Integrated high-resolution tomography Marco Perez CREWES

Introduction Tomography is a statistical method that determines subsurface velocities from recorded traveltimes An insufficient amount of data often leads to poorly distributed raypaths and non-unique solutions More reliable solutions can be obtained by using the quasi-null space to integrate surface and crosswell data

Tomography Theory surface Divide subsurface into cells and model traveltimes and raypaths through each cell An initial velocity estimate is updated, minimizing the error between the recorded and modelled traveltimes

Tomography The minimization is expressed mathematically as Solving for m requires inverting the matrix D which is done by Singular Value Decomposition Where matrix D represents the distances traveled by a ray through each cell, m is the parameter update value and t is the discrepancy between recorded and modelled traveltimes Here U is an np matrix whose columns contain p of the total n orthonormal observation eigenvectors ui V is a pp matrix whose columns contain the p orthonormal parameter eigenvectors

Tomography The inversion problem is ill-posed because of two main factors: 1) The matrix D is often overdetermined 2) Some cells within D contain no data or lack the required data to determine the model parameter uniquely

Quasi-null space surface an infinite number of solutions to the equation exist since any linear combination including u0 will yield the correct answer Cells with no data will satisfy the minimization equation with any velocity function making such a cell unreliable for inversion The corresponding columns of zero singular values in the V matrix constitute an orthonormal basis of the null space.

Quasi-null space Vesnaver, 1994 The quasi-null space is defined as the sum of squares of the entries in the columns of V whose corresponding singular values are above a predefined threshold Vesnaver proposed a change in cell size to limit the size of the quasi-null space however this reduces the resolution. The quasi-null space highlights the cells that are most reliable for traveltime inversion.

Integrated Tomography Using the quasi-null space to integrate two data sets, a more reliable velocity model can be determined No data Ray coverage is a function of both acquisition geometry and the velocity model No data

Models Model 1 Model 2 Model 3

Model 1 Minimal error for both cases transmission reflection

Model 2 Horizontal layer identified Horizontal layer smeared along ray path transmission reflection

Model 3 Anomaly smeared horizontally Anomaly smeared vertically transmission reflection

Quasi-null space of Model 1 Acquisition imprint visible in the quasi-null space transmitted reflection Threshold 0.1

Integrated Model 1

Quasi-null space of Model 2 transmitted reflection Threshold 0.1

Integrated Model 2 Correctly identifies more reliable inversion

Quasi-null space of Model 3 transmission reflection Threshold 0.1

Integrated Model 3 Combination of both experiments yield superior results

Residual traveltime error Reflection tomography yielded lowest residual error, however the quasi-null space correctly identified the solution as unreliable Model 3 Model 2 Integrated solution yields best residual error and is the most reliable result

Conclusion Quasi-null space integration correctly identifies more reliable solution Integrating two different data sets using the quasi-null space criterion yields more accurate tomograms

Future Work Implementing a dynamic smoother which uses the quasi-null space.

Acknowledgements Crewes Sponsors Dr. Larry Lines, Shauna Oppert, Pat Daley