### EgregiousEuler

```Egregious Euler Errors – the use and abuse of
Euler deconvolution applied to potential
fields
“Potential Field Methods II" session,
Thursday 7 June 2012, 15:55-16:20.
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
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
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 Euler’s
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)
What’s 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)
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
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.
Conventional 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.
Single-point sources
Sphere (eg Orebody).
Mag SI=3. Grav SI=2.
Depth to C of M/G
Pipe (eg kimberlite).
Mag SI = 2. Grav SI=1. Grav gradient SI=2.
Depth to centre of top.
Thin-bed fault. Mag SI = 2.
Depth to midpoint (d + 0.5 Δd).
Single-point sources
Dyke top, sill edge
Mag SI = 1
Gravity hard to detect
Edge must be isolated
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
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.
The “Moving Window”
Euler window
Move the window over the grid and at each
position, solve the Euler differential equation
What are the critical parameters ?
• Well chosen geological problem
• Adequately prepared and sampled magnetic or gravity field
(no aliasing)
• Grid interval
• Rational window size
• Meaningful Structural Index
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.
-follows Reid (1980)
To avoid aliasing
Field
Profile/sample spacing
Magnetic
<= Depth
<= Depth
Gravity
<= 2 x Depth
Good “rule of life” – your data spacing sets your data resolution
Rational grid interval
• Grid interval >= 0.25 x line spacing
• Over-gridding slows down the computer
• Over-gridding often yields misleading error estimates
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).
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 > 2 * window width
Profile data
Point data
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
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
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, 1–9.
Reid,A.B., Ebbing, J., and Webb, S.J.,
[2012]
Comment on ‘A crustal thickness map of Africa derived from
a 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, 1217–1222.
Well chosen geological problem ?
Estimate crustal thickness.
Assumes base of crust is a smoothly varying surface.
No “edges”.
One of our “intractable problems”
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° (λ ~ 100 km).
Free air gravity, so the effect of topography was ignored (but
We suggest they should have used the Bouguer anomaly.
EIGEN-GL04C gravity model
Taoudeni
Basin
Congo Basin
1000 km high-pass filtered
satellite gravity
calculated from the
grid.
No “ringing” – OK
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
Rational window size ?
Used a 20 x 20 km window size.
20% of the shortest wavelength in the data.
< original grid interval.
Not OK.
Cannot be expected to produce valid results.
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?
Did it work?
Southern Africa
Tedla et al 2011.
Webb et al, 2009 (Seismic).
Difference
Scoresheet
•
Well chosen geological problem? No. Intractable model
•
Adequately sampled magnetic or gravity field ? Low resolution, free air gravity,
•
Grid interval ? Over-gridded
•
•
Rational window size ? Much too small (< data interval)
•
Meaningful Structural Index ? No. SI = 0.5, chosen by “tuning”
•
Final result? Unreliable
poor projection
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.
The End
Thin sheet edge -> SIM = 1
Faulted thin bed -> SIM = 2
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.
Kuttikul (ITC,1995): Sprays and dip
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