Egregious Euler Errors – the use and abuse of Euler deconvolution applied to potential fields “ Potential Field Methods II" session, Thursday 7 June 2012,

Egregious Euler Errors – the use and abuse of

Euler deconvolution applied to potentialfields

“Potential Field Methods II" session,Thursday 7 June 2012, 15:55-16:20.

EAGE Copenhagen ‘12

Alan Reid* Jörg Ebbing** Susan Webb***

*Reid Geophysics & Univ. of Leeds, UK

** Geological Survey of Norway (NGU)

*** University of the Witwatersrand, South Africa

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”

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)• Many academic versions

What’s behind Euler?

SI = - degree (i.e. n = - N)

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.

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 etcBackground 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

Thin-bed fault. Mag SI = 2.

Depth to midpoint (d + 0.5 Δd).

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.

Single-point sources

Dyke top, sill edge

Mag SI = 1 Gravity hard to detect Grav. Gradient SI = 1

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 Grav. Gradient SI = 0.

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

• Valid gradient data

• 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.

Adequate sampling for Euler work (includes gradients)

-follows Reid (1980)

Good “rule of life” – your data spacing sets your data resolution

Field Profile/sample spacing

Magnetic <= Depth

Gravity Gradient <= Depth

Gravity <= 2 x Depth

To avoid aliasing

Rational grid interval

•Grid interval >= 0.25 x line spacing

•Over-gridding slows down the computer

•Over-gridding often yields misleading error estimates

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.

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 IndexStructural 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

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 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,

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”

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° (λ ~ 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.

EIGEN-GL04C gravity model

1000 km high-pass filteredsatellite gravity

Congo Basin

Taoudeni Basin

Valid gradient data ?

Gradients were 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, poor projection

• Grid interval ? Over-gridded

• Valid gradient data ? OK

• Rational window size ? Much too small (< data interval)

• Meaningful Structural Index ? No. SI = 0.5, chosen by “tuning”

• Final result? Unreliable

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