Simultaneous and joint inversions

The simultaneous inversion is implemented such that the same numerical kernel is used for inversion in different physical domains and different data types. Gyulai and Ormos (1997 and 1999) developed a simultaneous Series Expansion (SE) inversion of DC sounding curves with power and periodical basis functions (referred to as 1.5-D inversion). M. Kis (1998) applied simultaneous and joint SE inversion for the interpretation of DC geo-electrical and seismic refraction data with examination of the possibility of improvement to the approximate 1-D forward modelling applied in SE inversion methods and introduction of the integral mean concept to the SE inversion. Simultaneous and joint SE inversions have also been applied by other authors (such as Herwanger J.V. et al, 2002) for the interpretation of geo-electrical and seismic data, although with the use of a different approach. Their approach is also interesting because they consider an anisotropic and inhomogeneous media. Finite elements are used to discretize the anisotropic Laplace equation governing the forward problem. The inverse problem is solved using a variant of the popular Marquardt-Levenberg algorithm, with additional terms for smoothness, structural and anisotropy constraints.

(

)

(

⁽

⁾ )

where ∆m denotes model updates, J is the Jacobian, W is the data covariance matrix, C is the model covariance matrix and M is the matrix controlling step length. Matrix C contains the structure and anisotropy penalty.

In order to solve large-scale problems, parallel computer and domain decomposition techniques were used. Data from an electrical tomographic study between boreholes at a hydrological test–site were compared to the results with an anisotropic seismic tomography study carried out at the same location. Both the electrical and the seismic experiments scan a depth interval of 20–115 metres between two wells spaced at 25 metres. The number of data is approximately 8,000 for each survey and the sub-surface in the inter-well region is discretized in elements of approximately 1.5 metres in both x and z directions.

Figure 54. Anisotropic resistivity tomogram.

In the left image average resistivity is displayed, while on the right electrical anisotropy is shown.

Figure 55. Anisotropic velocity tomogram. In the left image average seismic velocity is shown, while the right image displays seismic anisotropy ε.

According to the authors, a comparison of anisotropic seismic velocity distribution and electrical conductivity distribution shows an extraordinary correlation between the two tomograms (Figures 54 and 55). Both methods clearly delineate an anisotropic body of highly layered and fractured siltstones underlain by an isotropic sandstone body. Zones of fractured rock and zones of highly layered sedimentary rock both result in electrical and seismic anisotropy.

Musil et al applied a joint inversion in order to overcome the limits of typical 1-D geo-electrical inversion (Musil et al, 2003). They found that it often has internal non-uniqueness and ambiguity problems. This is because source and receiver arrays are usually restricted to the surface or a small number of shallow boreholes and critical parts of the target media may be only sparsely sampled, resulting in ambiguities in the tomographic inversions. To compensate for the limitations of the recorded data, additional constraints are generally required. An efficient way to overcome internal ambiguities is the use of the joint inversion, which means the integration of various groups of data records (arising from physically or geometrically different methods and surveys) into a single inversion algorithm (Musil et al, 2003). Originally, the joint inversion algorithm was introduced by Vozoff and Jupp for magnetotelluric (MT) and DC resistivity data. These difficulties can also be reduced be by various regularization procedures. One option is to assume that spatial variations of the sub-surface physical properties are smooth. This may be implemented using an inversion algorithm that minimizes the curvature of the model space. A potential disadvantage of such a procedure is that the resultant images may be blurred and important small scale features may remain unresolved. Another way of compensating for sparse data is to introduce a priori information in the form of damping. In this approach, model parameters are not allowed to deviate greatly from a given starting model. Clearly, this requires that the starting model should be a close representation of the true sub-surface structure. Although smoothing and damping are powerful mathematical tools, it is much better to minimize the ambiguities by applying appropriate data constraints. This has led to the concept of joint inversions, whereby different types of data are inverted simultaneously (Vozoff & Jupp 1975). A necessary requirement for a joint inversion is to have a factor that is common to the two data sets. The most straightforward approach is to invert data sets that are sensitive to the same physical property. For example, direct-current electrical resistivity and electromagnetic data are both sensitive to electrical resistivity. A variety of studies have demonstrated the substantial reduction in ambiguity that may result from joint inversions (Vozoff & Jupp 1975). It follows from the paper of Musil et al (2003) that jointly inverting data sets that are sensitive to different physical properties is a more difficult problem. Coupling of the two data sets must involve common structural elements. In 1-D applications, the common elements may be layer thicknesses. This concept can be extended to 2 and 3-D data sets, as long as the targets can be represented by different physical models with common geometries. Besides smoothing, damping and joint inversion, a further option is open for reducing model ambiguity: a priori knowledge may enable the model parameters to be restricted to a few narrow ranges of values. If this type of information can be included in an inversion algorithm, the model space and thus the ambiguities, can be significantly reduced relative to standard least-square inversions that allow the model space to be continuous and unlimited.

Herrman R. B. et al have shown that teleseismic P-wave receiver functions and surface-wave dispersion measurements can be employed to infer simultaneously the shear-wave velocity distribution with depth in the lithosphere. Receiver functions are primarily sensitive to shear-wave velocity contrasts and vertical traveltimes and surface-shear-wave dispersion measurements

are sensitive to vertical shear-wave velocity averages, so that their combination bridges resolution gaps associated with each individual data set. The inversions are performed using a joint, linearized inversion scheme which accounts for the relative influence of each set of observations and allows a trade-off between fitting the observations and the smoothness of the model. Additional constraints on mantle structure are also incorporated during the inversion procedure since requiring the data to blend smoothly into an appropriate deep structure affects the estimate of the lower crust velocities, yielding models which are more consistent with expectations than those resulting from unconstrained inversions. The authors found that a priori knowledge of upper mantle velocities are required to predict the dispersion up to a 50-second period and that stability constraints are required. When dispersion is limited to periods greater than 15 seconds a priori information on the upper crustal velocities may also be required. The results of applying this technique to data from different tectonic environments in the Arabian Plate and North America are presented in the paper. A "jumping" algorithm is employed to jointly invert receiver functions and surface-wave dispersion observations for shear-wave velocity. The jumping scheme allows smoothness constraint to be implemented in the inversion by minimizing a model roughness norm that trades off with the goodness of fit.

The goodness-of-fit criterion takes into account the different units, magnitudes, noise and number of observations of the data and enables an influence parameter 'p' to be set before inversion to balance the relative importance of each data set of observations. In particular, a value of p=0 only uses the receiver function data and a value of p=1 only uses the dispersion data. The system of equations to be inverted is given by

x0

where D_r and D_s are partial derivative matrices for the dispersion measurements and the receiver function estimates respectively, rs and rr are the corresponding vectors of residuals, x is the vector of S-wave velocity, x0 is the starting model, and A is a matrix that constructs the second order difference of the model x. The factor q equals 1-p and the factor s balances the trade-off between data fitting and model smoothness. Additional a priori information is required to stabilize the results of the models in the upper mantle. One possibility is to require the deepest layers in the model to be similar to predetermined values, such as PREM. This can be achieved by adding the following set of equations to the original system

Wxa

Wx= (2.36)

where W is a diagonal matrix of weights and the vector x_a contains the a priori predefined velocity values.

Finally, the authors conclude that the combination of surface wave dispersion data and receiver functions provides constraints on the shear velocity of the propagating medium that improve those provided by either of the data sets considered separately and helps to avoid over-interpretation of single data sets.

The problem of solution appraisal, mentioned in the Introduction, is considered in many papers, including that by van Wijk et al (van Wijk K., et al, 2002). Moreover, Sharma and Kaikkonen (Sharma S. P. and P. Kaikkonen, 1999) discuss the problem of appraisal of equivalence and suppression problems in 1-D EM and DC measurements using global

optimization and joint inversion. According to these authors, individual inversion of the EM data set can resolve a conducting layer reasonably well but it fails when the layer is either thin or resistive with respect to the surroundings. On the other hand, the individual inversion of the DC resistivity data suffers from an inherent equivalence problem. In general, when a thin conducting layer is encountered, inversion results resolve the product conductivity × thickness or resistivity × thickness, rather than the exact values of conductivity and thickness separately.

Further, when a middle layer has values for the physical parameters between those of the overlying and underlying layers, then the presence of such layers is suppressed in the data.

Various researchers have provided a theory for the computation of forward responses of the horizontal co-planar coil system over stratified earth. Following the digital linear filtering approach to the computation of resistivity sounding curves, filter sets have been designed to compute EM sounding curves for various dipole-dipole configurations. The expression for the mutual impedance ratio for a horizontal co-planar coil system can be written as a convolution integral

where r is the distance between the transmitter and the receiver, J0 is the Bessel function of zero order and R() is the complex EM kernel function. The input and filter functions are given in the first and second brackets, respectively. The ratio of mutual impedances can be computed easily with the help of filter coefficients developed by Koefoed et al. (1972). In the following, the phase ϕ of the mutual impedance obtained from

( )

is considered as a representation of the EM response.

Similarly, the convolution form of the relationship has been developed for the apparent resistivity measured using the Schlumberger array

( ) _∫

( ) [

( ) ]

where T(y) is the resistivity transform and J1 is the Bessel function (further details of the original relationship are given in the sub-section entitled The deterministic approach).

The resistivity transform is the input of the filter and the second term in the above integral is the filter function. The two integrals complete a forward problem. Joint inversion is carried out using the following objective function, combining the EM phase data and DC apparent resistivity data

where ϕ_i^0, ϕi^c are the observed and computed phases, while ρ_i⁰ ρi^c are the observed and computed apparent resistivities respectively. NF and NS are the numbers of frequencies and observation points in the EM and DC measurements respectively.

Figure 56. The H-type model. From left to right: comparison between observed and computed responses, resistivity versus depth sections and inverted h2 versus ρ^{2 results,} arising from ten very fast simulated annealing runs after inversion of the following: phase (a), (b), (c); apparent resistivity (d), (e), (f); both data sets together (g), (h), (i). (source: Sharma S. P. and P. Kaikkonen, 1999, Appraisal of equivalence and suppression problems in 1-D EM and DC measurements using global optimization and joint inversion, Geophys. Prosp., 47, pp. 219 –249)

According to the authors, the study reveals that global optimization of individual data sets, the phase or apparent resistivity, cannot solve inherent equivalence or suppression problems.

Joint inversion of EM and DC measurements can overcome the problem of equivalence very well. However, a suppression problem cannot be solved even after the combination of data sets. It is also concluded that the equivalence associated with a thin resistive layer can be solved better by joint inversion than that for a thin conducting layer. Similar studies concerning 2-D and 3-D structures for genetically-related and non-related observations would be necessary to understand the circumstances in which the joint inversion is really meaningful in reducing ambiguities of interpretation.