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物探与化探, 2021, 45(1): 159-164 doi: 10.11720/wtyht.2021.1251

方法研究·信息处理·仪器研制

时移电阻率反演模拟研究

苏鹏,, 杨进

中国地质大学(北京) 地球物理与信息技术学院,北京 100083

Simulation study of inversion of time-lapse resistivity

SU Peng,, YANG Jin

School of Geophysics and Information Technique, China University of Geosciences, Beijing 100083, China

责任编辑: 沈效群

收稿日期: 2020-05-11   修回日期: 2020-11-20   网络出版日期: 2021-02-20

基金资助: 国家自然科学基金项目“深部多金属矿勘探中天然场激电法正反演计算和野外试验研究”.  41374133

Received: 2020-05-11   Revised: 2020-11-20   Online: 2021-02-20

作者简介 About authors

苏鹏(1989-),男,中国地质大学(北京)地球物理学专业在读博士,主要研究方向为地球物理电磁法正反演。Email:spvfly@126.com

摘要

电阻率法可用于地表浅层的探测,也可用于对动态地下目标进行监测。对于监测数据的反演,不同数据集的单独反演存在一定缺陷,为此本文在常规电阻率反演算法的基础上,推导了时移电阻率反演公式,实现了时移反演算法程序;为了论证时移反演算法对动态地下目标成像的优越性,建立一组多个正演模型,利用模拟数据进行单独反演和时移反演,并对比两种方法的结果。研究表明,尽管两种算法都能圈定动态地下目标体,但时移反演算法可以消除不同观测数据集中所包含的随机误差,减少假异常的出现。

关键词: 电阻率法 ; 时移反演 ; 数值模拟

Abstract

The resistivity method can be used for the detection of near ground surface, and can also be employed to monitor dynamic underground targets. For the inversion of monitoring data, the single inversion of different data sets has certain defects. For this reason, on the basis of the traditional resistivity inversion algorithm, the time-lapse resistivity inversion formula was derived and the time-lapse inversion program was realized; for the purpose of demonstrating the superiority of the time-lapse inversion algorithm for imaging of dynamic underground targets, a set of multiple forward models was established, and the simulation data were used for single inversion and time-lapse inversion. The results show that, although both algorithms can draw the dynamic underground targets, the time-lapse inversion algorithm can eliminate random errors contained in different observation data sets and reduce the occurrence of inversion artifacts.

Keywords: resistivity method ; time-lapse inversion ; numerical simulation

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本文引用格式

苏鹏, 杨进. 时移电阻率反演模拟研究. 物探与化探[J], 2021, 45(1): 159-164 doi:10.11720/wtyht.2021.1251

SU Peng, YANG Jin. Simulation study of inversion of time-lapse resistivity. Geophysical and Geochemical Exploration[J], 2021, 45(1): 159-164 doi:10.11720/wtyht.2021.1251

0 引言

电阻率法广泛应用于水文、工程、环境地质问题的探测工程中。通常,电阻率法用于对静态的地下电性结构进行成像,然而在环境监测、地质灾害监测及地下水污染监测中,由于地下介质处于动态变化的状态,如沿海地区地下水的海水入侵、山区滑坡岩移等,使用动态监测的方法更为有效。针对上述这些动态问题,需要通过不同时间的多次观测数据,利用时移电阻率法(time-lapse,ERT)来解决。目前,时移电阻率法正处于快速发展阶段,在水文、环境领域内的应用范围逐渐扩大,例如:利用时移监测方法研究渗流运移情况[1,2,3],研究水文地质参数变化规律[4,5]、溶质运移[6,7]及研究地下热传导情况[8,9,10];在国内,李飞等[11]利用时移高密度法研究地下煤层开采所引起的上覆岩层变形破坏,孙大利等[12]使用高密度时移监测分析堤坝隐患的时移特征。

时移电阻率反演算法基于常规2D/3D电阻率反演算法,但更加关注不同时刻观测数据差异以及相应地下电性结构模型参数的差异情况,旨在消除多个数据中的随机误差,凸显地下介质随时间变化而发生的真实变化。Daily等[13]反演初始数据集和后续数据集的比值以凸显地下电性变化;LaBrecque等[14]试图最小化初始数据集和后续数据集的数据差值及其与响应模型差值;Kim等[15]提出将数据集和模型参数放入时空域(4D)中进行离散,以进行全4D反演;Hayley等[16]将多个不同时间点的数据集和模型参数同时反演迭代;Karaoulis等[17]对全4D时移电阻率反演方法进行改进,将时间正则项设置为可变的,并对超参数调优方式加以改善;Loke等[18]利用光滑约束的最小二乘法结合L-curve的参数调优方法来提高4D反演速度。

本文对高密度电法监测数据时移反演做了初步研究,在对比分析上述不同时移算法的基础上,采用Karaoulis的思路推导并实现了时移反演算法;通过模拟数据对比了常规电阻率反演和时移反演的效果,展示了时移反演算法在对动态地下目标体的监测追踪中的优越性。

1 时移电阻率反演

1.1 电阻率法反演理论

对于常规的2D/3D电阻率法而言,反演是通过所采集数据获得可靠地下电性结构的过程。在电阻率反演中,地下电性结构为离散的模型参数电阻率,其反演问题是一个病态问题,故需要采取正则化的方法来避免反演的病态问题:

Φ(m)=Φd+βΦm

式中:Φd为数据拟合差;Φm为模型正则项;β为正则化参数,用以平衡两项的贡献。上式具体展开形式如下[19]:

Φ(m)=12Wddm-dobs2+β2Wmm-mref2

式中:Wd是一个数据加权矩阵,它是一个对角矩阵;d(m)为正演所得预测数据;dobs为观测数据;Wm为模型加权矩阵;m为迭代模型;mref为参考模型。

最小化目标函数: Φ(m)m=0,利用高斯牛顿法求其模型迭代公式,可得

(JTWTdWdJ+βWTmWm)δm=-{JTWTdWddm-dobs+βWTmWmm-mref

式中:J为雅可比矩阵,形式为Jij= di(m)mj

1.2 时移电阻率反演

相较于常规电阻率方法,时移电阻率法反演的目标函数中添加了一个时间约束项,即目标函数为:

Φ(m˙)=Φd˙+βΦm˙+αΦt

式中: Φd˙为多个数据集的数据拟合差,采用Tikhonov正则化; Φm˙为多个模型的空间模型正则项,采用光滑约束;Φt为时间模型正则项,采用平滑约束;αβ是时间约束和模型约束的正则化参数。

式(4)可展开为如下形式:

Φ(m˙)=12Wd˙[d˙-f(m˙)]2+β2Wm˙(m˙-m˙ref)2+α2Wtm˙2

式中: d˙是多个数据集的集合, d˙= d1dt; m˙是对应的不同模型, m˙= m1mt; Wm˙为二阶差分算子,对应于光滑约束,Wt为一阶差分算子,对应于平滑约束。

最小化目标函数: Φ(m˙)m˙=0,可得

(J˙TWTd˙Wd˙J˙+βWTm˙Wm˙+αWTtWt)δm=-J˙TWTd˙Wd˙Δd˙+βWTm˙Wm˙+αWTtWtm˙-m˙ref

式中: J˙为不同时间模型的雅可比矩阵的组合,可表示为

J˙=J1J2Jt;

WTd˙Wd˙WTm˙Wm˙J˙形式相同,是不同时间数据集的数据加权项和模型加权项的组合。

对于时间加权项Wt,形式如下(以3个模型为例):

$\boldsymbol{W}_{t}=\left[\begin{array}{ccccccccc}1 & \cdots & 0 & -1 & \cdots & 0 & 0 & \cdots & 0 \\\vdots & 1 & \vdots & \vdots & -1 & \vdots & \vdots & \ddots & \vdots \\0 & \cdots & 1 & 0 & \cdots & -1 & 0 & \cdots & 0 \\0 & \cdots & 0 & 1 & \cdots & 0 & -1 & \cdots & 0 \\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\0 & \cdots & 0 & 0 & \cdots & 1 & 0 & \cdots & -1 \\0 & \cdots & 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\0 & \cdots & 0 & 0 & \cdots & 0 & 0 & \cdots & 0\end{array}\right]$

对比式(3)与式(6),两者不同之处在于时移反演算法中需要构建时间加权矩阵,改加权矩阵的作用是体现出不同时间的模型差异对目标函数的贡献。笔者编写了时移电阻率反演程序,实现2D高密度电法时移反演。由于需要对模型约束和时间约束两个正则化参数调优,对模型约束项使用ACB方法[20]进行自动优化。实际上,模型参数在时间域上的变化要小于空间域的分布,而且各不同时间点的基本地下结构是保持不变的,所以时间正则项对目标函数的贡献要小于空间正则项,本文假设模型参数在时间域上的变化程度是中等的,时间正则化参数α=0.01[17]

2 模拟数据反演分析

为了研究时移电阻率法反演的效果,首先建立一组(多个)正演模型(图1)。其中,T0模型(图1a)由两部分组成,位于0~19 m,埋深4.44~7.88 m区域的电阻率设置为200 Ω·m,其余区域设定为100 Ω·m;后续的T1、T2、T3在T0模型基础之上,设置一个随着时间变化的目标低阻体,电阻率为10 Ω·m,随着时间的推移,目标体逐渐扩散,在横向和纵向均有变化,以便于检验反演算法的横向(纵向)分辨能力。模拟常用的温纳阵列类型的数据,测线共有51个电极,电极距为1 m,在模拟数据中加入了5%的随机噪声。

图1

图1   不同时间点的模型

Fig.1   The models of different time steps


在此基础上,对以上各个模拟数据分别进行单独反演和时移反演,经过对所编写的反演程序反复实验发现,通常情况下反演迭代过程拟合差下降速度较快,一般仅需迭代6次左右即可收敛,故将最大迭代次数设置为6,两种不同反演方式的结果如图2所示。

图2

图2   模拟数据单独和时移反演结果

Fig.2   The single inversion and time-lapse inversion results on simulation data


图2可以看到,单独反演与时移反演两种方法对不同数据均有较好的约束。其中,不同数据单独反演的拟合差在3%~4%之间,而时移反演的总拟合差则小于1%。由于噪声和误差的影响,反演结果中存在与模型无法对应的假异常。对比两种反演方法的结果,时移反演的假异常明显比单独反演的要少,而且对目标异常体的约束效果较好。为了突出地下目标体的变化情况,两种方法均以第一个模型的反演结果ρT0为基准,通过ρTi-ρT0的方式来显示各个模型的变动情况,结果如图3所示。

图3

图3   不同模型反演结果的差异

Fig.3   The difference images of inversion results


图3中,代表了T1、T2、T3相对与T0时的地下目标变化在反演结果中的相应体现,由于所设定的变化目标为一低阻体,故随着该目标的体积不断增大,图3中数值-0.3~-0.5所占的面积不断增大,说明两种方法均可以追踪地下电阻率动态变化的目标体。但是,对比时移反演(图3b)和单独反演(图3a)的结果可以看出,时移反演对其变化的目标具有更好的效果,而单独反演的效果较差,其对变化的目标体的约束不够,并且依然存在不同程度的假异常。

3 结论

1) 通过对反演算法的理论推导可发现,时移反演算法有别于常规算法之处在于该算法在目标函数中添加了一个时间约束项,所以时移反演算法的效果主要取决于时间约束项构建得是否合理。

2) 监测的多个数据集中存在不同程度的随机噪声,是导致反演中出现假异常的原因之一。由于时移反演算法中时移约束项的存在,将不同时间点的数据集统一放入一个目标函数中进行综合考量,从而使得时间序列上相邻的响应模型构成一定的联系,在时空域(4D)尺度上进行最优化,而不是孤立地对各个数据集和模型进行单独反演。

3) 模拟数据的实验表明,对于随时间变化的电阻率异常体,电阻率法时移反演克服了多个数据集存在不同随机噪声的影响,能够较好地约束地下动态目标,追踪其随时间的变化情况。

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Hayley K, Pidlisecky A, Bentley L R.

Simultaneous time-lapse electrical resistivity inversion

[J]. Journal of Applied Geophysics, 2011, 75(2): 401-411. https://doi.org/10.1016/j.jappgeo.2011.06.035

DOI:10.1016/j.jappgeo.2011.06.035      URL     [本文引用: 1]

Time-lapse monitoring is a powerful tool for observing dynamic changes in the subsurface. In particular it offers the potential for achieving inversion results with increased fidelity through the inclusion of complementary information from multiple time-steps. This inclusion of complementary information can reduce the need for spatial smoothing, without adding inversion artifacts to the resulting images. Commonly used time-lapse inversion methods include the ratio method, cascaded time-lapse inversion, difference inversion and differencing independent inversions. We introduce two additional methods in which both time-lapse data sets are inverted simultaneously. In the first, called temporally constrained time-lapse inversion, inversion of both datasets is done under a single optimization procedure and constraints are added to the regularization to ensure that the changes from one time to another are smooth. In the second method, called simultaneous time-lapse inversion, the inversions at time 1 and time 2 are performed simultaneously and constraints of smoothness and closeness to a reference model are applied to the difference image produced at each iteration, and subsequently, the constraints are updated at each iteration. Through both a numerical and a field example we compare the results of common time-lapse inversion methods as well as the introduced approaches. We found that of the commonly used time-lapse inversion methods the difference inversion method produced the best resolution of time-lapse changes and was the most robust in the presence of noise. However, we found that the alternative approach of simultaneous time-lapse inversion produced the best reconstruction of modeled EC changes in the numerical example and easily interpretable high resolution difference images in the field example. Moreover, there was less tailoring of regularization parameters with our simultaneous time-lapse approach, suggesting that it will lend itself well to an automated inversion code. (C) 2011 Elsevier B.V.

Karaoulis M C, Kim J H, Tsourlos P I.

4D active time constrained resistivity inversion

[J]. Journal of Applied Geophysics, 2011, 73(1): 25-34. https://doi.org/10.1016/j.jappgeo.2010.11.002

DOI:10.1016/j.jappgeo.2010.11.002      URL     [本文引用: 2]

Here the time regularization is not considered to be constant between different time steps but is now allowed to vary depending on the degree of spatial resistivity changes occurring between different monitoring stages. Two methods are proposed to assign different time Lagrangian values, one based on a pre-estimation during execution time, and one using a-priori information. Both methods require a threshold to characterize the significance of the observed resistivity changes with time. We performed numerous numerical experiments using synthetic data to provide reasonable threshold values. Synthetic data tests illustrate that the new algorithm, named 4D Active Time Constrained (4D-ATC), produces in most cases improved time-lapse images when compared with existing techniques. Further the applicability of the new scheme is demonstrated with real data. Overall, the new algorithm is shown to be a useful tool for processing time-lapse resistivity data, which can be used with minor modifications to other types of time-lapse geophysical data. (C) 2010 Elsevier B.V.]]>

Loke M H, Dahlin T, Rucker D F.

Smoothness-constrained time-lapse inversion of data from 3D resistivity surveys

[J]. Near Surface Geophysics, 2014, 12(1): 5-24. https://doi.org/10.3997/1873-0604.2013025

DOI:10.3997/1873-0604.2013025      URL     [本文引用: 1]

Three-dimensional resistivity surveys and their associated inversion models are required to accurately resolve structures exhibiting very complex geology. In the same light, 3D resistivity surveys collected at multiple times are required to resolve temporally varying conditions. In this work we present 3D data sets, both synthetic and real, collected at different times. The large spatio-temporal data sets are then inverted simultaneously using a least-squares methodology that incorporates roughness filters in both the space and time domains. The spatial roughness filter constrains the model resistivity to vary smoothly in the x-, y- and z-directions. A temporal roughness filter is also applied that minimizes changes in the resistivity between successive temporal inversion models and the L-curve method is used to determine the optimum weights for both spatial and temporal roughness filters. We show that the use of the temporal roughness filter can accurately resolve changes in the resistivity even in the presence of noise. The L1- and L2-norm constraints for the temporal roughness filter are first examined using a synthetic model. The synthetic data test shows that the L1-norm temporal constraint produces significantly more accurate results when the resistivity changes abruptly with time. The model obtained with the L1-norm temporal constraint is also less sensitive to random noise compared with independent inversions (i.e., without any temporal constraint) and the L2-norm temporal constraint. Anomalies that are common in models using independent inversions and the L2-norm and L1-norm temporal constraints are likely to be real. In contrast, anomalies present in a model using independent inversions but that are significantly reduced with the L2-norm and L1-norm constraints are likely artefacts. For field data sets, the method successfully recovered temporal changes in the subsurface resistivity from a landfill monitoring survey due to rainwater infiltration, as well as from an experiment to map the migration of sodium cyanide solution from an injection well using surface and borehole electrodes in an area with significant topography.

Pidlisecky A, Haber E, Knight R.

RESINVM3D: A 3-D resistivity inversion package

[J]. Geophysics, 2007, 72(2): 1-10. https://doi.org/10.1190/1.2402499

URL     [本文引用: 1]

Yi M J, Kim J H, Chung S H.

Enhancing the resolving power of least-squares inversion with active constraint balancing

[J]. Geophysics, 2003, 68(3): 931-941. https://doi.org/10.1190/1.1581045

DOI:10.1190/1.1581045      URL     [本文引用: 1]

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