Polynomial Equivalent Layer

Polynomial Equivalent Layer

Valéria C. F. Barbosa*

Vanderlei C. Oliveira JrObservatório Nacional

Observatório Nacional

Contents

• Conclusions

• Classical equivalent-layer technique

• Polynomial Equivalent Layer (PEL)

• Real Data Application

• Synthetic Data Application

• The main obstacle

yxN E

zDep

th

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

yxN E

Dep

th


can be exactly reproduced by a continuous and infinite 2D physical-property distribution

Potential-field observations produced by a 3D physical-property distribution


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

yxN E

zDep

thD

epthLayer of equivalent

sources


Equivalent sources may be

magnetic dipoles, doublets,

point masses.Equivalent Layer

(Dampney, 1969).


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)


How ?

Classical

equivalent-layer technique

Classical equivalent-layer technique

yxN E

De

pth


d NR

We assume that the M equivalent sources are distributed in a regular grid with a constant

depth zo forming an equivalent layer

zo

Equivalent sources

Equivalent Layer


yx

N E

y

E

x

N

Ph

ysic

al-p

rop

ert

y

dis

trib

uti

on

Estimated physical-property

distribution

Equivalent Layer D

epth

Transformed potential-field data

p*

t T p*=

How does the equivalent-layer technique work?

?


Step 1: Step 2:

Why is it an obstacle to estimate the physical property

distribution by using the classical equivalent-layer technique?


A stable estimate of the physical properties p* is obtained

by using:

Parameter-space formulationp* = (GT G + I ) -1 GT d,

p* = GT(G GT + 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


(PEL)


kth equivalent-source window

with Ms equivalent sources

The equivalent layer is divided into a regular grid of Q equivalent-source windows

Ms <<< M

Inside each window, the physical-property distribution is described by a

bivariate polynomial of degree .

12

Q

dipoles (in the case of magnetic data)

Equivalent sources

point masses (in the case of gravity data).

Phy

sica

l-pro

pert

y di

strib

utio

n

The physical-property distribution within the equivalent layer is


Equivalent-source window

Polynomial function

assumed to be a piecewise polynomial function

defined on a set of Q equivalent-source windows.

Phy

sica

l-pro

pert

y di

strib

utio

n

Equivalent-source window


How can we estimate the physical-property distribution within the entire equivalent layer ?

Phy

sica

l-pro

pert

y di

strib

utio

nkth equivalent-source window


Physical-property distribution pk

Relationship between the physical-property distribution pk within the kth

equivalent-source window and the polynomial coefficients ck of the th-order polynomial function

Polynomial coefficients ck

kckB

kp

Phy

sica

l-pro

pert

y di

strib

utio

n


Physical-property distribution p

How can we estimate the physical-property distribution p within the entire equivalent layer ?

All polynomial coefficients cEntire equivalent layer

B c(H x 1)

p(M x 1) (M x H)

QB00

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 observationsD

epth

Equivalent layer with Q equivalent-source

windows

c*

Phy

sica

l-pro

pert

y di

strib

utio

n

Computed physical-property

distribution p*

EN

Transformed potential-field data

t T p*=

c*Bp*

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


(H x H)

A system of H linear equations in H unknowns

Polynomial Equivalent Layer requires much less computational effort

c dGB TT

R BRBIG BGB TTTT ] ) ( [ 10

-1

A stable estimate of the polynomial coefficients c* is obtained by


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


Large-equivalent source window andHigh degree of the polynomial

Small-equivalent source window and Low degree of the polynomial

EXAMPLES

Ph

ysic

al-p

rop

erty

di

strib

utio

n

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.


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


A

B

C

The number of observations is about 70,000

The geomagnetic field has inclination of -3o and declination of 45o.

The magnetization vector has inclination of -2o and declination of -10o.

Polynomial Equivalent LayerTwo 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


Differences (color-scale map) between the simulated (black contour lines)

and fitted (not show) total-field anomalies at z = -150 m.

Large windownT

Poor data fit


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



75,000 × 75,000 system

Computed magnetization-intensity distribution obtained by PEL

with first-order polynomials and small equivalent-source windows

A/m


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.


Polynomial Equivalent LayerTrue total-field anomaly at the pole

(True transformed data)

Polynomial Equivalent LayerReduced-to-the-pole anomaly (dashed white lines) using the


Application of


to real magnetic data

Upward continuation and

Reduction to the pole

São PauloRio 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 -21.5o and declination of -19o.

The magnetization vector has inclination of -40o and declination of -19o.

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



78,000 × 78,000 system

The PEL solves a 2,500 × 2,500

system

Real Test

Computed magnetization-intensity distribution obtained by


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

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.


H <<<<< N H <<<<< M

Thank youfor your attention

Published in GEOPHYSICS, VOL. 78, NO. 1 (JANUARY-FEBRUARY 2013)

10.1190/GEO2012-0196.1

Documents

Polynomial Equivalent Layer