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![Page 1: Evaluating the utility of gravity gradient tensor components Mark Pilkington Geological Survey of Canada.](https://reader035.fdocuments.net/reader035/viewer/2022062620/551b3d77550346d31b8b45be/html5/thumbnails/1.jpg)
Evaluating the utility of gravity gradient tensor components
Mark Pilkington
Geological Survey of Canada
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Tensor component choice
Txx Txy Txz
Tyz
Tyy
Tzz
Single components
Combinations
Concatenations
Which to use?
Qualitative interpretation
Quantitative interpretation
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Tensor component choice
Quantitative interpretation [Inversions]
(Txx, Txy, Txz, Tyy, Tyz) Li, 2001(Tuv, Txy), Tzz Zhdanov et al., 2004(Txz, Tyz, Tzz, Tuv) Droujinine et al., 2007(Tuv, Txy) Li, 2010(Tuv, Txy), Tzz, (Tzz, Tuv, Txy) Martinez & Li, 2011Tzz, (Txz, Tyz, Tzz), (Txz, Tyz, Txz, Tyy, Txx) Martinez et al., 2013
Rating the solutions:goodness of fitsharp/smoothclose to geology
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Inversion versus component combinations
Martinez et al., 2013
Tzz
Txz, Tyz, Tzz
Txz, Tyz, Txz, Tyy, Txx
Txz, Tyz, Txz, Tzz, Tyy, Txx
Components inverted:
RMS error Txx Txy Txz Tyy Tyz Tzz1-C 23.9 23.2 31.8 23.1 26.1 16.53-C 17.5 16.0 15.9 16.0 12.4 22.55-C 16.6 12.6 16.3 15.8 12.2 24.36-C 15.7 13.0 17.9 13.8 13.8 21.4
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Outline
Aim: quantitative rating of component/combinations
Approach: inversion using a simple model – estimate parameter errors
Method: linear inverse theory – analyse model/data relations
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Inversion method used
Inversion Parametric[underdetermined inversionproblem]
n datam parameters m >> n m << n
Model 3-D volume Specified shapequantity
Solution Physical property Parameters (density …) (depth, dip…)
Methodology Regularized inversion Overdeterminedleast – squares
Solution Resolution, covariance Parameter errorsappraisal
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Prism model
z
t
bw
xcyc
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Inverse theory
Forward problem: b = f (x) b = data
x =
parameters
(linearized) db = Adx A = Jacobian
[model dependent]
aij = dbi/dxj
Inverse problem : dx = A+db
A = UVT singular
value
decomposition
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Inverse theory
A = UVT singular value decomposition
U = data eigenvectors
V = parameter eigenvectors
= singular values
R = VVT Resolution matrix (=I)
S = UUT Data information matrix
C = CdV-2VT Covariance matrix
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Model parameter errors
C = CdV-2VT Parameter covariance matrix
Cd = Data covariance
=singular values
small large C large small C
Cd = e2I Equal data errorCd = D Variable data error
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Variable component errors
Components have different error levels: e.g., e(Txx) = e(Txz) only relative levels requiredestimate based on FFT or equivalent source methodratio Tzz : Txz, Tyz : Txy : Txx, Tyy = 1 : 0.70 : 0.37 : 0.59
Component quantities are combined: e.g., H1 = sqrt(Txz2+Tyz2) combine errors: e(Tuv) = [0.5 (e(Txx)2+e(Tyy)2)]1/2
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Component quantities tested
Single components:
Txx Tyy Tzz Txy Tyz Txz Tuv
Invariants:
I1 = TxxTyy+TyyTzz+TxxTzz-Txy2-Tyz2-Txz2
I2 = Txx(TyyTzz-Tyz2)+Txy(TyzTxz-TxyTzz)+Txz(TxyTyz-TxzTyy)H1 = sqrt(Txz2+Tyz2) H2 = sqrt[Txy2+0.25(Tyy-Txx)2]
Concatenations:
(Tuv, Txy)(Txz, Tyz, Tzz)
(Txy, Tyz, Txz)
(Txx, Tyy, Txy) (Txz, Tyz, Txz, Txy, Txx)(Tyy, Tyz, Txz, Txy, Txx)
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Inversion tests
Procedure:
•Specify model and evaluate matrix A [db=Adx]•Calculate covariance matrix C•Get parameter standard deviations (p.s.d.)•Rank p.s.d. for each parameter versus component quantity
Models tested:
xc yc z t w b
32 32 4 1,3,6,13,43 12 12 0.2
32 32 4 13 0.1,0.5,2,6,9 12 0.2
32 32 0.1,1,3,6,12 40 12 12 0.2
32 32 2 1 1 1 0.2
32 32 2 4 4 4 0.2
32 32 2 8 8 8 0.2
32 32 0.5,1,2 4 1 1 0.2
32 32 0.5,1,2,4 1 8 8 0.2
32 32 0.5,1,2,4 2 2 2 0.2
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Eigenvector matrix V
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Eigenvector matrix V
Invariants:
I1 = TxxTyy+TyyTzz+TxxTzz-Txy2-Tyz2-Txz2
I2 = Txx(TyyTzz-Tyz2)+Txy(TyzTxz-TxyTzz) +Txz(TxyTyz-TxzTyy)
H1 = sqrt(Txz2+Tyz2)
H2 = sqrt[Txy2+0.25(Tyy-Txx)2]
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Eigenvector matrix V
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Correlation matrix
corrij = covij
[ covii covjj ]1/2
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Parameter errors
xc,yc = locationz = deptht = thicknessw = widthb = breadth = density
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Parameter errors
xc,yc = locationz = deptht = thicknessw = widthb = breadth = density
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Parameter errors
xc,yc = locationz = deptht = thicknessw = widthb = breadth = density
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Parameter error ranking [29 models]
error
high
low
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Parameter errors versus averaging
No averagingcorrection
With averaging correction
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Conclusions
Concatenated components produce smallest parameter errors
Invariants I1, I2 best performers in combined component category
Purely horizontal components poor performers
Tzz best single component
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Parameter rankings
I1Txz
higher error higher error
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Width error versus coordinate rotation
coordinateaxis
bodyaxis
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Information density matrix
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Information density versus eigenvector