Using GPR Data as Constraints in RMT Data Inversion for Water Content Estimation: A Case Study in Heby, Sweden

Abstract

This study uses ground penetrating radar (GPR) data as constraints in the inversion of radio-magnetotelluric (RMT) data, to provide an improved model at shallow depth. We show that modification of the model regularization matrix using all GPR common-offset (CO) reflections can mislead the constrained inversion of RMT data. To avoid such problems, common mid-point (CMP) GPR data are translated to a resistivity model by introducing a new petrophysical relationship based on a combination of Topp’s and Archie’s equations. This model is updated through a semi-iterative method and is employed as an initial and prior model in the subsequent inversion of RMT data. Finally, a water content model that fits the GPR CMP and RMT data is derived from the resistivity model computed by the constrained inversion of RMT data. To assess the proposed scheme, it is applied to a synthetic data set. Then, real RMT data collected along an 870 m-long profile across a known aquifer situated in the north of Heby, central Sweden, are inverted. By removing the smoothness constraints across GPR CO interfaces or using CMP-based inversion, thick (> 10 m) vadose and saturated zones are resolved and shown to correlate with logs from nearby boreholes. Nevertheless, the application of our CMP-based inversion was the only efficient scheme to retrieve thin (~ 3 m) saturated zones and the water table at a depth of 7–15 m in the RMT models. The estimated models of water content are in good agreement with the available hydrogeological information in the study area.

Appendices

Appendix 1: RMT Theory

In the RMT method, two horizontal components of the electric field (e_x and e_y) and the three magnetic field components (h_x, h_y, and h_z) are measured simultaneously. These electromagnetic field components are connected through the impedance tensor Z and the the tensor of vertical magnetic transfer functions (tipper vector) T. Pedersen et al. (2006) showed that the distribution of transmitters in Europe is appropriate for the estimation of RMT transfer functions that are given in the frequency domain as follows:

$$\left[\begin{array}{*{20}c} E_{x}(f) \\ E_{y}(f) \end{array} \right] = \left[\begin{array}{*{20}c} Z_{xx}(f) & Z_{xy}(f) \\ Z_{yx}(f) & Z_{yy}(f) \end{array}\right] \left[\begin{array}{*{20}c} H_{x}(f) \\ H_{y}(f) \end{array} \right] {\text{ or }} {\mathbf{E}}(f) = {\mathbf{Z}}(f){\mathbf{H}}(f),$$

(8)

$$H_{z}(f) = \left[ T_{x}(f) T_{y}(f)\right]\left[ \begin{array}{*{20}c} H_{x}(f) \\ H_{y}(f) \end{array} \right]{\text{ or }}H_{z} (f) = {\mathbf{T}}(f){\mathbf{H}}(f),$$

(9)

where E and H are the Fourier transforms of the electric (e) and magnetic (h) components, respectively, at signal frequency f, Z(f) denotes the two-by-two complex impedance tensor, and T(f) is the one-by-two tensor of vertical magnetic transfer functions (VMTF) or tipper vector. Z and T contain information about the subsurface resistivity distribution. Ideally, in a two-dimensional case, with structures striking in one direction (for example x-direction), zero values are expected for the diagonal elements of the impedance tensor. When the strike is in the x-direction, $Z_{xy}$ corresponds to the TE mode, where the electric currents flow in the strike direction, and $Z_{yx}$ represents the TM mode, where the electric currents flow in a plane perpendicular to the strike direction. For each impedance tensor element and each frequency, the apparent resistivity and phase can be deduced as

$$\rho_{a} \left( f \right) = \frac{1}{{\omega \mu_{0} }}\left| {Z\left( f \right)} \right|^{2} ,$$

(10)

$$\varphi = \arg \left( {Z\left( f \right)} \right),$$

(11)

where ω is angular frequency, and µ₀ is the magnetic permeability of free space. The distribution of electric resistivity in the subsurface can be estimated through inversion of the appropriate impedance data, tipper data or apparent resistivities and phases.

As a result of time-series processing, estimates of the transfer functions (impedance and tipper data) together with their uncertainties in form of standard deviations are retrieved. However, the inversion code by Kalscheuer et al. (2010) uses apparent resistivities and phases of the complex-valued impedances rather than the impedances as input data. For a given standard deviation of the impedance, it can be shown through Gaussian uncertainty propagation that 1% relative uncertainty on the impedance corresponds to 2% relative uncertainty on the apparent resistivity and 0.57 degrees absolute uncertainty on the phase. Moreover, in the inversion code, apparent resistivities are transformed to logarithms of apparent resistivities, such that the corresponding uncertainties essentially are relative uncertainties of the apparent resistivities. In the inversion of field data, the uncertainty floors for the impedance and tipper data are subjective choices based on an assessment of the general data quality and tentative inversions.

Appendix 2: Damped Occam Inversion

In the damped Occam inversion algorithm, the Lagrange multiplier λ for the smoothness constraints is fixed at a user-defined value, and a Marquardt–Levenberg damping term is added to the cost function U. An optimal Marquardt–Levenberg damping factor β is searched for in every iteration of the inversion. In this study, we used the resistivity model and corresponding Lagrange multiplier found optimal in the Occam inversion as an initial model and fixed λ, respectively, in a subsequent damped Occam inversion. For our synthetic example in Figs. 3 and 4, Fig. 12 shows the produced artifacts due to a suboptimal fixing process in Step 4 and the influence of the damped Occam method in Step 5 for smearing them out.

Fig. 12

a 2D resistivity model from Step 4 of our CMP-based RMT inversion scheme with a suboptimal selection of fixed cells in the saturated zone (white polygon) and incorrect velocity analysis in the vadose zone (black rectangles) for synthetic model shown in Fig. 3a. b Model from 3rd iteration of damped Occam inversion (output of Step 5) used to remove artifacts in a and c model of water content deduced from b. Note that the effect of the suboptimal fixing process is relatively small by comparing the results in c and Fig. 4e

Appendix 3: RMT Strike Analysis of and Fit to Heby Data

By analyzing the impedance tensor, the Swift skew (Swift 1976) provides an overall measure of dimensionality and 3D effects. The measured RMT data seems to be affected by noise, especially along P1-1. Upon closer inspection, this effect could be related to cables buried underneath the ditches along the road and low-quality data at a frequency of 113 kHz. After the rejection of noisy data, the Swift skews of the RMT data measured in the Heby area are lower than 0.25 (Fig. 13a, c). Therefore, subsurface structures can be expected to be predominantly 1-D and 2-D. In this study, we utilized Zhang et al.’s (1987) method for dimensionality, distortion, and strike analysis. Independent strike angles were extracted for each station and frequency of RMT data by calculating distortion parameters and strike directions simultaneously. Generally, strike angles computed by analysis of the impedance tensor are ambiguous by 90°, such that complementary information in the form of the directions of induction arrows computed from vertical magnetic transfer functions or geological maps is needed to assign a unique strike direction. We have predominantly considered the geological map to determine the strike angles. The results of the strike analyses show that there is no consistent strike direction along P1. However, one preferred strike direction can be distinguished from cumulative rose diagrams for each profile (Fig. 13). Relative to our measurement coordinate systems, these preferred strike directions are − 15 and 10 degrees East of North for P1-1 and P1-2, respectively. These strike directions correspond to 30° and 5° West of geographic North, for P1-1 and P1-2, respectively. Therefore, for each profile on Fig. 5, we have rotated the reference coordinate system to the mentioned strike direction minimizing the least-squares Q value (Zhang et al. 1987) and projected the station locations accordingly.

Fig. 13

a and c Swift skews for the edited RMT data of profiles P1-1 and P1-2, respectively. b and d Strike directions calculated with the method proposed by Zhang et al. (1987) for profiles P1-1 and P1-2, respectively

Figure 14 shows the observed and calculated data, as well as their misfits for TE-mode impedances, TM-mode impedances and tipper data using smoothness-constrained inversion of the RMT data collected along P1 (Fig. 7b). White cells represent data rejected in the inversion due to either low quality or high misfit values in initial test inversions. Similarly, the data fits of the inverted models using the proposed CMP-based scheme along P1-1 (Fig. 8d) and P1-2 (Fig. 10d) are illustrated in Figs. 15 and 16, respectively.

Fig. 14

Observed data, calculated data, and misfits, normalized by the corresponding uncertainties of a TE-mode impedances, b TM-mode impedances and c tipper data along profile P1 using the smoothness-constrained inversion (Fig. 7b). White cells in the misfit plots indicate data that were rejected either because of noise or because of higher misfits (> 6) in tentative inversions

Fig. 15

Misfits normalized by the corresponding uncertainties of a TE-mode impedances, b TM-mode impedances and c tipper data along profile P1-1 using the CMP-based inversion (Fig. 8d). White cells indicate data that were rejected either because of noise or high misfits (> 6) in tentative inversions

Fig. 16

Misfits normalized by the corresponding uncertainties of a TE-mode impedances, b TM-mode impedances and c tipper data along profile P1-2 using the CMP-based inversion (Fig. 10d). White cells indicate data that were rejected either because of noise or high misfits (> 6) in tentative inversions