Polynomial Equivalent Layer
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Transcript of Polynomial Equivalent Layer
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Polynomial Equivalent Layer
Valéria C. F. Barbosa*
Vanderlei C. Oliveira JrObservatório Nacional
Observatório Nacional
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Contents
• Conclusions
• Classical equivalent-layer technique
• Polynomial Equivalent Layer (PEL)
• Real Data Application
• Synthetic Data Application
• The main obstacle
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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
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yxN E
Dep
th
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
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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
Potential-field observations
Equivalent sources may be
magnetic dipoles, doublets,
point masses.Equivalent Layer
(Dampney, 1969).
Equivalent-layer principle
Equivalent sources
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• 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 ?
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Classical
equivalent-layer technique
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Classical equivalent-layer technique
yxN E
De
pth
Potential-field observations
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
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Classical equivalent-layer technique
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?
?
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?
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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.
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Objective
We present a new fast method for performing any linear
transformation of large potential-field data sets
Polynomial Equivalent Layer
(PEL)
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Polynomial Equivalent Layer
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).
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Phy
sica
l-pro
pert
y di
strib
utio
n
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.
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Phy
sica
l-pro
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Equivalent-source window
Polynomial Equivalent Layer
How can we estimate the physical-property distribution within the entire equivalent layer ?
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Phy
sica
l-pro
pert
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strib
utio
nkth equivalent-source window
Polynomial Equivalent Layer
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
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Phy
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l-pro
pert
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Polynomial Equivalent Layer
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
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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
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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*?
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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 TT
R BRBIG BGB TTTT ] ) ( [ 10
-1
A stable estimate of the polynomial coefficients c* is obtained by
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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
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Gravity data set Magnetic data set
Polynomial Equivalent Layer
Large-equivalent source window andHigh degree of the polynomial
Small-equivalent source window and Low degree of the polynomial
EXAMPLES
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Ph
ysic
al-p
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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
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Application of
Polynomial Equivalent Layer (PEL)
to synthetic magnetic data
Reduction to the pole
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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 -3o and declination of 45o.
The magnetization vector has inclination of -2o and declination of -10o.
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Polynomial Equivalent LayerTwo applications of Polynomial Equivalent Layer (PEL)
Large-equivalent-source window Small-equivalent-source window
First-order polynomials
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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
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Computed magnetization-intensity distribution obtained by PEL
with first-order polynomials and large equivalent-source windows
A/m
First Application of Polynomial Equivalent Layer
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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
First Application of Polynomial Equivalent Layer
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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
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Computed magnetization-intensity distribution obtained by PEL
with first-order polynomials and small equivalent-source windows
A/m
Second Application of Polynomial Equivalent Layer
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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
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Polynomial Equivalent LayerTrue total-field anomaly at the pole
(True transformed data)
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Polynomial Equivalent LayerReduced-to-the-pole anomaly (dashed white lines) using the
Polynomial Equivalent Layer (PEL)
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Application of
Polynomial Equivalent Layer
to real magnetic data
Upward continuation and
Reduction to the pole
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São PauloRio de Janeiro
Aeromagnetic data set over the
Goiás Magmatic
Arc, Brazil.
Brazil
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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
The classical equivalent layer
technique should solve
78,000 × 78,000 system
The PEL solves a 2,500 × 2,500
system
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Real Test
Computed magnetization-intensity distribution obtained by
Polynomial Equivalent Layer (PEL)
N
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Real Test
N
Observed (black lines and grayscale map) and
predicted (dashed white lines) total-field anomalies.
Acceptable data
fit.
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Real Test
N
Transformed data produced by applying the upward continuation and the
reduction to the pole using the Polynomial Equivalent Layer (PEL)
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Conclusions
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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
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Thank youfor your attention
Published in GEOPHYSICS, VOL. 78, NO. 1 (JANUARY-FEBRUARY 2013)
10.1190/GEO2012-0196.1