is similar to modeling drawdown in confined settings, we remove the need for specifying poorly known boundary condition values and natural source/sink terms within the problem domain. Dipole pumping tests are complementary to this strategy in that they can be more realistically modeled than single-well tests due to their conservative nature, quick achievement of steady state, and the insensitivity of near-field response to far-field boundary conditions. After developing the mathematical theory, our approach is first validated through a synthetic example. We then apply our method to the inversion of data from a field campaign at the Boise Hydrogeophysical Research Site. Results from inversion of nine pumping tests show expected geologic features, and uncertainty bounds indicate that hydraulic conductivity is well constrained within the central site area.     

论CT科技在工程项目中的某些应用

本文论述由工业ＣＴ发展到工程ＣＴ的科学思路与意义,在精密仪器无损检测的过程中,在检测技术的基础上发展了工业ＣＴ的技术分支,共所检测的对象,它们的尺寸在１ｍｍ～１０００ｍｍ之间,近年来,中国在建造房,桥梁与堤坝等水工建筑方面出现了许多质量问题,造成重大伤亡事故,已引起各界人士的重视,由电磁波。γ射线和超声波等ＣＴ技术能得以楼房,桥梁地基和墙体内部的精细的剖面图像,因而检查这些建筑物质量时,开始起到日     

Combining 3D Hydraulic Tomography with Tracer Tests for Improved Transport Characterization

Hydraulic tomography (HT) is a method for resolving the spatial distribution of hydraulic parameters to some extent, but many details important for solute transport usually remain unresolved. We present a methodology to improve solute transport predictions by combining data from HT with the breakthrough curve (BTC) of a single forced‐gradient tracer test. We estimated the three dimensional (3D) hydraulic‐conductivity field in an alluvial aquifer by inverting tomographic pumping tests performed at the Hydrogeological Research Site Lauswiesen close to Tübingen, Germany, using a regularized pilot‐point method. We compared the estimated parameter field to available profiles of hydraulic‐conductivity variations from direct‐push injection logging (DPIL), and validated the hydraulic‐conductivity field with hydraulic‐head measurements of tests not used in the inversion. After validation, spatially uniform parameters for dual‐domain transport were estimated by fitting tracer data collected during a forced‐gradient tracer test. The dual‐domain assumption was used to parameterize effects of the unresolved heterogeneity of the aquifer and deemed necessary to fit the shape of the BTC using reasonable parameter values. The estimated hydraulic‐conductivity field and transport parameters were subsequently used to successfully predict a second independent tracer test. Our work provides an efficient and practical approach to predict solute transport in heterogeneous aquifers without performing elaborate field tracer tests with a tomographic layout.     

Geophysical investigation of tumuli by means of surface 3D Electrical Resistivity Tomography

Tumuli are artificially erected small hills that cover monumental tombs or graves. In this work, the surface three-dimensional (3D) Electrical Resistivity Tomography (ERT) method, composed of dense parallel two-dimensional (2D) tomographies, was used to investigate the properties of the tumuli filling material and to resolve buried archaeological structures inside the tumuli.The effectiveness of the method was investigated by numerical modeling and through 3D inversion of synthetic apparent resistivity data. A resistivity model that simulates the inhomogeneous tumulus material and the tombs that are buried inside the tumulus was assumed. The Dipole–Dipole (DD), Pole–Dipole (PD), Pole–Pole (PP), Gradient (GRAD), Midpoint-Potential-Referred (MPR) and Schlumberger Reciprocal (SCR) arrays, which are suitable for multichannel resistivity instruments, were tested. The tumulus topography (pyramid or capsized cup) was incorporated into the inversion procedure through a distorted finite element mesh. The inversion procedure was based on a smoothness constrained Gauss–Newton algorithm in which the Active Constraint Balancing (ACB) method was also applied in order to enhance the least-squares resolving power and stability.Synthetic modeling showed that the different tumulus layers and the horizontal contact of the artificial tumulus material with the natural background soil were reconstructed by all of the tested electrode arrays. Generally, PD and the GRAD arrays comprise the optimum choices to investigate the subsurface properties of a tumulus and locate buried tombs. The MPR model was inferior to the GRAD model, while the DD, PP and SCR models had the poorest resolution. It was also shown that the inversion models are practically independent from the survey direction and the topography shape of the tumulus.The real field data collected employing the PD array along a small tumulus from the archaeological site of Vergina in northern Greece enhanced the synthetic modeling results. The inversion model outlined a number of archaeological structures that exhibit a high possibility to correlate with graves. Overall, this work signifies that the surface 3D ERT method can provide a valuable tool in the non-destructive archaeological exploration of tumuli.     

X线CT的性能评估

Ｘ线ＣＴ自问世以来，以飞快的快速向前发展。世界上各主要生产厂家不断地采用新技术，不断地推出新产品，不断地更新换代。目前我国引进的与国产的各种类型的Ｘ线ＣＴ已经达到一千数百台，而且这个数字还在不断地而且快速地增长。对于如何评价Ｘ线ＣＴ的性能指标无疑是非常重要的。在这篇文章中，作者根据自己的经验与体会对Ｘ线ＣＴ的性能如何评估作了扼要的阐述以供读者参考，特别是对那些正订购或即将订购Ｘ线ＣＴ的医院或许会有     

Hydraulic Tomography Offers Improved Imaging of Heterogeneity in Fractured Rocks

Fractured rocks have presented formidable challenges for accurately predicting groundwater flow and contaminant transport. This is mainly due to our difficulty in mapping the fracture‐rock matrix system, their hydraulic properties and connectivity at resolutions that are meaningful for groundwater modeling. Over the last several decades, considerable effort has gone into creating maps of subsurface heterogeneity in hydraulic conductivity (K) and specific storage (S_s) of fractured rocks. Developed methods include kriging, stochastic simulation, stochastic inverse modeling, and hydraulic tomography. In this article, I review the evolution of various heterogeneity mapping approaches and contend that hydraulic tomography, a recently developed aquifer characterization technique for unconsolidated deposits, is also a promising approach in yielding robust maps (or tomograms) of K and S_s heterogeneity for fractured rocks. While hydraulic tomography has recently been shown to be a robust technique, the resolution of the K and S_s tomograms mainly depends on the density of pumping and monitoring locations and the quality of data. The resolution will be improved through the development of new devices for higher density monitoring of pressure responses at discrete intervals in boreholes and potentially through the integration of other data from single‐hole tests, borehole flowmeter profiling, and tracer tests. Other data from temperature and geophysical surveys as well as geological investigations may improve the accuracy of the maps, but more research is needed. Technological advances will undoubtedly lead to more accurate maps. However, more effort should go into evaluating these maps so that one can gain more confidence in their reliability.     

Effect of a Starting Model on the Solution of a Travel Time Seismic Tomography Problem

In the problems of three-dimensional (3D) travel time seismic tomography where the data are travel times of diving waves and the starting model is a system of plane layers where the velocity is a function of depth alone, the solution turns out to strongly depend on the selection of the starting model. This is due to the fact that in the different starting models, the rays between the same points can intersect different layers, which makes the tomography problem fundamentally nonlinear. This effect is demonstrated by the model example. Based on the same example, it is shown how the starting model should be selected to ensure a solution close to the true velocity distribution. The starting model (the average dependence of the seismic velocity on depth) should be determined by the method of successive iterations at each step of which the horizontal velocity variations in the layers are determined by solving the two-dimensional tomography problem. An example illustrating the application of this technique to the P-wave travel time data in the region of the Black Sea basin is presented.     

基于三维层状介质模型地震层析成像正演网格研究

地震层析成像技术是研究地层速度结构的一个重要手段,该技术是利用震源到接收点的地震波走时或波形的观测数据,结合建立的数学物理模型来重建地层速度结构。而层析成像正演模拟的精度将直接影响反演出的速度与地层真实速度的拟合度,因此找出一种精度较高的层析成像正演模拟算法是非常重要的。本文针对三维层状介质,通过网格化模型,采用最小走时射线追踪方法展开层析成像正演研究,分别采用三角形网格、矩形网格和六边形网格进行模型参数化,并对比分析各自正演结果的精度,总结出针对三维层状介质模型的最佳网格划分方式。