1. Interactive 2D magnetic inversion: a tool for aiding forward modeling and testing geological hypotheses Valéria C. F. Barbosa LNCC - National Laboratory for Scientific Computing João B. C. Silva UFPA- Federal University of Pará

2. Contents Introduction and Objectives Methodology Real Data Inversion Result Conclusions Interactive Interpretation in a User-Friendly Environment

3. Introduction and Objectives The processing and interpretation of gravity and magnetic data on the x-y plane is well developed y x z It is performed by Linear Transformation Techniques such as:

4. Introduction and Objectives Derivative mapsAnalytic Signal maps Potential-field interpretations on the x - y plane

5. Introduction and Objectives Upward and Downward Continuations Reduced to the pole map Apparent Density and Magnetization Mapping These techniques are usually implemented in a user-friendly software Other examples of linear transformations in potential-field interpretations on the x - y plane

6. Introduction and Objectives Potential-field interpretations on the x - y plane The multiple, complex, and closely separated magnetic sources are easily handled by the available potential-field interpretation methods on the x - y plane. x y However, they estimate only the sources horizontal projections.

7. Introduction and Objectives The interpretation of gravity and magnetic data assuming that the physical property varies along the z -direction requires the solution of a strongly underdetermined problem. Potential-field interpretations on the x - z plane z x

8. Introduction and Objectives The reconstruction of a subsurface magnetic intensity distribution from the magnetic data has neither unique nor stable solution. Potential-field interpretations on the x - z plane Magnetic data -10010203040506070 0 1 2 3 4 x (km) z (km)

9. Potential-field interpretations on the x - z plane Introduction and Objectives Inversion Interactive Forward Modeling ALTERNATIVES: The interactivity and flexibility of introducing geological information The facility of automatically fitting the observations Advantages: Disadvantages: Tedious, time- consuming, and usually poor data fitting Poor results on interfering anomalies

10. Introduction and Objectives Combines the interactive forward modeling and the traditional inversion Interactive magnetic inversion Potential-field interpretations on the x - z plane interfering magnetic anomalies interactivity and guarantee of data fitting.

11. Methodology Interactive 2D magnetic inversion Magnetic Observations 2D magnetic sources 2D prism Interpretation Model y x z h o N R  Estimate the 2D magnetization intensity distribution

12. Methodology 2 G mh o  1  N h  The unconstrained Inverse Problem The linear inverse problem of estimating the magnetization intensity distribution, m, using just the magnetic data, h o, can be formulated by minimizing The problem of obtaining a vector of parameter estimates, m, that minimizes this functional is an ill-posed problem. ^

13. Methodology The prior information introduced: 1.The estimated magnetization sources must be homogeneous and compact 2.The outlines of the magnetic sources may be constructed by a combination of geometric elements: Magnetic Observations 2D magnetic sources y x z Point Axes Point Interactive 2D magnetic inversion axes and points

14. y x Magnetic Observations Point Axis Point Each geometric element is assigned a target magnetization intensity and a magnetization direction m  1.5 A/m I = 60 o D = 0 o m  1.7 A/m I = 30 o D = 0 o m  1.2 A/m I = 20 o D = -5 o Axis m  1.5 A/m I = 10 o D = -2 o Methodology z Interactive 2D magnetic inversion Each magnetic source must be assigned one or more geometric elements

15. Methodology The Iterative Constrained Inverse Problem Starting from the minimum-norm solution the method, at the kth iteration, looks for a constrained parameter correction ˆˆ ˆ )( F mΔm m   ) 1 ( k  k )( k and updates the magnetization distribution estimate by  )(11)( ˆ )( ˆ k F T )(k T )(k k mG h IGG WGWmΔ o  1     ˆ m or )( k Frozen elements o hIGGGm 1o ) ( ˆ   TT,

16. Methodology Interactive 2D magnetic inversion The method estimates iteratively the constrained parameter correction by Minimizing W Δ m 2 )( k )( k G h o  1 N 2  Subject to Δ m ) (m + )( k )( k djdj axis j     )( 2 ˆ k j j jj m d w

17. The method estimates the constrained Δ m which affects mainly the prisms close to the geometric elements Small weights Δ m j may be large Interactive 2D magnetic inversion Methodology y x z Large weights Δ m j will be small

18. True magnetic sources Geometric element Noise-corrupted magnetic anomaly Iteration 0102 Iteration 03 Iteration Data fit 04 Iteration 05 Iteration 06 Iteration 07 Iteration 08 Iteration 09 Iteration 10 Iteration 11 Iteration 12 Iteration 13 Iteration Simulated Example The evolution of the magnetization distribution estimates along the iterations Magnetization with intensity of 2 A/m, Inclination of 20° and Declination of 0°.

19. Interactive 2D magnetic inversion Interactivity in a User-Friendly Environment

20. Magnetization with intensity of 1.2 A/m, Inclination of 60° and Declination of 0°. Noise-corrupted magnetic anomaly True sources axes Fitted magnetic anomaly Estimated sources First Hypothesis

21. Interactivity in a User-Friendly Environment Second Hypothesis New axes

22. Interactivity in a User-Friendly Environment Third Hypothesis Further hypothesis framework Data fit Estimated sources

23. Fourth Hypothesis The ambiguity: Magnetization Intensity x Volume Interactivity in a User-Friendly Environment True Magnetization with intensity of 1.2 A/m Assumed target magnetization with intensity of 0.6 A/m Correct axes Estimated sources

24. Interactivity in a User-Friendly Environment 5th Hypothesis The correct prior information

25. The ore body has remanent magnetization Real Data Example Northwest Ore Body at Iron Mountain Mine, Missouri (Leney, 1966). The geomagnetic field has inclination of 67 o and declination of 0 o. magnetization vector of 150 A/m, inclination of -5 o and declination of 180 o. SWNE x (m) Depth (m) 76% hematite 4% magnetite 20% gangue Geometric elements Data fit Estimated solution 76% hematite 4% magnetite 80% of the iron ore Total-field anomaly The hook-shaped geometry (Leney,1966). 4% magnetite

26. Conclusions The proposed method combines the best features of the interactive modeling and of the automatic inversion procedures INTERACTIVE INVERSIONINTERACTIVE MODELING REQUIRES JUST THE SPECIFICATION OF THE SOURCES’ FRAMEWORKS REQUIRES THE SPECIFICATION OF THE COMPLETE SOURCE GEOMETRY AUTOMATIC DATA FITTING DATA FITTING BY TRIAL-AND- ERROR x (km) Depth (km) x (km) Depth (km) Interactive magnetic inversion

27. AS COMPARED TO FULLY AUTOMATIC INVERSION METHODS, THE PRESENTED METHOD: Makes it feasible and easier interpreting interfering magnetic anomalies anomalies produced by sources with complex geometries Interactive magnetic inversion Conclusions

28. Thank You I hope to see you in San Antonio with the 3D interactive inversion !

29. Extra Figures

31. 1.0 0.5 0.0 Depth (km) (b) (a) x (km) (A/m) 1.20 0.00 0.60 1.21 1.25 e1e1 e3e3 e2e2 e4e4 (nT) o h

32. The convergence criterion is based on the proximity of the magnetization estimate of each cell to the associated target magnetization by checking the inequality A typical value assigned to  is 0.01, corresponding to an estimated magnetization differing from the target magnetization by less than 1% of the latter. Convergence Criterion

33. ˆˆ ˆ )( F mΔm m   ) 1 ( k  k )( k The method updates the magnetization distribution estimate by ˆ m or )( k Frozen elements if the magnetization of the jth cell does not violate the target magnetization assigned to the jth cell. the jth element is replaced by the violated boundary (i.e., the target magnetization assigned to each cell) m )( k ˆ Each cell is assigned the target magnetization intensity and direction of the closest geometric element (axis or point). First, we define an M-dimensional vector whose jth element is the target magnetization assigned to the jth cell.

36. Li and Oldenburg’s (1996) and (1998) methods produce a blurred picture of the source geometry, i.e., the true source is not delineated (even though its center can still be located). However, Li and Oldenburg’s method produce poor delineations and unacceptable locations in the case of deep sources

37. where B the matrix is called first difference discrete operator and the other terms have the same definition used by those authors. According to them W is a diagonal matrix of depth-dependent weights whose ith diagonal element is given by. Li and Oldenburg (1996) proposed a 3D method to estimate the susceptibility distributions by minimizing Subject to They used this weighting function decays with depth to presumably counteract the decay rate of the magnetic kernel with and counteract, in this way, the natural decay of the kernel function. These authors also employed the same approach using gravity data (Li and Oldenburg, 1998). Both approaches produced good result in locating the source and just reasonable results in delineating the sources if they are shallow (see Li and Oldenburg’s examples). It is an advance when compared with other methods employing the first-order Tikhonov regularization.

38. Portniaguine and Zhdanov (2002) presented a regularizing method called “focusing inversion”. The difference between their functional and Last and Kubik’s (1983) is the introduction of an iterative weighting matrix in the regularizing functional. Specifically, Last and Kubik’s (1983) regularizing functional is given by and Portniaguine and Zhdanov’s (2002) regularizing functional is given by Portniaguine and Zhdanov’s (2002) method also introduced bounds to the magnetization intensity. This is stated on page 1535: “Also note that our algorithm includes constraints on material properties, implemented via a penalization algorithm”.

39. The Portniaguine and Zhdanov’s (2002) method does not require an isolated source but it does not resolve the geometry of several magnetic sources that give rise to a complex and strong interfering magnetic field. Figures 1c and 2f (shown below) are illustrating two true isolated shallow sources (Figure 1c) and the estimated solution, respectively. Note that the estimated solution (Figure 2f) shows a compact image of the magnetic sources which is unable to resolve closely separated magnetic sources. Portniaguine and Zhdanov (2002) Figure 1c (the true sources) Portniaguine and Zhdanov (2002) Figure 2F (the estimated susceptibility distribution)