0 引言
传统高密度电阻率法的数据处理广泛采用二维反演技术[7-8]。该技术结合了2.5D正演与2D反演方案,以实现对实测视电阻率数据的快速稳健反演[9-10]。在2.5D正演模型中,通常假设电阻率在垂直于测线方向(y方向)上保持恒定,而电极则沿x方向线性布置,通过傅里叶变换进行高效的正演计算[11]。然而,在复杂地形条件下,电极的布置往往难以保持直线,这导致部分电极在y方向上的位置出现偏移。如果继续采用2.5D正演模型来近似实际的电极布置,可能会导致正演结果出现明显偏差,进而影响反演电阻率的准确性。随着计算机算力的快速增强,三维反演技术已逐步应用于高密度电阻率法的地质调查任务中[12-13]。三维反演方案通过有限元方法或有限差分方法实现三维正演,并利用平滑约束等先验信息优化反演过程[14]。该技术能够显著提升在实际地质调查中,任意电极布局下的视电阻率正演计算精度,并能有效处理更为复杂的地下结构的反演问题[15-16]。然而,算法的计算成本随着模型维度的增加而呈指数级增长,这使得其在时间消耗上面临较大挑战[17]。此外,当使用二维测线数据进行三维反演时,数据在y方向(垂直测线方向)的分辨率受限,这可能会影响到三维模型的准确性和可靠性。
本研究针对不规则测线反演问题的特点,提出了一种优化的三维反演方案。通过寻找测线最小包围面积矩形的坐标转换算法,实现3D模型的最少网格节点剖分,降低计算成本。此外,通过增加y方向平坦度矩阵的权重系数,提升反演结果的稳健性。数值模拟与白水河滑坡上的实测结果表明,该技术可有效提升在复杂地形条件下高密度电阻率法的应用效果。
1 不规则测线视电阻率数据的3D反演方法
1.1 最小化网格剖分节点
在实际调查中,滑坡体复杂的地形、广泛分布的植被和地表散落的巨石使得测线难以沿直线布置。此时,任意分布的电极节点在布置时必须使用GNSS RTK技术进行精确定位,其定位误差一般在10 cm以内[18]。在使用国内通用CGCS2000系统时,一系列电极的位置被记录为投影坐标(东坐标,北坐标)格式(见图1a)。此时,为了使模型尽可能小,需要寻找测线的最小包围面积矩形,并通过坐标转换将矩形的左下顶点平移至坐标原点,两条相邻边旋转至x方向与y方向(图1b)。然后,以新的坐标系统作为xy平面进行网格剖分(图1c)。其中,主要难点包括寻找最小包围面积矩形以及坐标转换。坐标旋转可通过构造旋转矩阵与线性算法实现,见式(1):
式中:x'、y'为新坐标;x、y为初始坐标最小包围面积矩形,可采用开源的Andrew's monotone chain算法实现[19]。在实际调查中,矩形也可根据地质情况选取,y方向应该选择电阻率变化更小的方向。
图1
图1
不规则测线视电阻率数据的3D反演技术流程示意
a—投影坐标系下的电极分布;b—最小包围面积矩形与转换后的坐标系;c—转换坐标后的网格剖分;d—最大化y方向正则化参数后3D反演得到的电阻率模型;e—提取到的测线底部的电阻率数据;f—将e中坐标转为2D测线长度与高程的电阻率断面
Fig.1
Schematic of the 3D inversion process for resistivity data from irregular survey lines
a—the electrode distribution in the projected coordinate system; b—the minimum enclosing rectangular area and the transformed coordinate system; c—the grid division after coordinate transformation; d—the resistivity model obtained from 3D inversion after maximizing the regularization parameter in the y direction; e—the resistivity data extracted at the bottom of the survey line; f—the resistivity cross-section diagram where the coordinates in e are converted into 2D survey line length and elevation
1.2 3D反演方法
式中:J为雅克比矩阵,可通过互易定理计算。λ为阻尼系数向量,g为数据残差。FR为平坦度矩阵,可写为式(3):
2 数值模拟
2.1 模型建立
建立滑坡体模型如图2所示,滑坡体可分为堆积层与基岩两部分,其中堆积层电阻率设置为30 Ω·m,基岩电阻率设置为100 Ω·m。堆积层在滑坡上缘厚度较小,约为1.25 m,在下缘厚度逐渐增大至11 m。滑坡3D模型的x方向长度20 m,y方向长度2.5 m,假定滑坡结构电阻率在y方向上是无变化的。模型共设置电极21个,极距为1.03 m。受地形与障碍物的影响,测线在x=10 m,y=0 m位置处未能继续沿直线布置,转角约38°。
图2
图2
滑坡体数值模型与不规则测线示意
Fig.2
Schematic of numerical model of landslide bodies and irregular survey lines
2.2 模型计算
图3
图3
数值模拟结果
a—测线底部电阻率模型,其中蓝色区域为30 Ω·m,红色为100 Ω·m;b—将3D正演视电阻率数据转为2D断面显示;c—平滑约束2D反演断面,其中黑色线条为滑带位置在反演断面上投影位置;d—本研究中3D反演方法得到的电阻率断面
Fig.3
Numerical simulation results
a—shows the resistivity model at the bottom of the survey line, where the blue area represents 30 Ω·m and the red area represents 100 Ω·m; b—shows the 2D cross-sectional display of the 3D forward modeling apparent resistivity data; c—shows the smooth-constrained 2D inversion cross-section, where the black lines represent the projected positions of the fault zones on the inversion cross-section; d—shows the resistivity cross-section obtained by the 3D inversion method in this study
3 工程实例
白水河滑坡是三峡库区内典型的堆积型滑坡,位于湖北省宜昌市秭归县沙镇溪乐丰村。滑坡位于百福坪背斜北翼与秭归向斜交汇的长江南岸向斜坡上。出露岩层主要为侏罗系下统香溪组(J1x)中厚层状粉砂岩、夹薄层状泥质粉砂岩。滑坡整体形态呈“舌”状(见图4),后缘以岩性分界为界限(高程约400 m),前缘直抵长江。滑坡剖面上陡下缓,总体坡度约为30°,主滑方向为N32°E。钻孔资料显示,白水河滑坡发育有两层滑带,浅层滑带为坡体堆积物(碎石土或块石)与下部块裂岩的接触带,厚度一般为2 m左右,埋深一般在16 m左右。滑带物质为碎石土,结构紧实,不透水。深部滑层为块裂岩与下伏基岩(粉砂岩)的接触面,厚度一般在1 m左右,埋深在20~35 m左右[22]。
图4
图4
白水河滑坡地表地形与实测测线位置
Fig.4
Surface topography of the Baishui River landslide and positions of actual survey lines
为了调查滑坡结构,2条高密度电阻率法测线被布置于滑坡体上,受滑体上广泛分布的植被与复杂地形影响,测线扭曲严重,测线详见图4。ERT-1测线沿垂直于滑动方向布置,测线极距为5 m,全长595 m。ERT-2测线沿滑动方向布置,测线极距为5 m,全长445 m。图5为本研究提出的3D反演方法对扭曲测线实测数据的反演结果。结果显示,反演电阻率断面从浅到深可以被划分为3个电性层。其中,第一层大致位于地表到大约20 m的深度,电阻率通常在30~80 Ω·m之间,主要对应于浅层的碎石土层。第二电性层深度大致位于15~30 m,电阻率通常在80~200 Ω·m之间,主要对应了块裂岩层。第三电性层埋深一般大于30 m,电阻率值200~400 Ω·m,对应基岩层。反演电阻率断面的电性分层特征与测线周边ZK1、ZK4、ZK5揭露的地层情况吻合良好。
图5
图5
不规则测线实测数据反演结果
a—ERT-1测线结果;b—ERT-2测线结果
Fig.5
Inversion results of actual measured data from irregular survey lines
a—shows the results for ERT-1 survey line; b—shows the results for ERT-2 survey line
4 结论
针对复杂地形滑坡上的不规则测线视电阻率数据的处理问题,提出了一种基于3D反演的处理方案。通过数值模拟与工程实例研究,得到如下结论:
1)数值模拟结果表明,扭曲测线会导致视电阻率数据出现明显偏差,影响传统2D反演效果,使电性分层出现虚假扭曲异常,最多可偏差1倍极距以上。本研究提出的3D反演方法,可显著改善滑坡体结构的成像效果,避免虚假扭曲异常出现。
2)现场实测结果表明,本研究提出的3D反演方法在处理白水河滑坡上不规则测线实测数据时,反演结果可清晰识别碎石土、块裂岩、基岩的分界面信息。测线周边的3个钻孔验证了电阻率结果的准确性。
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