Interpolating electrode position from surveying peg

Hollin Hill is instrumented with a series of 5 electrical resistivity survey lines, roughly running downslope and parallel to landslide movement. To understand the dynamics of the landslide system an array of 45 survey pegs were inserted into the land surface, positioned coincident with ERT survey lines, and has been monitored by highly accurate Real-Time Kinematic GPS (Leica Viva System). It should be noted that pegs are not co-located with electrodes.

Electrical resistivity tomography is increasingly being applied to both characterise and monitor landslide internal structure and physical property variation (Suzuki et al., 2001; Lebourg et al., 2005; Jomard et al., 2006). When the intention of investigation is to improve the understanding of landslide hydrogeological dynamics – such as those related to landslide activation – monitoring by ERT is not a straight forward task. Monitoring of landslide systems by electrical resistivity surveying requires known electrode positions as relative electrode position is a fundamental input when computing both apparent resistivity (using the geometric factor) and when modelling resistivity using a suitable software package such as Res2DINV, or in this case Res3DINV. Active landslide systems are dynamic natural phenomena and as a result the positions of buried electrodes do not remain constant as landslide displacements evolve through time. The exact positions of the electrodes are not known because they are buried at

approximately 0.1 m below land surface; therefore a method to derive the best-estimate of electrode positions is required.

Several methods to derive best-estimates of electrode positions have been proposed. Wilkinson et al (2010) predicts the movement of permanently installed electrodes on an active landslide by analysing electrical signatures which manifest as electrodes mobilise. Wilkinson et al (2010) found that movement artefacts over-print and obscure genuine time-lapse resistivity changes taking place in the subsurface. Their method is applied to predict electrode movement based on the time-lapse artefacts which appear as a result of changing electrode array geometry. This approach is has not been implemented for full 2D and 3D array monitoring, and so has not been used in this study.

Instead, we estimate electrode positions using the positions of a known set of reference points available at the field site. In the case of this investigation, the RTK-GPS monitored 45 survey peg array provide the necessary reference points (RPs).

Several approaches exist to derive electrode positions from movements of a known set of reference points (Uhlemann et al., 2013; Uhlemann et al., in press) and as a result attempt to avoid the considerable artefacts which would exist in time-lapse resistivity models. Deriving electrode positions from known reference points is the only option of determining electrode positions of the electrodes of the ERT monitoring array at Hollin Hill because the electrodes were buried during installation to protect the arrays from damage and due to strict planning regulations.

These approaches are described in Uhlemann et al (2013), and are briefly summarised here:

Velocity Approach (VA)

Movement of a set of reference points (RPs) from their initial positions (where is the position of an RP) at time 0 (at ( )) – in the case of this investigation the array of GPS monitored surveying pegs – to another position (at ( )) at a future time n is divided into two directional movement vectors ( ) and ( ).

( ) ( ) ( )

Equation 4.10. Determining directional movement vectors ( )

To estimate the positions of electrodes between two reference points a linear interpolation of and to the initial electrode positions is applied. At a time n, the directional movement of an electrode along the x-axis is determined by

( ) ( ) [( ( ) ( ))

( ) ( )

( ) ( )]

Equation 4.11. Directional movement of an electrode at time n

The positions of the electrodes ( ) at a time , where initial electrode positions are known is estimated by applying the following equation

( ) ( ) ( )

Equation 4.12. Position of electrode at time n

For times when no RP positions are known, electrode positions are interpolated by determining the velocities dvx and dvy from RP positions

directly before and after the time of interest . General Approach (GA)

Three non-collinear points spanning a basis can be used to describe any point in a plane. The point of interest can be described

( ) ⃗ ⃗ ‖ ⃗ ‖

Equation 4.13. Electrode position at time t0

Where , and are weights and the final factor is the normalised normal vector which is perpendicular to ⃗ and . If the positions of the three nearest RPs and the point of interest, here, an electrode ( ( )), are known the three weights , and can be derived as well as the vectors and . The weights remain the same if we assume that the relative position of the electrode to the vectors ⃗ and remain constant. Therefore, to calculate the new electrode position at time , ( ), the RP positions at time are

used to determine and and are inputted into the equation above along with the three weights. To derive electrode positions for times when no RP positions are known, RP positions are derived by linear interpolation between times of known RP positions.

Uhlemann et al (in press) reports that the most accurate of the two methods to apply to interpolate electrode positions from a sparse number on reference points is the general approach, their comparison found the RMS error of the general approach to be substantially lower than that from the velocity approach, 0.0042 m compared to 1.12 m. For that reason the general approach is applied to interpolate electrode movement during this investigation.

The method described to interpolate electrode positions from a sparse number of reference points was applied in this geophysical investigation to generate an electrode location file for every field resistivity survey performed during the monitoring campaign. These files are termed ‘.geom’ files and were incorporated into the ‘pre-inversion’ raw resistance data file which was later read into Res3DINV.

Figure 4.11. Example of interpolation of electrode positions from sparse array of GPS monitored survey peg positions.

As will be discussed in Section 4.3.2, these .geom files – which contain electrode position information – were also used when calculating temperature correction factor to correct transfer resistance results for the effects of temperature.