An accelerated time domain finite difference simulation scheme for three-dimensional transient electromagnetic modeling using geometric multigrid concepts

Authors


Abstract

[1] The fact that the transient electromagnetic (TEM) field is smoothed gradually in space with time allows for a reduced spatial sampling rate of the EM field. On the basis of concepts known from multigrid methods, we have developed a restriction operator in order to map the EM field and the material properties from a fine to a coarser finite difference mesh during a forward field simulation with an explicit time-stepping scheme. Two advantages follow. First, the grid size can be reduced. Field restriction involves reducing the number of grid nodes by a factor of 2 for each Cartesian direction. Second, as can be seen from the Courant-Friedrichs-Levy condition, the larger grid spacing allows for proportionally larger time step sizes. After field restriction, a material averaging scheme is employed in order to calculate the underlying effective medium on the coarse simulation grid. Example results show a factor of up to 5 decrease in solution run time, compared to a scheme that uses a constant grid. Key to the accuracy of the approach is knowledge of the proper time range to restrict the fields. An adequate criterion to decide during run time when to restrict involves an error measure for the locations of interest between the fields on the fine mesh and the restricted fields.

1. Introduction

[2] The finite difference time domain (FDTD) method belongs to the most widely used numerical methods, employed to solve complex electromagnetic scattering problems. FDTD is designed to handle large problems, conformal to structured grids, using an explicit leapfrog time-stepping scheme [Yee, 1966]. The modeling of increasingly complex structures has created a need to improve the inherently high computational time requirements of the FDTD method. Engineering applications in particular require modeling very complex and electrically large structures. For such applications, various schemes for FDTD subgridding algorithms have been presented [Zivanovic et al., 1991; Prescott and Shuley, 1992; White et al., 1997, 2001]. These algorithms have in common that a local mesh refinement is carried out to allow for increased resolution in regions of interest. Simultaneously, a computationally more economic coarse grid is maintained for the rest of the model domain.

[3] In contrast to engineering problems, typically characterized by high-frequency field simulations, our solution is designed to treat transient electromagnetic (TEM) problems arising in geophysical exploration, such as mineral and hydrological reconnaissance, or general geological mapping. The frequency range of such applications allows us to neglect displacement currents; thus we solve the diffusive forms of Maxwell's equations in the quasi-static limit.

[4] One highly efficient three-dimensional (3-D) solver for modeling diffusive EM fields is based on the spectral Lanczos decomposition method (SLDM) [Druskin and Knizhnerman, 1994; Knizhnerman, 1995; Druskin et al., 1998]. It has been widely applied and continuously improved for geophysical applications, such as for example 3-D EM induction modeling in boreholes [Druskin et al., 1999]. Compared to FDTD, SLDM and its recent improvements [e.g., Davydycheva et al., 2003] has the outstanding advantage of providing faster solutions. On the other hand, FDTD is more effective in the case of a large and spatially distributed number of receptors to be simulated, as is typical in large-scale EM inverse problems. It is also reliable for producing accurate results over large dynamic time ranges.

[5] In this study, we aim at improving one of the major drawbacks of FDTD, its large computational requirements when simulating responses at late times. Our FDTD method's capabilities and the implementation for parallel computers have been described in two recent works [Commer and Newman, 2004; Newman and Commer, 2005]. We simulate the TEM fields due to a step response signal, impressed by two different transmitter types. These are loops with an inductive coupling to the ground and galvanically coupled grounded wires. The employed explicit time-stepping algorithm is of second-order accuracy and allows for a variation of the electrical conductivity in the three dimensions of a Cartesian grid. A modified version of the DuFort-Frankel method [DuFort and Frankel, 1953] is used to achieve unconditional stability.

[6] To achieve a further speedup of the explicit time-stepping scheme, we use concepts known from classical multigrid (MG) schemes. Classical MG methods [e.g., Brandt, 1973; Trottenberg et al., 2000] solve a partial differential equation of interest on a hierarchy of grids. These methods perform a certain number of (pre)smoothing iterations to the initial equation system. The change to a coarser grid level is done after a sufficient reduction of the solution's high-frequency errors. Applying this concept to our explicit time-stepping scheme translates to the frequency content of the EM field. While the decaying TEM field diffuses in the conductive Earth, which is a lossy medium, high frequencies are damped out first. Hence the spatial sampling rate can be decreased accordingly with progressing simulation time. After a certain simulation time on the initial grid, the electromagnetic field is transferred to a coarser mesh by a field restriction procedure. Similar to classical MG, where the optimal number of presmoothing steps is problem-dependent, the optimal restriction time needs to be individually evaluated for a given modeling problem.

2. Method

[7] For a detailed description of the FDTD scheme for the diffusion equation for both inductive and galvanic source types, the reader is referred to Oristaglio and Hohmann [1984], Wang and Hohmann [1993], and Commer and Newman [2004]. As mentioned previously, we employ the explicit DuFort-Frankel method, where the field unknowns en+1 at a time (n + 1)Δt are updated directly from unknowns en from the preceding time nΔt. By introducing an artificial displacement term into the diffusive form of Maxwell's equations, the DuFort-Frankel method becomes unconditionally stable and thus allows for a coarser time sampling of the fields compared to the traditional Euler method [Oristaglio and Hohmann, 1984]. The stability of this modified system of Maxwell's equations is governed by the Courant-Friedrichs-Levy condition [Richtmyer and Morton, 1967] and allows increasing the time step size Δt with time [Wang and Hohmann, 1993], specifically,

equation image

where Δmin, σmin and μmin are the minima of the model's grid spacing, conductivity, and magnetic permeability, respectively and α is a constant less than one (we usually choose α = 0.1). Equation (1) defines an approximate upper bound for the time step size, in order to have diffusion dominating at the earliest time of interest t.

[8] Similar to the time sampling, the initial spatial sampling rate needs to be sufficient to suppress numerical dispersion, which would be caused by improper simulation of the high-frequency EM field. Over longer times, the field obtains a more diffusive nature due to the domination of low frequencies and allows reducing the spatial sampling rate. Changing to a coarser mesh with a larger mesh spacing, Δ, after a certain time, allows us to continue the field update with a larger time step, as can be seen from (1). At the grid changeover, we double the mesh spacing and perform a mapping of the EM field to the new mesh, referred to as restriction in the following scheme.

2.1. Field Restriction Scheme

[9] We develop averaging schemes for the restriction of the electric and magnetic fields. The procedure also applies to nonuniform meshes. After restriction, we distinguish between the initial or fine simulation grid and the coarse simulation grid. We use the index variables i, j, k to refer to fine-grid cells in the x, y, z directions, respectively. Similarly, the indices I, J, K refer to a coarse-grid node position. The coarse grid is constructed for each Cartesian direction as follows: Starting at the first node and skipping every second, all uneven fine-mesh nodes define the coarse nodes. The actual position of the node I, J, K is at the cell center of the corresponding fine cell, which is denoted by i′, j′, k′ in the following. The relationship

equation image

maps a coarse node index to the corresponding fine node.

[10] The geometry for restricting the electric fields is sketched in Figure 1, for the example of the x component. Electric fields are sampled on edges, between two grid nodes, and magnetic fields are sampled on facets of grid cells. The shown two coarse-grid nodes define the edge where the new electric field Ex(I, J, K) will be sampled. Ex(I, J, K) is computed from the average

equation image
equation image

where the three components of l are fractions of the corresponding cell lengths Δxi. The inner summation involves an arithmetic average of the fine-grid electric fields ex(i, j, k) over the length of the coarse-grid edge. The new field Ex(I, J, K) then results from a simple average of the four edges surrounding the coarse edge. Derivation of similar formulas for the other two components is straightforward. For example, for Ez(I, J, K) the inner summation is carried out from k = k′ to k′ + 2 and the two outer summations involve the four edges over the nodes i′, i′ + 1 and j′, j′ + 1.

Figure 1.

Geometry for restricting electric fields. Coarse-grid nodes are in the center of fine-grid cells. The gray arrow represents the electric field on the coarse edge comprising the fine-mesh cells i′, j′, k′ to i′ + 2, j′, k′.

[11] Sampling magnetic fields on cell facets leads to another arithmetic averaging scheme, with the weights defined by facet areas. Sketched in Figure 2, the coarse x component is, for example,

equation image
equation image

where the coefficients a of the 3 × 3 matrix

equation image

denote the fractions of the facet areas ΔyjΔzk contributing to the arithmetic average. The two inner summations are an integration of the fine-grid magnetic field over the coarse facet and are carried out twice, for the planes through the nodes i′ and i′ + 1, and averaged. Again, the terms for By and Bz are very similar. In this case, the outer summation is done over j and k, respectively.

Figure 2.

Geometry for restricting magnetic fields. The coarse magnetic field (gray arrow) is sampled in the center between two facets of the fine mesh.

2.2. Field Restriction at a High-Contrast Interface

[12] When modeling diffusion in an inhomogeneous medium, it might happen that the material properties of certain parts in the medium allow only little diffusion through this region. This is particularly true for regions with jumping diffusion coefficients, in our case regions with high contrasts in the electrical conductivity σ. Large jumps of σ over small grid regions cause a nonsmooth electric field. Appropriate grid transfer operators have been developed for problems with significant, for example, several orders of magnitude, contrasts in the diffusion coefficients [Alcouffe et al., 1981; Knapek, 1998]. Such a transfer operator would involve the Helmholtz decomposition of the electric field into an edge-based and a node-based potential field [Haber et al., 2002]. In principle, this would allow to exploit the continuity of the product of σ with the gradient of a scalar potential. Since we only treat models with moderate conductivity contrasts, this aspect shall not be further investigated in this paper.

[13] When simulating measurements on the Earth's surface, a rapid field variation across the air-Earth interface is to be expected, because this typically accounts for the largest conductivity contrast in a model. For example, in the simulations shown below, we use a contrast of up to 1:500 to approximate the conductivity difference between air and Earth. To treat the problem of an inaccurate field restriction due to discontinuous fields, one approach would be to adequately refine the initial mesh at the interface, providing a “buffer” zone at the discontinuity [White et al., 1997]. Instead, we have modified the restriction operator to account for a planar interface with jumping conductivity. This modified restriction scheme is shown in Figure 3 for a horizontal interface at the vertical node k′. Here, the coarse-grid design differs from the rest of the mesh by the position of the node K with respect to the fine mesh. Instead of shifting this node position to the cell center of k′, it remains on its original vertical position. Thus we separate the vertical electric field restriction into one above (K − 1) and one below (K) the interface. Specifically, below the interface we have

equation image

where w(k) = Δzklkk′+1.

Figure 3.

Geometry for field restriction in the presence of an interface separating two layers with a high conductivity contrast. The interface is the horizontal plane through k′.

[14] Equation (7) is the same for computing Ez(I, J, K − 1) above the interface; the only difference is that the two components of l (1.0,0.5) are then to be replaced by (0.5,1.0).

[15] Similarly, the horizontal magnetic field is computed from

equation image

Here, the weights w(j, k) involve only the first/last two columns of (6) when calculating the restricted field Bx(I, J, K − 1)/Bx(I, J, K). The corresponding term for By follows from exchanging the summation over i and j.

[16] With this modified scheme, the horizontal electric and vertical magnetic coarse-grid field sampling remains in the horizontal plane through k′. For example, the computation of Ex only involves fields on the node level k′,

equation image

Exchanging the summation over i and j leads to the equivalent term for Ey.

[17] Finally, the vertical magnetic field on the interface is given by

equation image

with w(i, j) = ΔxiΔyjaii′+1, jj′+1 and the same area fractions a given by (6).

2.3. Restriction of the Conductivities

[18] Our FDTD formulation requires that the electrical conductivities be computed on the same edge positions where electric fields are sampled. Applying Ampere's law to the staggered grid, the discrete curl of the magnetic field around an edge defines the area for a proper averaging of the cell-based conductivities σ to an edge-based directional conductivity σd, where d is the x, y or z direction. This is exemplified for σx in Figure 4a, for the case when both the simulation grid and the grid defining the material properties are equal. The effective conductivity σx(i′, j′, k′) is the arithmetic average of the four adjoining cells, connected by the magnetic curl loop around the x edge at i′, j′, k′, and is computed by [Wang and Hohmann, 1993; Alumbaugh et al., 1996]

equation image

with the weights w(j, k) = equation imageΔyjΔzk.

Figure 4.

(a) Geometry for the calculation of the effective material property σx(i′, j′, k′) on the initial grid's cell edges. (b) Averaging scheme to compute the restricted Σx(I, J, K) on the coarse grid.

[19] The larger integration area for the material averaging on the coarse grid is illustrated in Figure 4b, again for the x component. Calculating the effective conductivity Σd(I, J, K) on the coarse grid can be described in terms of a serial circuit of three parallel circuits of resistors. A parallel/serial circuit involves an arithmetic/harmonic average. Exemplified for the x component, it follows that

equation image

The two inner summations represent one parallel series of resistors in the y-z plane and involve the modeling grid cells bounded and intersected by the coarse-grid magnetic curl loop that is shown by the dotted rectangle in Figure 4b. In a uniform mesh, one has equation image = j′ − 1, equation image = j′ + 1 and equation image = k′ − 1, equation image = k′ + 1. However, in a nonuniform mesh, if the fine-mesh spacings between two coarse-grid nodes vary, the summations may start at equation image = j′ − 2 and end at equation image = j′ + 2; the same holds for equation image and equation image. The contribution of each cell conductivity of the inner average is weighted by the corresponding cell volume fraction dV(i, j, k), with Vi its sum. The reciprocal values of three such combinations in a series along i′, i′ + 1, i′ + 2 form the outer (harmonic) average along the length ΔXI of the coarse edge, with the corresponding weights w(i) as in (3). The scheme leads to the “integro-interpolation” method mentioned by Moskow et al. [1999].

[20] We also want to mention and will demonstrate below our FDTD scheme's flexibility to have the initial simulation grid different to the one defining the material properties. Computing the underlying effective conductivity σd for the initial simulation grid from an arbitrary material property grid, will involve the same averaging scheme exemplified by (12).

2.4. Restriction Criteria

[21] A critical issue is the determination of the time for changing to a coarser mesh. One wants to make sure that the restriction time is as early as possible to minimize computation time. On the other hand, a sufficient spatial sampling needs to be maintained. We have experimented with spatial fast Fourier transforms (FFT) to investigate the field's frequency behavior. Assuming that the highest frequency content of the field is adequately sampled by the initial mesh, we computed the amplitude spectrum corresponding to the frequency range from equation image to fmax, where fmax is the grid's maximum spatial sampling frequency. With the diffusion of the field, the decrease of the amplitude spectrum below a predefined level may serve as a criterion to double the mesh spacing. To be accurate, this would involve a 3-D FFT over the whole grid range, which might become computationally prohibitive for large meshes. A spatially limited FFT over selected model regions may vary depending on the material distribution in the model; for example the field decay is slower in conductive regions. Here, we do not further pursue this approach.

[22] An alternative approach uses the computationally cheap field restriction, evaluating the error term (in percent)

equation image

Equation (13) is an average of the differences between the total electric fields, ∣ef∣ on the fine mesh and ∣ec∣ on the coarse mesh, evaluated at each receiving position i by linear interpolation. N is the number of receiver locations to be simulated. Weighting by the fine-grid field accounts for the field decay with time.

[23] It is demonstrated for the simulation examples in the next section how a certain percentage threshold may be employed as a restriction criterion. We simulated a borehole scenario and two surveys on the Earth's surface, referred to as surface model 1 and 2. With more details given in the next section, Figure 5 shows the run of the error curve with progressing simulated time for each model. We evaluated (13) periodically on a predefined time raster in order to decide for a grid changeover, where a threshold of 1% (borehole model) and 2% (surface models) was chosen. The restriction times are marked by the dotted lines on the time axis. The borehole example and surface model 1 involve a two-grid scheme, with one coarse grid in addition to the initial one. Two coarse levels and thus three simulation grids were used for the surface model 2. Therefore the corresponding curve shows a jump after the first restriction at 2 × 10−3 s. Afterward, ∣ef∣ and ∣ec∣ in (13) are computed on the first and second coarse-grid level, respectively.

Figure 5.

Error between the total electric field on the initial and coarse grid, computed using (13), for the three models. The vertical dotted lines on the time axis denote the times of a grid changeover. The jump in the curve of the second surface model is owing to the second restriction at this time.

3. Results

[24] In the following, we show the simulations of three different models, representing measurement configurations that might be typical in the field. Figure 6 shows the setup for the first example. The inductive transmitter is a magnetic dipole with vertical loop axis. The transmitter signal is a shutoff in a stepwise fashion. A total of N = 104 receivers are distributed over eight wells surrounding the source position at the origin. Two layers (0.1 S/m and 0.02 S/m) with a curved bounding interface form the model background and two rectangular conductive (2 S/m) blocks are embedded into the upper layer. The initial grid has 71 × 71 × 71 cells and is nonuniform with a smallest spacing of 1 m in the center part that gradually increases to 200 m toward the outer boundaries. The simulated measurement time ranges from 0.01–1 ms. Shown in Figure 5, after an initially exponential decrease, the error err(N) starts to converge after ≈1 × 10−5 s. After the grid changeover, which was carried out at 0.1 ms, the coarse grid comprises 34 × 34 × 34 cells.

Figure 6.

Borehole simulation with a curved layered background and two conductive anomalies above the curved layer boundary. The magnetic dipole transmitter is located at the origin.

[25] For a comparison, we use data calculated by the spectral Lanczos decomposition method, using the time domain version of the 3-D solver presented by Druskin and Knizhnerman [1994]. The solution approximates Maxwell's equations on a spatial staggered grid, resulting in a system of ordinary differential equations. The matrix functional representing the time domain solution is a product of an exponential function of the spatial finite difference operator and the vector describing the initial conditions.

[26] In Figure 7, the SLDM and FDTD simulations are plotted together for nine selected sensor positions, marked by solid rectangles in the bottom part of Figure 6. Shown are the electric field voltages for the x component. The shapes of the transients at position a are characterized by an exponential decay with time and are representative for most of the locations in the receiver wells. More complicated field responses arise at positions b and c, because of the close conductive anomalies. Here, the field decay is distorted by sign reversals, with the most observed at the position x = 25.5 m, y = 0.5 m, and z = −15 m. For all receiver positions, we obtain a good agreement between the two solutions. Using (1), the initial time step is set to 2 × 10−8 s. This amounts to a total of 5954 time steps, compared to 9296 time steps for the constant-grid scheme.

Figure 7.

Comparison between the SLDM and FDTD electric field solutions for the receiver positions denoted by a, b, and c in Figure 6. Shown is the x component. Field restriction was carried out after 1 × 10−4 s (vertical dotted line). Close to the anomalies at receivers b and c, sign reversals occur, marked by the signs in the graphs.

[27] The second model example simulates a survey on the Earth's surface and thus involves an air-Earth interface. Sketched in Figure 8, the fields are sourced by a 50 m long grounded wire over a half-space (0.01 S/m). The half-space contains two near-surface bodies (0.05 S/m and 0.005 S/m) at a depth of 50 m and a deeper dipping structure (0.1 S/m). Note that the contours of the structures are projected to the sections. For this model, a conductivity of 2 × 10−4 S/m is chosen to approximate the infinitely resistive air space at z < 0 m. The field responses are extracted at three profiles along the x direction at y = −100 m, y = 0 m and y = 75 m, comprising 114 receiver locations.

Figure 8.

Field setup and model for a survey example on the Earth's surface (z = 0 m). The model structures are projected to the (a) x-y plane and (b) x-z plane. The transmitter at the origin is a grounded-wire horizontal electric dipole. The background half-space has a conductivity of 0.01 S/m and contains three different anomalies. Three profiles along the x direction at y = −100, 0, 75 m are simulated.

[28] In Figure 9 the simulated data is exemplified for three receiver positions, one close to the transmitter and two close to the near-surface blocks (a, b, c in Figure 8a). The transients are the voltage curves for the electric field's y component parallel to the transmitter line (top graphs) and the vertical magnetic induction time derivative equation image (bottom graphs). In this example, the comparison fields for a constant grid (solid lines) were also calculated with the FDTD code, using a very fine model grid with 97 × 89 × 69 cells. We used such a fine grid to ensure accuracy and to allow modeling the fine structures given by the dipping conductor. The initial grid for the MG results (dotted curves) has 36 × 36 × 32 cells. Its spacings range from 15 to 1000 m. In the area around the dipping conductor, the horizontal grid spacings are larger than the finest model structures. Similarly to the restriction of the material properties, outlined in 2.3, we use the same material averaging procedure to initialize the underlying effective conductivities for this example. After restriction, carried out at 2 × 10−4 s, the simulation is continued with a mesh size of 17 × 17 × 15 cells. In view of this relatively coarse mesh, both the electric and the equation image responses show a good agreement to the fine-grid solution.

Figure 9.

FDTD results computed from a (constant) fine mesh with 97 × 89 × 69 cells compared with the MG solution for a restriction after 2 × 10−4 s. Shown are (top) the electric field responses for the component parallel to the transmitter and (bottom) the vertical magnetic induction time derivatives. The letters a, b, and c refer to the sensor locations in Figure 8a.

[29] The third presented example is another survey simulation on the Earth's surface, similar to the previous one, and is shown in Figure 10. Here, the stepwise source signal is generated by a rectangular loop with size 25 × 25 m2 over a 0.1 S/m background, where the air layer is again set to 2 × 10−4 S/m. The two shallow anomalies both have a conductivity of 0.005 S/m, and the dipping structure has 1 S/m. The responses on three profiles of 1500 m length, including 144 receivers, are simulated for a measurement time range of 5 × 10−4 − 1 s. Figure 11 shows the voltages of the electric field parallel to the y direction (top graphs), vertical magnetic induction Bz (middle graphs), and its time derivative equation image (bottom graphs) for four time points along the profiles. At 0.002 s, after the average error (13) has dropped below 2%, the first restriction was carried out. Afterward, the error err(N) converged toward a slightly higher value of 2.3%. Therefore we used an alternative criterion for the second grid changeover. This is given by the convergence of (13) within a certain number of time steps. Here we have chosen a time range of 10000 time steps, leading to the second changeover at 0.05 s.

Figure 10.

Field setup and structure for the second surface model. The background half-space has a conductivity of 0.1 S/m and contains three different anomalies, which are shown in a projection to the (a) x-y plane and (b) x-z plane. A magnetic source at the origin generates the fields simulated at the three receiver profiles (dotted lines) along the x direction.

Figure 11.

FDTD results computed from a (constant) fine mesh with 136 × 128 × 96 cells compared with the MG solution using two coarse-grid levels. Shown are (top) the electric field Ey, (middle) magnetic induction Bz, and (bottom) the time derivative of the vertical induction equation image. The letters a, b, and c refer to the profiles in Figure 10a.

[30] Deviations can be observed for the electric field curves after the second restriction (0.1 and 1 s). Compared to the fine-grid solution, the second coarse grid appears to smooth the responses due to the shallow anomalies, observed around 300 m and 100 m from the source on profiles a and c, respectively. This can be explained by the size of the anomalies with respect to the spatial sampling rate of the grids. For example, while the first coarse grid's vertical sampling of the region below the surface, with nodes at 0,15,35,55,75 m, is still sufficient to reproduce the anomalous responses, the second coarse grid, with nodes only at 0,25,65 m, causes a smoothing effect. Nevertheless, in a real TEM survey study, this may be of no further concern, since the information of such shallow structures are primarily contained in the early time responses.

[31] Table 1 summarizes the computational requirements for the three simulations. For the borehole model, we used eight (1 GHz) processors of a parallel cluster and achieved a factor of 3.2 decrease in computing time. The surface model 1 was calculated on a single CPU (1.5 GHz). Because of the relatively coarse initial mesh, a smaller speedup factor of 2.6 is achieved for this example. The largest performance increase, a factor of 5.3, is achieved for the second surface model, owing to the late simulated time of 1 s. Here, we used 16 (3.6 GHz) processors of another parallel cluster.

Table 1. Summary of Modeling Parameters and Computational Requirements for the Simulated Models
 Borehole ModelSurface Model 1Surface Model 2
Initial time step size t0, s2 × 10−81 × 10−71 × 10−7
Latest simulation time, s1 × 10−31 × 10−21.0
Restriction time, s1 × 10−42 × 10−42 × 10−3
5 × 10−2
Error level, %122
Minimum/maximum grid spacing, m1/20015/100010/250
Initial grid size71 × 71 × 7135 × 35 × 31135 × 127 × 95
Reduced grid size34 × 34 × 3417 × 17 × 1567 × 63 × 47
33 × 31 × 23
Time steps, constant grid929619541306634
Time steps, multigrid595413645144353
Computation time, constant grid22 min, 10 s21 min, 17 s13 hours, 24 min
Computation time, multigrid7 min, 2 s8 min, 8 s2 hours, 30 min
CPU speed, GHz11.53.6
Number of processors8116

[32] To conclude, we can state that the presented method is particularly useful for simulation scenarios, where one is interested in accurate early time results that might involve a very dense and thus computationally demanding initial mesh, while also a large time range shall be simulated. The presented results have shown that the solutions remain stable after the grid changeover. To demonstrate the necessity of an initially fine mesh for the shown examples, we have carried out simulations with the grids of the first coarse level used from the beginning. The results are exemplified in Figure 12 for all three models. The coarse mesh for the borehole model is not capable of reproducing the complex electric field responses at receivers b in the vicinity of the conductor. However, as shown before (Figure 7), it is adequate for the later part of the transient, when used in combination with the fine mesh. The 17×17×15 nodes of the coarse mesh for the surface model 1 are still sufficient to reproduce the later part of the electric field transient at receiver b, yet the earliest times show significant differences (b). Similarly, as exemplified by the shifted sign reversal, the coarse grid of the second surface model does not produce accurate early time responses for equation image at some locations, while the later times are reproduced correctly (c). Here, we verified the fine-mesh response using a SLDM solution.

Figure 12.

Simulation responses using the grids of the first coarse level for the (a) borehole case, (b) first surface model, and (c) second surface model.

[33] Finally, a computational speed comparison between SLDM and MG-FDTD for the borehole simulation example is provided. Several aspects are worth mentioning for a reasonable comparison. First, since we do not have a parallel implementation of SLDM, both solutions are produced on a serial computer. Second, we assume to be only interested in the response at a single receiver, thus the FDTD mesh size could be further decreased to a mesh of 63 × 63 × 63 cells, which is the same size as used for SLDM in this example. At last, both codes were generated using the same compiler options. For this example, the MG-FDTD solution requires 31 min of computing time, versus 22 min for SLDM.

4. Conclusions

[34] We have extended the FDTD time-stepping method to reduce the spatial and temporal sampling rate with progressing simulation time. This makes use of the fact that the TEM field is dominated by lower frequencies at a later stage after field excitation. Significantly decreased computation times result from a reduction of the total number of time steps and a smaller number of grid nodes. We have demonstrated the capabilities of the method at three different survey simulations, representing typical scenarios in geophysical exploration.

[35] A simulation grid needs to be individually designed in order to accurately simulate the responses due to model inhomogeneities for a given time range. The time step size strongly depends on the mesh spacing. Its strong limitation in order to retain a diffusive solution accounts for the expensive computational requirements of the explicit FDTD scheme. In conjunction with its parallel implementation, the presented scheme improves the performance of FDTD, particularly when simulating late measurements. While a very fine initial grid may be employed in order to account for models with finely structured regions of interest, changing to a coarser at later simulation times avoids unnecessary time steps with the fine mesh.

[36] This exploratory study has been carried out to investigate the potential of improving the speed of finite difference schemes by using series of coarser meshes. We note that design of adequate simulation grids for the presented model problems has been based on our experience. Obviously, there exists a further potential in using optimized grid refinements, such as presented by Ingerman et al. [2000], particularly for well logging simulations, where sources and receivers are usually within a confined area.

[37] Another topic of ongoing research includes the incorporation of the presented scheme into an inversion framework. In a time domain inversion study [Newman and Commer, 2005], we employed data migration concepts in conjunction with FDTD, in order to invert data from a large number of spatially distributed receivers for realistic models. One inversion iteration involves one forward and one adjoint solution for the necessary data sensitivities, and several forward solutions for the line search procedure, carried out during the minimization of the global error functional. Thus forward simulations dominate the inverse solution, which accounts for a large potential to speed up inversion schemes using a sequence of coarser grids.

Acknowledgments

[38] We wish to thank the German Alexander-von-Humboldt (AvH) Foundation for support of this work through a Feodor-Lynen research fellowship granted to Michael Commer. We are indebted to Leonid Knizhnerman and two anonymous referees for improving the quality of the manuscript. We also thank Leonid Knizhnerman and Vladimir Druskin for providing a temporary code license of the SLDM code. This work was carried out at Lawrence Berkeley National Laboratory, with base funding provided by the United States Department of Energy, Office of Basic Energy Sciences, under contract DE-AC02-05CH11231.

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