2D INVERSION OF DATA FROM UNDERWATER ELECTRICAL IMAGING SURVEYS M.H.Loke, Universiti Sains Malaysia, Penang, Malaysia

Abstract The 2D electrical imaging method has now become a standard exploration tool in many environmental and engineering surveys. An interesting extension of the electrical imaging method is surveys where the some or all of the electrodes are placed underwater on a river or sea bed. The smoothness-constrained least-squares method that is used for normal land surveys can be adapted for underwater surveys. To accommodate the underwater topography, the finite-element method with a distorted grid is used to calculate the apparent resistivity values for the inversion model. The first five or six rows of nodes are used to model the water layer, while the lower part of the grid is used for the subbottom resistivity distribution. For an accurate inversion, the water resistivity as well as the river or sea bottom topography must be accurately known since a large proportion of the current flows through the water layer. The section of the earth below the bottom surface is subdivided into a large number of rectangular cells, and the inversion program attempts to determine the resistivity of the cells that would most accurately reproduced the observed measurements. The water resistivity is kept fixed during the inversion of the data, and the inversion program tries to determine the resistivity of the sub-bottom materials. The inversion results for data sets from two field surveys and a synthetic model are shown. While in theory it is possible to allow the water resistivity to vary as well, it has been found that this produces significantly poorer results.

Introduction The resistivity surveying method has undergone dramatic changes over the last decade. Traditionally the resistivity sounding method that gives a 1D picture of the subsurface had been used since the 1930’s (Keller and Frischknecht, 1966). A major improvement since the early 1990’s have the development of two-dimensional (2D) electrical imaging surveys (Griffiths and Barker, 1993) that provide a more realistic model of the subsurface in complex areas. Three-dimensional (3D) surveys have also been used (Loke and Barker, 1996b), particularly in mineral exploration surveys (Denne et al., 2001) where the high costs involved is justified. However, for most surveys, a 2D resistivity survey is probably the most cost effective method (Dahlin, 1996). Two-dimensional resistivity and I.P. imaging surveys are now widely used for various engineering and environmental studies (Barker, 1996; Ritz et al., 1999; Seaton and Burbey, 2000; Acworth and Dasey, 2001). An interesting application of the 2D electrical imaging method is in water-covered areas (Wynn and Grosz, 2000). In these surveys, some or all of the electrodes are underwater. The following section gives a brief description of the survey and data interpretation methods used for such surveys. This is then followed by one synthetic data and two field data examples.

Method 2D electrical imaging surveys Figure 1a shows a typical arrangement used for an on-land 2D electrical imaging survey. Twodimensional electrical imaging surveys are usually carried out using a large number of electrodes, 25 or 1

more, connected to a multi-core cable (Griffiths and Barker 1993). A laptop microcomputer (or an internal micro-controller within the resistivity meter unit) together with an electronic switching unit is used to automatically select the relevant four electrodes for each measurement. To interpret the data from such surveys, a cell-based model that subdivides the subsurface into a large number of rectangular cells (Figure 1b) is commonly used (Loke and Barker, 1996a). A static system with electrodes planted into the ground (Figure 1a) is commonly used for land surveys as well as in shallow water environments, such as in a situation where the survey line crosses a river (Acworth and Dasey, 2001). Mobile systems where the cable is towed manually or by a small vehicle have also been used (Sorenson, K., 1996; Panissod et al., 1998; Bernstone and Dahlin, 1999). One disadvantage of a mobile system for land surveys is the limited current that can be injected or induced into the ground. This limits the depth of investigation of such systems. An underwater environment provides an almost ideal situation for a mobile system since there is no problem in obtaining good electrode contact. Figure 2 shows a possible arrangement for an underwater mobile surveying system where a cable with a number of nodes is pulled along the river/lake/sea bottom by a boat. Two of the nodes are used as current electrodes, while the rest are used as potential electrodes. Such a system is usually coupled with an ultrasonic depth profiler and a water conductivity meter. Forward modeling method Figure 3 shows the two possible situations that might be encountered in surveys in water-covered areas. In the first situation, the entire survey line is underwater. This is typical of surveys in marine environments, lakes and along wide rivers. In the second situation, some of the electrodes are underwater while others are above the water. This is encountered where the survey line crosses a stream or narrow river. In order to model the response of the situation in Figure 3a, a distorted finite element grid is used (Silvester and Ferrari, 1990; Loke, 2000). A number of mesh rows are used to model the water layer (Figure 4). The top of the grid is flat to match the water surface while the nodes along bottom row of the mesh used to model the water layer are adjusted such that they match the water bottom topography (Figure 4). It is assumed that the water layer is homogeneous and the water resistivity as well as the bottom topography is known. This method of modeling the water layer can be easily modified to handle the situation in Figure 3b with some of the electrodes above the water by assigning a very high resistivity value for the finite-element cells above the water surface. Inversion method The inversion of resistivity data is inherently non-unique. To obtain a model that is closest to the real geology, other available information should be included in the inversion process in the form of constrains (Ellis and Oldenburg, 1994). In the case of an underwater resistivity survey, the depth of the water layer and the water resistivity can be easily measured during the course of a survey. Thus both parameters are fixed during the inversion process, and the purpose of the inversion program is to determine the resistivity of the sub-bottom materials. The sub-bottom region is divided into a number of rectangular cells in a similar way as in normal land surface surveys (Figure 1b). The inversion of the data is carried out to determine the resistivity of the model cells so that the calculated apparent resistivity values agree with the measured field data. The iteratively reweighted smoothness-constrained least-squares optimization method is used for the data inversion. The equation used is J Ti R d J i + λ i W T R m W Δri = J Ti R d g i − λ i W T R m Wri −1 , (1) where gi is the data misfit vector containing the difference between the logarithms of the measured and calculated apparent resistivity values, ∆ri is the change in the model parameters for the ith iteration and ri-1 is the model parameters (the logarithm of the model resistivity values) vector for the previous

(

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iteration. J is the Jacobian matrix of partial derivatives. A first order finite-difference operator (deGrootHedlin and Constable, 1990) is used for the roughness filter W. The damping factor λ determines the relative importance given to minimising the model roughness. Starting from a homogeneous earth model, the inversion routine usually takes about 4 to 6 iterations to converge (Loke and Dahlin, 1997). Rd and Rm are weighting matrices introduced to modify the weights given to the different elements of the data misfit and model roughness vectors (Wolke and Schwetlick, 1988). In the l2-norm based smoothness-constrained least-squares method, they are equal to the identity matrix I.

Figure 1. (a) Schematic diagram of a typical field setup for a 2D electrical imaging survey. (b) The arrangement of the data points in a pseudosection and the 2D rectangular model cells used for the inversion model. 3

Figure 2. Schematic diagram of a possible mobile underwater survey system. The cable has two fixed current electrodes and a number of potential electrodes so that measurements can be made at different spacings. The above arrangement uses the Wenner-Schlumberger type of configuration. Other configurations, such as the gradient and dipole-dipole arrays, can also be used.

Figure 3. A schematic diagram of the two possible situations for underwater surveys. (a) All the electrodes are underwater. (b) Some of the electrodes are underwater and some above the water level.

Figure 4. The finite element mesh used to calculate the apparent resistivity values for an underwater survey. The electrodes are marked by small red ticks, while the mesh elements used to model the water layer are drawn in blue. The number of mesh levels used to model the water layer depends on the ratio of the maximum water thickness to the spacing between adjacent electrodes.

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Results Synthetic model The first example is a synthetic model with two rectangular blocks, with resistivities of 1 and 100 Ohm.m respectively, embedded in a medium of 10 Ohm.m. The top of both blocks are 1 meter below the ground surface. Both have a thickness of about 2 meters and width of 4 meters. Figure 5b shows the apparent resistivity pseudosection for the Wenner-Schlumberger array (Pazdirek and Blaha, 1996) for a normal land survey. The apparent resistivity values range from 3.3 to 14.7 Ohm.m. This is much smaller than the range of the model resistivity values of 1 to 100 Ohm.m. When a 2 meter thick water layer with a resistivity of 5 Ohm.m is added to the model, the amplitude of the anomalies in the pseudosection (Figure 5c) are significantly reduced. The apparent resistivity values range from 5.8 to 10.3 Ohm.m. Increasing the water thickness to 5 meters (Figure 5d) and 15 meters (Figure 5e) generally results in a gradual reduction in the amplitude of the anomalies in the pseudosection. Beyond 15 meters, the effect of increasing the water thickness results in a gradually smaller reduction in the amplitude of the anomalies. The reduction in the amplitude of the anomalies is due to the flow of part of the current into the water layer. The apparent resistivity data set for the model with a 5 meters thick water layer (Figure 5d) is then used as the input data for the inversion program. Since both rectangular blocks in the original model (Figure 5a) have sharp boundaries with the surrounding medium, the l1-norm based smoothnessconstrained least-squares inversion method was used (Ellis and Oldenburg, 1994). The resulting model produced by the inversion program is shown in Figure 6. The shape, size and depths of both rectangular blocks are accurately reproduced in the inversion model. The model resistivity value reaches a minimum of 1.04 Ohm.m at the location of the low resistivity block which is close to the true value of 1.0 Ohm.m. At the location of the high resistivity block, it reaches a maximum of 49 Ohm.m that is much lower than the true value of 100 Ohm.m. However, it is still much higher than the maximum value of 8.9 Ohm.m in the apparent resistivity pseudosection. River bed survey, Belgium This survey was carried out by Sage Engineering of Belgium to map the near surface lithology of the river bed where there were plans to lay a cable. The depth to the river bed is about 3 to 7 meters while the water resistivity is 29 Ohm.m. This survey was carried out by dragging a measuring line with a number of nodes behind a boat, much like the arrangement shown in Figure 2, except a gradient type of array was used. This results in a highly asymmetrical electrode arrangement for many of the measurements. Figure 7.9a shows the data from the first two kilometers of an eight kilometers survey line along the river. The particular data subset in Figure 7a is slightly more than 2 kilometers long and has an electrode position at almost every meter. It has a total of 1994 electrode positions and 1760 data points, whereas the inversion model used has 5312 blocks. On a 550 Mhz Pentium III computer, it took slightly over 14 hours to process this data set. In the inversion model (Figure 7b), most of the river bed materials have a resistivity of less than 120 Ohm.m. There are several areas where the near-surface materials have significantly higher resistivities of over 150 Ohm.m. Unfortunately, geological information in this area is rather limited. In the high resistivity areas, the divers faced problems in obtaining sediment samples. The lower resistivity materials are possibly more coherent sediments (possibly sand with silt/clay), whereas the higher resistivity areas might be coarser and less coherent materials (Bucknix, pers. comm.). Tidal creek survey, Australia This study was carried out by the University of New South Wales Groundwater Centre (Australia) to map the saline intrusion into a coastal sand-dune aquifer at Hat Head in northern New 5

South Wales (Acworth and Dasey, 2001). Resistivity measurements were made using electrodes installed on the ground surface, in boreholes as well as underwater on a creek bed. Figure 8a shows the apparent resistivity pseudosection from one of the survey lines (using the Wenner array) that crosses the Korogoro Creek at about 2.5 kilometers from the open ocean. The tidal range at the site is approximately 1.5 m. The creek is approximately 40 meters wide at this survey line. At the points where the survey line crosses the creek, the electrodes were planted on the creek bed beneath the water surface. The water is saline with a resistivity of 0.18 Ohm.m, and thus it has a very large effect on the measured apparent resistivity values. The inversion model (Figure 8b) shows that the saline intrusion is largely limited to below the creek, with a small lobe extending to the dune side of the creek (Acworth and Dasey, 2001).

Discussion and Conclusions An inversion was also attempted with the synthetic data set where the resistivity of the water layer was allow to vary. It was found that the program attempts to fit the data set by varying the resistivity of the water layer and the rectangular blocks were not accurately reproduced in the inversion model. This is probably because the water layer has a very large influence of the calculated apparent resistivity values that is much larger than the effect of the subsurface resistivity distribution. To obtain accurate results, the resistivity and thickness of the water layer must be known and fixed in the inversion process. The examples show that the resistivity imaging method that is widely used for ground surface surveys can be adapted to underwater environments. Similarly, the smoothness-constrained least-squares inversion method can also be modified for such a situation.

Acknowledgements The author would like to acknowledge the support given by the School of Physics, Universiti Sains Malaysia. The Redas river survey data set was kindly provided by Jef Bucknix of Sage Engineering, Belgium. Dr. Ian Acworth and Greg Dasey of the Water Research Laboratory, School of Civil and Environmental Engineering, The University of New South Wales, Australia kindly provided the tidal creek survey data set.

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Figure 5. Effect of a water layer on the apparent resistivity pseudosection. (a) The model with a 5 meters thick water layer. Apparent resistivity pseudosections for the WennerSchlumberger array with (b) no water layer, a (c) 2 meters thick, (d) 5 meters thick and (e) 15 meters thick water layer. 7

Figure 6. Inversion model obtained from the inversion of the apparent resistivity pseudosection data set with a 5 meters thick water layer.

Figure 7. (a) The apparent resistivity pseudosection for the first two kilometers of an underwater survey along a river bed by Sage Engineering, Belgium. (b) The inversion model after three iterations.

Figure 8. (a) Apparent resistivity pseudosection (Wenner array) from the Hat Head tidal creek survey and (b) the inversion model. Note the creek between the 65 and 95 meters marks where the electrodes are underwater. 8

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