Egregious Euler Errors

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Egregious Euler Errors the use and abuse
ofEuler deconvolution applied to
potentialfieldsPotential Field Methods II"
session,Thursday 7 June 2012, 1555-1620.
Alan Reid Jörg Ebbing Susan Webb
Reid Geophysics Univ. of Leeds, UK
Geological Survey of Norway (NGU) University
of the Witwatersrand, South Africa

• EAGE Copenhagen 12

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Synopsis

• What is Euler deconvolution?
• What are the critical parameters?
• How to select optimum values for the critical
  parameters
• A horrible (egregious) example of misuse
• egregious outstandingly bad

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What is Euler deconvolution?

• Is used to estimate position and depth of
  magnetic or gravitational source body edges,
  using profile or gridded data
• Exploits a moving window, measured or calculated
  gradients and Eulers differential equation
• Assumes Euler homogeneity
• It is not a deconvolution sensu stricto, but is
  similar to Werner deconvolution
• Profile version - Thompson (1982). Grid version
  Reid et al (1990)
• Subsequent developments by Mushayandebvu,
  Stavrev, Barbosa, Nabighian Hansen, FitzGerald
  and others
• Two commercial releases (Intrepid Geosoft)
• Many academic versions

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Whats behind Euler?
A function f(x,y,z) is homogeneous of degree N
if f(sx, sy, sz) s N f(x,y,z), Where
N is an INTEGER and s is a scale factor. i.e. f
scales sensibly. This is fundamental to Euler.
If f(x,y,z) C/r n (n 1,2,3), then f is
homogeneous of degree N-n. n is the
Structural Index (SI). Again, n is an integer.
This is a special case where a single
measurement-source vector r may be used and the
source body has no relevant spatial dimensions.
SI - degree (i.e. n - N)
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Grid Based Euler

• (x-xo) ?T/?x (y-yo) ?T/?y (z-zo) ?T/?z
  N(B-T)
• Measurement Point (X, Y, Z)
• Source Location (Xo, Yo, Zo)
• Measured field and gradients T, ?T/?x etc
• Background field B
• N Structural Index (SI) depends on source
  type
• Solve (IN MOVING WINDOWS) for
• (Xo, Yo, Zo) and B

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Single-point and multi-point sources
A single-point source (infinite
step/fault/contact). Only one relevant vector
r from the sensor to the source critical
location. No other critical length-parameters. Co
nventional Euler methods assume this kind of
source.
A two-point source (finite step/fault/contact).
More than one r, or else a critical length
parameter (e.g. thickness). Conventional Euler
methods cannot handle this kind of source.
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Single-point sources
Sphere (eg Orebody). Mag SI3. Grav SI2.
Depth to C of M/G
Pipe (eg kimberlite). Mag SI 2. Grav SI1. Grav
gradient SI2. Depth to centre of top.
Thin-bed fault. Mag SI 2. Depth to midpoint (d
0.5 ?d).
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Single-point sources
Dyke top, sill edge Mag SI 1 Gravity hard
to detect Grav. Gradient SI 1 Edge
must be isolated
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Single-point sources
Infinite Faults/contacts. A block has infinite
depth extent if the thickness is much greater
than the depth to top. Mag SI 0 Gravity
is infinite not a useful model Grav.
Gradient SI 0.
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Intractable sources
Euler methods are EDGE DETECTORS. Smoothly
varying surfaces are not appropriate targets.
Multi-point sources require a more sophisticated
Euler treatment (Stavrev Reid 2007, 2009). No
commercial implementation. The thick step
(fault, contact) is not amenable to simple Euler
methods.
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The Moving Window
Euler window
Move the window over the grid and at each
position, solve the Euler differential equation
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What are the critical parameters ?

• Well chosen geological problem
• Adequately prepared and sampled magnetic or
  gravity field (no aliasing)
• Grid interval
• Valid gradient data
• Rational window size
• Meaningful Structural Index

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Well chosen geological problem

• Must be capable of splitting into SI-friendly
  local sources.
• Any one window should see only one simple
  source edge.
• Cannot be used to estimate the depth of smoothly
  varying interfaces.

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Adequate sampling for Euler work (includes
gradients)-follows Reid (1980)
To avoid aliasing
Field Profile/sample spacing
Magnetic lt Depth
Gravity Gradient lt Depth
Gravity lt 2 x Depth

• Good rule of life your data spacing sets your
  data resolution

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Rational grid interval

• Grid interval gt 0.25 x line spacing
• Over-gridding slows down the computer
• Over-gridding often yields misleading error
  estimates

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Valid gradient data

• If the original data are undersampled and
  aliased, the grids will be even more aliased.
• If the gradients are calculated using Fourier
  methods, beware of Miller effect (edge ringing).
• Always check the gradient grids.

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Rational window size

• Window - as small as possible to avoid seeing
  adjacent structure.
• Window large enough to define curvature properly.
• Useful solutions are seldom returned from gt 2
  window width

Point data
Profile data
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Meaningful Structural Index

• Structural Index is not a tuning parameter, to
  be varied until the depth returned is right on
  average.
• It has structural/geological meaning.

Source SI Mag SI Grav grad. SI Grav
Sphere 3 3 2
Vertical pipe 2 2 1
Hor. Cylinder (pipeline) 2 2 1
Thin bed fault 2 2 1
Thin sheet edge 1 1 0
Infinite contact/fault 0 0 Not useful
I am not aware of any other valid source types
or SI values for conventional Euler methods
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A horrible (egregious) example of Euler abuse

• Tedla, G. E., van der Meijde, M., Nyblade, A. A.
  and van der Meer, F. D., 2011
• A crustal thickness map of Africa derived from a
  global gravity field model using Euler
  deconvolution. Geophysical Journal International,
  187, 19.
• Reid,A.B., Ebbing, J., and Webb, S.J.,
• 2012
• Comment on A crustal thickness map of Africa
  derived froma global gravity field model using
  Euler deconvolution by Getachew E. Tedla, M. van
  der Meijde, A. A. Nyblade and F. D. van der Meer.
  Geophysical Journal International, 189,
  12171222.

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Well chosen geological problem ?

• Estimate crustal thickness.

Assumes base of crust is a smoothly varying
surface. No edges. One of our intractable
problems
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Adequately prepared and sampled magnetic or
gravity field ?

• Used EIGEN-GL04C gravity model (Förste et al
  2008).
• Is a spherical harmonic model of order and degree
  360.
• Contains wavelengths longer than 1 (? 100 km).
• Free air gravity, so the effect of topography was
  ignored (but what about isostasy ?).
• We suggest they should have used the Bouguer
  anomaly.

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EIGEN-GL04C gravity model

• 1000 km high-pass filtered
• satellite gravity

Taoudeni Basin
Congo Basin
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Valid gradient data ?

• Gradients were calculated from the grid.
• No ringing OK

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Rational grid interval ?

• Spherical harmonic model was represented by a
  grid at an interval of 0.25 (OK to represent
  wavelengths of 1).
• Reprojected to World Mercator (OK-ish).
• Cartesian projection is necessary, but Mercator
  brings scale distortions up to 15 at Cairo and
  20 at Cape Town. Will yield similar depth
  distortions (not the best).
• Was regridded to 5 km. Since 0.25 is about 25
  km, the regridding to 5 km adds nothing. (Not OK)
  
• 5 x over-gridding

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Rational window size ?

• Used a 20 x 20 km window size.
• 20 of the shortest wavelength in the data.
• lt original grid interval.
• Not OK.
• Cannot be expected to produce valid results.

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Meaningful Structural Index ?

• Chose an SI of 0.5. It is not an integer.
• For gravity this is somehow intermediate between
  a thin sheet edge and a horizontal line source.

From Tedla et al, (2011). SI chosen for best
fit with other depth estimates.
Missing points?
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Did it work?
Southern Africa
Tedla et al 2011. Webb et al, 2009
(Seismic). Difference
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Scoresheet

• Well chosen geological problem? No. Intractable
  model
• Adequately sampled magnetic or gravity field ?
  Low resolution, free air gravity, poor projection
• Grid interval ? Over-gridded
• Valid gradient data ? OK
• Rational window size ? Much too small (lt data
  interval)
• Meaningful Structural Index ? No. SI 0.5,
  chosen by tuning
• Final result? Unreliable

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Conclusion
GIGO (Garbage In Garbage Out) BATS (But At
Tremendous Speed) Do not use sophisticated
commercial software unless you understand the
assumptions, requirements and pitfalls.
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The End
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Thin sheet edge -gt SIM 1 Faulted thin bed
-gt SIM 2
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Common misunderstandings

• Mag SI 0.5 is often used for a thick dyke or
  thick step.
• BUT a non-integer SI is not constant. It varies
  with distance.

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Kuttikul (ITC,1995) Sprays and dip