Polynomial Equivalent Layer Valéria C. F. Barbosa* Vanderlei C. Oliveira Jr Observatório Nacional

Contents Conclusions Classical equivalent-layer technique Polynomial Equivalent Layer (PEL) Real Data Application Synthetic Data Application The main obstacle

y x N E z Depth 3D sources Potential-field observations produced by a 3D physical-property distribution Potential-field observations Equivalent-layer principle can be exactly reproduced by a continuous and infinite 2D physical-property distribution

y x N E Depth Potential-field observations can be exactly reproduced by a continuous and infinite 2D physical-property distribution Potential-field observations produced by a 3D physical-property distribution Equivalent-layer principle z 2D physical-property distribution

This 2D physical-property distribution is approximated by a finite set of equivalent sources arrayed in a layer with finite horizontal dimensions and located below the observation surface y x N E z Depth Layer of equivalent sources Potential-field observations Equivalent sources may be magnetic dipoles, doublets, point masses. Equivalent Layer (Dampney, 1969). Equivalent-layer principle Equivalent sources

Interpolation To perform any linear transformation of the potential-field data such as: Upward (or downward) continuation Reduction to the pole of magnetic data (e.g., Silva 1986; Leão and Silva, 1989; Guspí and Novara, 2009). (e.g., Emilia, 1973; Hansen and Miyazaki, 1984; Li and Oldenburg, 2010) (e.g., Cordell, 1992; Mendonça and Silva, 1994) Noise-reduced estimates (e.g., Barnes and Lumley, 2011) Equivalent-layer principle How ?

Classical equivalent-layer technique

Classical equivalent-layer technique y x N E Depth Potential-field observations d N R  We assume that the M equivalent sources are distributed in a regular grid with a constant depth z o forming an equivalent layer zozo Equivalent sources Equivalent Layer

Classical equivalent-layer technique y x N E y E x N Physical-property distribution Estimated physical-property distribution Equivalent Layer Depth Transformed potential-field data p * tT = How does the equivalent-layer technique work? ? Potential-field observations Step 1:Step 2: Why is it an obstacle to estimate the physical property distribution by using the classical equivalent-layer technique?

Classical equivalent-layer technique A stable estimate of the physical properties p * is obtained by using: Parameter-space formulation p* = (G T G +  I ) -1 G T d, p* = G T (G G T +  I ) -1 d Data-space formulation or The biggest obstacle (M x M) (N x N) A large-scale inversion is expected.

Objective We present a new fast method for performing any linear transformation of large potential-field data sets Polynomial Equivalent Layer (PEL)

Polynomial Equivalent Layer k th equivalent-source window with M s equivalent sources The equivalent layer is divided into a regular grid of Q equivalent-source windows M s <<< M Inside each window, the physical-property distribution is described by a bivariate polynomial of degree . 1 2 Q dipoles (in the case of magnetic data) Equivalent sources point masses (in the case of gravity data).

Physical-property distribution The physical-property distribution within the equivalent layer is Polynomial Equivalent Layer Equivalent-source window Polynomial function assumed to be a piecewise polynomial function defined on a set of Q equivalent-source windows.

Physical-property distribution Equivalent-source window Polynomial Equivalent Layer How can we estimate the physical-property distribution within the entire equivalent layer ?

Physical-property distribution kth equivalent-source window Polynomial Equivalent Layer Physical-property distribution p k Relationship between the physical-property distribution p k within the kth equivalent-source window and the polynomial coefficients c k of the  th-order polynomial function Polynomial coefficients c k k c k B k p 

Physical-property distribution Polynomial Equivalent Layer Physical-property distribution p How can we estimate the physical-property distribution p within the entire equivalent layer ? All polynomial coefficients c Entire equivalent layer B  c (H x 1 ) p (M x 1) (M x H)                Q B00 0B0 00B Β     2 1 Q equivalent-source windows

Estimated polynomial coefficients How does the Polynomial Equivalent Layer work? Polynomial Equivalent Layer Step 1: N E Potential-field observations Depth Equivalent layer with Q equivalent-source windows c* Physical-property distribution Computed physical-property distribution p * E N Transformed potential-field data t T p * = c* B p*  Step 3: Step 2: ? How does the Polynomial Equivalent Layer estimate c * ?

H is the number of all polynomial coefficients describing all polynomial functions H <<<< M H <<<< N Polynomial Equivalent Layer (H x H) A system of H linear equations in H unknowns Polynomial Equivalent Layer requires much less computational effort c dGB T T   R BRBIG BGB TTTT  ] ) ( [ 1 0  A stable estimate of the polynomial coefficients c* is obtained by

Polynomial Equivalent Layer the smaller the size of the equivalent-source window THE CHOICES: The shorter the wavelength components of the anomaly the lower the degree of the polynomial should be. A simple criterion is the following: and Size of the equivalent-source window Degree of the polynomial

Gravity data set Magnetic data set Polynomial Equivalent Layer Large-equivalent source window and High degree of the polynomial  Small-equivalent source window and Low degree of the polynomial  EXAMPLES

Physical-property distribution How can we check if the choices of the size of the equivalent- source window and the degree of the polynomial were correctly done? Acceptable data fit. Polynomial Equivalent Layer A smaller size of the equivalent- source window and (or) another degree of the polynomial must be tried. Unacceptable data fit. Estimated physical-property distribution via PEL yields

Application of Polynomial Equivalent Layer (PEL) to synthetic magnetic data Reduction to the pole

Simulated noise-corrupted total-field anomaly computed at 150 m height Polynomial Equivalent Layer A B C The number of observations is about 70,000 The geomagnetic field has inclination of -3 o and declination of 45 o. The magnetization vector has inclination of -2 o and declination of -10 o.

Polynomial Equivalent Layer Two applications of Polynomial Equivalent Layer (PEL) Large-equivalent-source window Small-equivalent-source window First-order polynomials

First Application of Polynomial Equivalent Layer Large window Large-equivalent-source windows and First-order polynomials M ~75,000 equivalent sources H ~ 500 unknown polynomial coefficients The classical equivalent layer technique should solve 75,000 × 75,000 system The PEL solves a 500 × 500 system

Computed magnetization-intensity distribution obtained by PEL with first-order polynomials and large equivalent-source windows A/m First Application of Polynomial Equivalent Layer

Differences (color-scale map) between the simulated (black contour lines) and fitted (not show) total-field anomalies at z = -150 m. Large window nT Poor data fit First Application of Polynomial Equivalent Layer

Small-equivalent-source windows and First-order polynomials Small window M ~ 75,000 equivalent sources H ~ 1,900 unknown polynomial coefficients Second Application of Polynomial Equivalent Layer The PEL solves a 1,900 × 1,900 system The classical equivalent layer technique should solve 75,000 × 75,000 system

Computed magnetization-intensity distribution obtained by PEL with first-order polynomials and small equivalent-source windows A/m Second Application of Polynomial Equivalent Layer

Differences (color-scale map) between the simulated (black contour lines) and fitted (not show) total-field anomalies at z = -150 m. Small window nT Acceptable data fit. Second Application of Polynomial Equivalent Layer

Polynomial Equivalent Layer True total-field anomaly at the pole (True transformed data)

Polynomial Equivalent Layer Reduced-to-the-pole anomaly (dashed white lines) using the Polynomial Equivalent Layer (PEL)

Application of Polynomial Equivalent Layer to real magnetic data Upward continuation and Reduction to the pole

São Paulo Rio de Janeiro Aeromagnetic data set over the Goiás Magmatic Arc, Brazil. Brazil

Real Test Aeromagnetic data set over the Goiás Magmatic Arc in central Brazil. The geomagnetic field has inclination of o and declination of -19 o. The magnetization vector has inclination of -40 o and declination of -19 o. N M ~ 81,000 equivalent sources H ~ 2,500 unknown polynomial coefficients N ~ 78,000 observations Small-equivalent-source windows and First-order polynomials Small-equivalent source window The classical equivalent layer technique should solve 78,000 × 78,000 system The PEL solves a 2,500 × 2,500 system

Real Test Computed magnetization-intensity distribution obtained by Polynomial Equivalent Layer (PEL) N

Real Test N Observed (black lines and grayscale map) and predicted (dashed white lines) total-field anomalies. Acceptable data fit.

Real Test N Transformed data produced by applying the upward continuation and the reduction to the pole using the Polynomial Equivalent Layer (PEL)

Conclusions

We have presented a new fast method (Polynomial Equivalent Layer- PEL) for processing large sets of potential-field data using the equivalent-layer principle. The PEL divides the equivalent layer into a regular grid of equivalent-source windows, whose physical-property distributions are described by polynomials. The PEL solves a linear system of equations with dimensions based on the total number H of polynomial coefficients within all equivalent-source windows, which is smaller than the number N of data and the number M of equivalent sources The estimated polynomial-coefficients via PEL are transformed into the physical- property distribution and thus any transformation of the data can be performed. Polynomial Equivalent Layer H <<<<< N H <<<<< M

Thank you for your attention Published in GEOPHYSICS, VOL. 78, NO. 1 (JANUARY-FEBRUARY 2013) /GEO