1. Integrated high-resolution tomography
Marco Perez
CREWES

2. 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

3. 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

4. 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

5. 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

6. 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.

7. 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.

8. 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

9. Models
Model 1
Model 2
Model 3

10. Model 1
Minimal error for both cases
transmission
reflection

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

12. Model 3 Anomaly smeared horizontally Anomaly smeared vertically
transmission
reflection

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

14. Integrated Model 1

15. Quasi-null space of Model 2
transmitted
reflection
Threshold 0.1

16. Integrated Model 2
Correctly identifies more reliable inversion

17. Quasi-null space of Model 3
transmission
reflection
Threshold 0.1

18. Integrated Model 3
Combination of both experiments yield superior results

19. 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

20. 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

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

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