HR: 0800h
AN: NG31B-0870    [Abstracts]
TI: Nonlinear Joint Inversion to 3D Geological Models
AU: * Horowitz, F G
EM: frank.horowitz@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Australian Resources Research Centre, POB 1130, Bentley, WA 6102 Australia
AU: Hornby, P
EM: peter.hornby@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Australian Resources Research Centre, POB 1130, Bentley, WA 6102 Australia
AU: Wilson, G A
EM: glenn.wilson@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Australian Resources Research Centre, POB 1130, Bentley, WA 6102 Australia
AU: Boschetti, F
EM: fabio.boschetti@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Australian Resources Research Centre, POB 1130, Bentley, WA 6102 Australia
AU: Poulet, T
EM: thomas.poulet@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Australian Resources Research Centre, POB 1130, Bentley, WA 6102 Australia
AU: King, A
EM: andrew.king@csiro.au
AF: C.S.I.R.O Division of Exploration and Mining, Queensland Centre for Advanced Technologies, POB 883, Kenmore, QLD 4069 Australia
AB: We formulate an inverse problem to jointly incorporate all available observations from multiple geophysical surveys, geological data, and spatial statistics. The goal is to estimate the subsurface lithology distribution within a common earth model. For the $i$th particular geophysical surveying technique (e.g., gravity) write a discrete, linearized forward problem as the matrix equation ${\bf L}_i m_i = d_i$, where ${\bf L}_i$ maps a physical property $m_i$ (e.g., density distribution) into the corresponding observed data $d_i$ (e.g., $g_z$). While uniqueness theorems exist for some special cases, in practice finding $m_i$ from $d_i$ is ill-posed leading to a need for prior information to achieve a solution. Typically, priors are imposed via an assumed regularization of $m_i$. By merging and concatenating the $m_i$ as well as concatenating the $d_i$ into longer vectors $m$ and $d$, the joint forward operator now can be written as ${\bf L}$. The operator ${\bf L}$ is almost block diagonal with entries ${\bf L}_i$. However, merging operations on the $m$ vector lead to physical coupling terms appearing off diagonal for related phenomena (e.g., gravity and linearized seismic). Next, introduce a permutation operator ${\bf P}$ which re-orders rows of $m$ such that adjacent entries in the vector correspond to physical properties at the same region of space. (Call this the spatial grouping of the properties $m_v$.) From this we construct an operator ${\bf L}^v = {\bf L P}$ which relates the region grouped property vector $m_v$ to the data $d$. Next, consider a set of column vectors constructed as follows. The coordinate basis vector $e_i$ (zeros everywhere, except for a one in the $i$th row) represents lithology type $i$. Introduce a matrix ${\bf R}$ whose $i$th column represents the property values for lithology type $i$. Then the property values in a region are given by ${\bf R} v$ where $v$ is one of the basis vectors $e_i$ representing the composition of that region. We can represent the lithological parameterization of the model $m'$ as the concatenation of the $v$s corresponding to each region. By forming a block diagonal matrix ${\bf D}$ from repeated diagonal entries ${\bf R}$, one for each region, we construct the matrix relating the regional \textit{lithology} parameterization $m'$ to the region grouped \textit{property} vector $m_v$. Putting all of this together we obtain ${\bf L}' m' = d$ where ${\bf L}' = {\bf L P D}$. This is a constrained linear relationship between a lithological parameterization of the model and the data. This system restricts the degrees of freedom (DOFs) of $m'$ to that of single lithologies for each region. Geological sampling further restricts DOFs by directly constraining components of $m'$. Spatial statistics such as indicator variograms for lithology can also serve as further priors. The $m'$ are discrete vector data in each region that represent categorical variables. This makes the problem constrained, and hence nonlinear. Nevertheless, allowing linear mixing of lithologies during the solution process enables us to take advantage of linearity. If a linear mixing hypothesis is correct, such mixed lithology solutions might even be admissible. Nonlinear ${\bf L}$ could be treated in a similar fashion.
DE: 3210 Modeling
DE: 3260 Inverse theory
DE: 3299 General or miscellaneous
DE: 0900 EXPLORATION GEOPHYSICS
DE: 0910 Data processing
SC: Nonlinear Geophysics [NG]
MN: 2004 AGU Fall Meeting