Airborne Gravity 2010 - Geoscience Australia

Airborne Gravity 2010
where is the vector of model parameters (densities, ) of length ; is a vector of observed
data of length ; and is a rectangular matrix of size formed by the corresponding
kernels. The integral form of the gravity fields and their gradients are convolutions of the
corresponding Green’s functions and density inside each cell. As the background density is assumed
to be homogeneous and the Earth model is discretized into a regular grid, then becomes a Toeplitz
or block-Toeplitz matrix, whose number of different entries is much less than the number of elements.
This makes storage of more economical. Moreover, matrix multiplications involving are of an
order of rather than . This results in considerable speed up for iterative inversion
based on repetitive matrix multiplications (Zhdanov, 2002).
Regularized Focusing Inversion
Gravity field and gradient data is reduced to the solution of the matrix equation (21). This inverse
problem is ill-posed; i.e., the solution can be non-unique and unstable. Therefore, we have to use
methods of regularization in order to recover unique and stable density models (Tikhonov and Arsenin,
1977; Zhdanov, 2002). This is achieved by minimizing the Tikhonov parametric functional, :
where is a misfit functional, is a stabilizing functional, and is a regularization
parameter that balances (or biases) the misfit and stabilizing functional. The stabilizing functional
incorporates information about the class of models used in the inversion. Equation (22) can be
minimized any number of ways. The inverse problem is a linear one. We choose to use the reweighted
regularized conjugate gradient method (Zhdanov, 2002), since it reduces the iterative scheme to a
series of matrix-vector multiplications that can be evaluated very quickly with FFT-based matrix-vector
multiplications.
The choice of stabilizing functional should be based on the user’s geological knowledge and prejudice.
Traditionally, the stabilizers have been based on minimum norm or first or second derivatives of the
density distribution, which recover models with smooth density distributions (e.g., Li and Oldenburg,
1998). Smooth models bear little resemblance to economic geology. Moreover, smooth stabilizers can
result in spurious oscillations and artifacts when the density distribution is discontinuous. As such, it is
useful to search for unique and stable solutions within those models that have sharp density contrasts.
This can be accomplished by introducing the so-called focusing functionals (Portniaguine and
Zhdanov, 1999, 2002; Zhdanov, 2002, 2009b). First, we present the minimum support (MS) stabilizer:
where is a focusing parameter introduced to avoid singularity when . The minimum
support stabilizer minimizes the volume with non-zero departures from the a priori model. Thus a
smooth distribution of all model parameters with a small deviation from the a priori model is penalized.
A focused distribution of the model parameters is penalized less. Similarly, we present the minimum
gradient support (MGS) stabilizer:
which minimizes the volume of model parameters with non-zero gradient.
Case study – Nordkapp Basin
The Nordkapp Basin is located in the Norwegian sector of the Barents Sea, and is an intra-continental
salt basin containing over 30 salt structures. The salt is of an Early Permian age, and was mobilized
by Early Triassic sedimentation. Tertiary uplift and erosion removed nearly 1400 m of Cretaceous and
younger sediments (Neilsen et al., 1995). The petroleum plays are mainly salt-related traps. Only two
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