Evaluating the utility of gravity gradient tensor components

Mark Pilkington

Geological Survey of Canada

Tensor component choice

Txx Txy Txz

Tyz

Tyy

Tzz

Single components

Combinations

Concatenations

Which to use?

Qualitative interpretation

Quantitative interpretation

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

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

Outline

Aim: quantitative rating of component/combinations

Approach: inversion using a simple model – estimate parameter errors

Method: linear inverse theory – analyse model/data relations

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

Prism model

z

t

bw

xcyc

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

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

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

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

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)

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

Eigenvector matrix V

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]

Eigenvector matrix V

Correlation matrix

corrij = covij

[ covii covjj ]1/2

Parameter errors

xc,yc = locationz = deptht = thicknessw = widthb = breadth = density

Parameter errors

xc,yc = locationz = deptht = thicknessw = widthb = breadth = density

Parameter errors

xc,yc = locationz = deptht = thicknessw = widthb = breadth = density

Parameter error ranking [29 models]

error

high

low

Parameter errors versus averaging

No averagingcorrection

With averaging correction

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

Parameter rankings

I1Txz

higher error higher error

Width error versus coordinate rotation

coordinateaxis

bodyaxis

Information density matrix

Information density versus eigenvector