T. G¨unther, Leibniz Institute for Applied Geosciences, Hannover, Germany
C. R¨ucker, University of Leipzig, Germany
We present a new joint inversion approach that allows for a combined inversion of independent physical parameters by exchanging structural information. The technique is based on the ideas of robust modeling. Occurring gradients of one parameter facilitate the development of gradients in the other but does not enforce this. In the presence of boundaries that can be seen by both methods it leads to sharp contrasted models. Finally, a combined image of the subsurface is obtained by cluster analysis.
The technique is applied to the inversion of dc resistivity and seismic refraction data. Two synthetic data sets show how different boundary types are resolved with and without structural coupling. It is demonstrated how the quality of the inversion results is improved by the new approach.
The joint interpretation of different measurement types is a basic principle to confine the ambiguity of the inverse problems in geophysics. In the existing approaches one inverts for one parameter using an augmented data vector (Vozoff and Jupp, 1975). This is only possible if all measurements depend on the same parameter or if the parameters are interconnected by some petrophysical relationship. However, often such a relationship does not exist, as for electrical conductivity and seismic velocity. Nevertheless, we expect at least similar structures in the resulting models. Therefore we like to combine the otherwise independent inversions to allow for structural similarities.
The challenge is to facilitate similar structures without enforcing it. Gallardo and Meju (2004) presented an algorithm where a combined data functional based on the cross-gradients of both models is minimized. However, one problem is the weighting of the individual data and model updates for different data numbers and convergence properties. Here we present an approach where two inversion runs are carried out separately. The combination of both models is accomplished by mutually controlled structural weights based on the principles of robust modeling.
First we describe the minimization problem and the regularization procedure. Thereafter we introduce robust modeling and depict how structural information may be interchanged. We apply this technique to the dc resistivity and seismic refraction data on unstructured meshes. Two synthetic models using different model boundaries shows how the joint inversion improves the model concept.
Minimization procedure
Assume D data points d i subsumed in a data vector discretized by M basis functions ψ j d = ( and their coefficients d m
1 j
, . . . , d
D
,
) . The parameter distribution p ( ~ ) is p ( ~ ) =
M
X m j
ψ j j =1

the m j yield model vector m = ( m
1
, . . . , m
M
) . The vector containing the forward response of a model m is denoted by f ( m ) . We try to minimize the discrepancy between data and model response with respect to a certain norm, i.e. the `
2
-Norm yielding the method of least squares.
To account for different data accuracy, their errors i are used as a weighting function yielding the minimization of
Φ d
( m ) =
D
X i =1 d i
− f i
( m )
2
= k D ( d − f ( m )) k 2
2 i with D = diag(1 / i
) .
(1)
Beginning from a starting model m 0 , a new model m k +1 is obtained by an iterative procedure m k +1
= m k
+ ∆ m k
.
We choose the Gauss-Newton algorithm to determine ∆ m k due to its fast convergence. Additionally the model has to be constrained to yield a unique solution.
Regularization
We introduce a model functional Φ m which is weighted by a regularization parameter λ
Φ = Φ d
+ λ Φ m
← min .
To find a general formulation we build the model functional by three matrices
Φ m
= k W c
Cm k 2
2
.
(2)
The constraint matrix C is generally a discrete derivation matrix. Possible is a first order, second order or zeroth order (identity matrix). The coefficients depend on the element distances and may be varied by the choice of reference points or boundary conditions (2nd order). Also, for unstructured grids such a matrix may be constructed.
Generally C is skewed, i.e. for a 1D parameterization C ∈ R ( M − 1) × M
. The very popular 1 st constraints using in the 3D case C T x
C x
+ C T y
C y
+ C T z
C z may be represented by order
C =


C x
C y
C z


, C x
∈ R
( M x
− 1) × M y
× M z etc.
w c i
The constraint weighting matrix W c is generally diagonal W c
= diag w c i
. The individual elements describe penal factors for the different model cell boundaries. For example, the model characteristic with respect to horizontal and vertical structures may be controlled. Very small values allow for sharp boundaries which may be known from boreholes or other methods (as GPR or reflection seismics). Furthermore, other tomographic methods may provide structural information by using the roughness distribution of one parameter to control the weights of the other.
Robust modeling
One may use the structure of m k to reinforce structures. This proceeding is usually applied to data weighting (Claerbout and Muir, 1973) but can also be applied to the model gradients. In fact, the weighting is determined such that an `
1
(Claerbout and Muir, 1973).
minimization is simulated which is referred to as robust modeling

Let c = CW m portions of c l in the `
1
( m − m 0 ) be the roughness vector. The weighting term w l c and `
2
-Norm is determined by the w c l
=
| c l
|
1 k c k
1
1
| c l
| 2 k c k
2
2
=
P c
| l c
| i
|
2 l c
2 i
=
P c
2 i
| c l
| P | c i
|
.
(3)
To allow for strong contrasts step-by-step, all w l c
> 1 are set to 1. Thus, we obtain models with more contrast by introducing an additional non-linearity. However, in some cases the weight can reach zero and lead to absurd parameters.
Instead of using the structure of m k for m k +1
, we apply the latter procedure to introduce structural information to a parameter vector p . Thus, the weights w c l from c = CW c ( m k − m 0 for determining
) . Simultaneously, the gradients of p k p k +1 are calculated after (3) are used to obtain the constraint weights for m k +1
.
The joint inversion scheme is illustrated in Figure 1 for DC and refraction tomography.
dc resistivity refraction tomo
DC Data Mesh Ra Data rho0
Constraint
Matrix C v0 rho1 C_rho C_v v1 rho2 rhon v2 vn clustermodel
Figure 1: Scheme of the joint inversion procedure for combined dc resistivity and refraction tomography
Note that we assume m and p existing on the same grid. If not, we have to introduce appropriate transformation operators. From both data sets a mesh incorporating topographical information is created.
Electrode positions and geophones/shotpoints must be included as node points. Then we choose a starting

resistivity model ρ
0 and velocity model derive the first iteration models ρ
1 and v v
1
0
. An appropriate constraint matrix independently.
C is created which is used to
In the next step the resistivity model rho1 is used to alter the smoothness matrix used for velocity inversion, C v
. Analogously, the smoothness matrix C
ρ for resistivity inversion is built up based on the velocity. Thus the exchange of structural information is achieved by the created matrices. We proceed with this until convergence is achieved, resulting in the final models ρ n and v n
.
Implementation
We used the above depicted minimization procedure, once with coupling and once without. For the resistivity method the data were the logarithms of the apparent resistivities, the model parameter we chose the logarithms of the cell resistivities to ensure positivity. For the forward operator and the sensitivity calculation the techniques of G¨unther and R¨ucker (2005) have been applied.
Data of the refraction part have been the first arrival travel times. The model parameter are the logarithmized velocities. To solve the forward problem we used a Shortest Path algorithm (Dijkstra, 1959) that restricts the ray paths to the model edges. Although this is only accurate for very fine meshes, it is sufficient to prove the concept.
Finally, a cluster analysis may help to find a model that combines both parameters. For this both inversion parameters represent the input. Additionally, the geometric position of the cell, i.e. x and z of the midpoint, is used to find contiguous clusters. In our experience, clustering based on euclidean distances and complete linkage provided appropriate results.
A synthetic example
In the following we like to test the approach using a very simple rectangular model. Figure 2 shows the synthetic model used consisting of two layers with an offset in the center of the model. Note that both resistivity and velocity show the same structure based on an rectangular mesh. To accomplish an appropriate forward calculation for the refracted rays, the mesh is refined by dividing each rectangle into
4 triangles.
15
0
−15
2 z/m
6
−10 −5
ρ
=500
Ω m, v=500m/s
0 5 x/m
ρ
=1000
Ω m, v=1000m/s
Figure 2: Simple synthetic example containing identical velocities and resistivities
First we invert the noisified data seperately, i.e. without structural coupling. The individual regularization parameters are selected such that the data are fitted within noise. Figure 3 shows on the left side the resulting models. Both exhibit the main features but resolve the sharp boundaries only by smooth gradients which are smoothest in regions of low resolution. The jump is expressed as a slope and artifacts can be seen in the velocity model.
Now we introduce structural coupling yielding the inversion results at the right side of Fig. 3. The boundaries, particularly the vertical ones, are reproduced much clearer which is due to the robust modeling background. To obtain a unified model combining both parameters, we apply a cluster analysis, whose

0
−15
2 z/m
6
501
−10
562
−5
631
0
708
5
794 x/m 15
0
−15
2 z/m
6
Ohmm 1000 501
−10
562
−5
631
0 5 x/m 15
708 794 Ohmm 1000
0
−15
2 z/m
6
−10 −5 0 5 x/m 15
0
−15
2 z/m
6
−10 −5 0 5 x/m 15
501 562 631 708 794 m/s 1000 501 562 631 708 794 m/s 1000
Figure 3: Inversion results (resistivity on top, velocity at bottom) for the synthetic model of Fig. 2 for separate inversion (left) and joint inversion (right) results are shown in Fig. 4 for two clusters. The structures and also their resistivities and velocities do almost coincide with the synthetic model.
15
0
−15
2 z/m
6
−10 −5
ρ
=486
Ω m, v=522m/s
0 5 x/m
ρ
=1102
Ω m, v=1018m/s
Figure 4: Model obtained by cluster analysis of the joint inversion results.
A more sophisticated example
We now want to investigate a more complicated model. At first we introduce a topography with a slope connecting two different levels. Consequently, triangular meshes are used to describe the model parameters. Figure 5 shows the synthetic model used for creating the data. It consists of two layers with an additional unit in between. Its velocity equals that of the upper layer, whereas the resistivity is identical to the one of the lower layer. Thus, common (layer) boundaries and different boundaries (of the red block) are present. Moreover, resistivity and velocity show different ranges and the direction of the gradient is oppositional.
0
−15 −10
−5
−5 0 5 10 15 20 v=500m/s,
ρ
=50
Ω m v=2000m/s,
ρ
=50
Ω m v=500m/s,
ρ
=200
Ω m
−10
25
Figure 5: Synthetic model with three units consisting of 2 different velocities and resistivities.

We assumed 41 electrodes/geophones with 1 m spacing between -15 and 25 m. A complete Wenner pseudo-section was applied to create the dc data. For the refraction data every 5th geophone was used as shot point with receivers at the other. Synthetic data have been created using a highly refined quality model to avoid errors. Additionally Gaussian errors of 3% (for dc data) and 0.5 ms (Ra data) standard deviation was added.
The inversion has been carried out once without and once with coupling. In Figure 6 the inversion results are displayed for separate (left) and joint (right) inversion. Both methods show the different structures of their synthetic models in the separate inversion. All existing gradients are quite smooth due to the smoothness constraints, especially near the boundaries. The resolution of the resistivity data is too low at depth to resolve the resistive bedrock well.
0
−15 −10 −5 0 5 10 15 x/m 25
0
−15 −10 −5 0 5 10 15 x/m 25 z/m z/m
−10 m/s
501
−15
0
631
−10 −5
794
0
1000
5 10
1259
15
1585 x/m
−10
1995 m/s
501
25 −15
0
631
−10 −5
794
0
1000
5 10
1259
15
1585 x/m
1995
25 z/m z/m
−10
Ω m
50
−10
200
Ω m
50 63 79 100 126 159 63 79 100 126 159 200
Figure 6: Inversion results (velocity on top, resistivity at bottom) for the synthetic model of Fig. 5 for separate inversion (left) and joint inversion (right)
Using the joint inversion (right hand side) the gradients become much sharper where resistivity and velocity have identical boundaries. At the single boundaries the gradients remain smooth and do not cause model boundaries in the other parameter. In regions of missing resolution as the right boundary the joint inversion cannot improve the results.
A final image of the combined inversion is obtained by a cluster analysis. We used Chebychev
(interior) distances and an average linkage method and restricted the number of clusters to 3 as in the synthetic model. Figure 7 displays the final model.
0
−15 −10 z/m
−5 0 5 10 15 x/m v=536m/s,
ρ
=197
Ω m v=1567m/s,
ρ
=50
Ω m v=807m/s,
ρ
=88
Ω m
−10
25
Figure 7: Final model obtained by cluster analysis of the joint inversion velocity and resistivity results.

Clearly the upper layer can be traced, their parameters coincide very well with the synthetic model
(Figure 5). Similarly this holds for the lower layer, although their velocity is underestimated. The intermediate block matches quite well with the synthetic model in the central part. However, both velocity and resistivity is overestimated. Near the right boundary the resolution is too low at depth to image the course of the layer boundary well. To do so, additional electrodes/geophones are needed.
We presented a new approach for the combined inversion of independent parameters by structural exchange based on the ideas of robust modeling. The experiments on synthetic models shows that at coinciding boundaries the ambiguity is reduced and the image becomes more contrasted. This is mainly due to the underlying robust modeling. On the other hand, boundaries that can only be seen by one parameter remain unchanged by the coupling process. Finally, a cluster analysis can provide a clearer view of the model structure.
In order to improve the accuracy, the Shortest Path algorithm may be replaced by solving the
Eikonal equation with Finite Elements. To meet an arbitrary topography, the inversion will be done on unstructured meshes, which provide the highest flexibility. Finally, the technique must be applied to field data to prove its applicability.
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Gallardo, L. A. and Meju, M. A. (2004). Joint two-dimensional dc resistivity and seismic travel time inversion with cross-gradients constraints.
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, 109:03311.
G¨unther, T. and R¨ucker, C. (2005). A triple-grid technique for the 3d inversion of dc resistivity data incorporating arbitrary topography. In EAGE Near Surface Conference , Palermo, Italy.
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, 42:977–
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