Jeffrey D. Phillips U.S. Geological Survey Presented at SEG 2002

1. Two-step processing for 3-D magnetic source locations and structural indices using extended Euler or analytic signal methods Jeffrey D. Phillips U.S. Geological Survey Presented at SEG 2002 U.S. Department of the Interior U.S. Geological Survey

2. Modifications to Euler and Analytic Signal Methods • The two-step approach Using a physical reference surface to constrain depths and estimate structural indices • The use of generalized Hilbert transform components Increases the number and reliability of solutions

3. Outline • Structural index values • Applying the two-step approach • Choosing a reference surface • Using generalized Hilbert transforms • The two-step extended Euler method • The two-step analytic signal method • Data example

4. Structural Index Values Contact Thick Dike Thin Sheet 0.0 0.5 1.0 Ribbon Pipe Finite Pipe Dipole 1.5 2.0 2.5 3.0

5. The Two-Step Approach 1. Solve for source locations assuming a specified structural index (usually zero to force the shallowest possible solutions), and retain solutions lying on or below a specified reference surface. X X X X X X X X X

6. The Two-Step Approach 2. Move the shallower solutions down to the reference surface and attempt to solve for a new structural index (Euler method), or assign a structural index using a pre-determined formula (analytic signal method). X X X X X X X X X X X X

7. The Reference Surface • represents an upper bound to the acceptable source locations. • Examples include the topography, the seafloor, or the seismic basement.

8. Generalized Hilbert Transforms Total Hx Field Hy Nabighian, 1984 Nabighian and Hansen, 2001

9. Two-Step Extended Euler Method 1. Within a data window, solve one or more sets of Euler equations for an initial structural index n (usually zero to force the shallowest possible solutions): Here (x0,y0,z0) is the unknown source location,  is an unknown constant, and  is either the total field, T, or one of its generalized Hilbert transform components, Hx or Hy. Reid and others, 1990 Nabighian and Hansen, 2001

10. Two-Step Extended Euler Method 2. Solutions on or below the reference surface z1 are retained. Solutions above the reference surface are moved down to z1 and a solution for (x0,y0,n) is attempted, using equations: If a valid solution exists and n<3, the solution is retained. If n>3, the solution is rejected.

11. Component Equations Unknowns T only M2 4 Hx only M2 4 Hy only M2 4 T and Hx 2M2 5 T and Hy 2M2 5 Hx and Hy 2M2 5 T, Hx, and Hy 3M2 6 Two-Step Extended Euler Method Within a small data window of size M by M, seven different solutions are possible:

12. Two-Step Extended Euler Method • A depth solution is successful if: • The system of equations can be solved • The error in the depth is below a specified threshold • The depth is above a specified maximum depth • The horizontal position is within a specified radius of the window center • A structural index solution is successful if: • The system of equations can be solved • The horizontal position is within a specified radius of the window center • The resulting structural index is between zero and three

13. Two-Step Extended Euler Method • Solutions with smaller average error in depth are considered more reliable. • Solutions averaged from a larger number of component combinations are also considered more reliable. • These factors are combined in the Euler Information Index:

14. Two-Step Analytic Signal Method • The horizontal location (x0,y0) of a contact source (structural index zero) is estimated from the crest of the analytic signal amplitude function: • The depth coordinate z0 is estimated from the curvature of |A(x,y)|2 at the crest: • If the depth solution lies above the reference surface, z1, the source can be moved down to z1 by increasing the structural index n using the formula:

15. Data Example: Buried volcanic flows and sedimentary faults in the Albuquerque basin Interpreted fault trace Igneous rocks at ~15 m Igneous rocks at ~150 m (Grauch, 2001)

16. Two-Step Euler Solutions from observed total field only

17. Two-Step Extended Euler Solutions Joint and separate solutions from Hx and Hy

18. Two-Step Extended Euler Solutions from all combinations of components

19. Two-Step Extended Euler Solutions T only Hx,Hy T,Hx,Hy

20. Two-Step Extended Euler Solutions Three thresholding methods: %Err < 5.56 #Soln > 2 EII > 0.4

21. Analytic Signal Pre-Filtering Low-pass signal High-pass noise

22. Analytic Signal Amplitude Functions Computed from Total Field Hx Hy

23. Two-Step Analytic Signal Solutions from T from Hx from Hy

24. Two-Step Analytic Signal Solutions Combined results from all three components

25. Flow Chart Mag.grd PLUGGRID Mag.plg Obsurf.grd Refsurf.grd SETEULER Hx.grd Hy.grd Mag.x32 Hx.x32 Hy.x32 Mag.y32 Hx.y32 Hy.y32 Mag.z32 Hx.z32 Hy.z32 Mag.a32 Hx.a32 Hy.a32 EULERAVE or EULER Eulerave.pst or Euler.pst x y p1 p2 p3 p4 p5 p6 x y z zerr -- EII #soln SI x y z zerr -- -- comp SI ASDEP or CURVDEP ASDEP or CURVDEP ASDEP or CURVDEP

26. Conclusions • A two-step solution process can be used to correct initial magnetic source locations lying above a specified reference surface, such as the topography. • The approach can be applied to Euler or analytic signal solutions, and it yields new structural indices for all solutions corrected to the reference surface. • The use of generalized Hilbert transform components adds significant new information to the solutions.

27. USGS vs. Geosoft Results – SI=0.0

28. USGS vs. Geosoft Results – SI=0.5