Strain Error Analysis
Consider sampling a prototype shear profile with displacement
u ,u  0, U arctan x Dx 
0
x
y
,
(S1)
which corresponds toa vertical fault strike-slipping U from infinite depth to depth D, and
locked from there to the surface, at four points arranged in a square, with side length S,
center of the square at distance x0 from the fault plane, rotation angle between the fault
plane and the square side . If the strain components are estimated from finding the
least-square plane fit to these samples,
uˆ y,x
 S
 S
 1  
cos   x 0   
 cos arctan 
 D 
 2
 2

 S

 S

1

sin    x 0  
 sin  arctan 
 D  
U  2
 2
  2 

S  S
 S
 1  
cos arctan  cos   x 0  
 2
 D  
 2


 S
 1  
 S
 sin    x 0  
 2 sin  arctan 
 D  
 2

(S2)
uˆ y,y
 S
 S
 1  
cos   x 0 
 sin  arctan 
 D 
 2
 2

 S
 S
 1  
sin    x 0  
 cos arctan 
 D  
U  2
 2
  2 

S  S
 S
 1  
sin  arctan  cos   x 0 
 2
 D  
 2


 S
 1  
 S
 sin    x 0  
 2 cos arctan 
 D  
 2

(S3)

1

Estimation by finite difference has a similar result.
For this special geometry, the estimated shear magnitude is
2
2
,
sˆ  uˆ y,y
 uˆ y,x
(S4)
Errors are determined by differencing with the true value (obtained by direct

differentiation of (S1)
s 
The estimated dilatation is
U
D
 D  x 02

2

(S5)

ˆ u  uˆ

y,x
(S6)
Note (2–3) converge quadratically to the correct strain for sample spacings smaller than

about D. This can be demonstrated by retaining the first two terms of the Taylor series
for arctan(z) (z–z3/3). The linear term (z) leads to complete cancellation of terms in (S2);
the cubic term leads to an error term, which is quadratic in S. Similar results obtain for
(S3), but rather than cancellation the result converges to (S5). The dilatation estimate
(and error) from this quadratic term has the form
ˆ u   US sin( 4 ) ,

24D3
2
(S7)
which implies four grid orientations of maximum error including 22.5 degrees. This

formula indicates a fault with maximum shear U/D will have an error in the dilatation
estimate (for a worst case sampling orientation) of approximately 5% the maximum shear
2
magnitude when the spacing S is twice the fault depth D, and 1.3% when S = D. Errors in
shear magnitude are of similar order.
Can this behavior be improved by including more distant points to form weighted leastsquare estimates? Clearly not: the arctan function flattens out to a horizontal asymptote,
so the addition of any points farther from the fault plane with positive weight will tend to
reduce the estimate of strain obtained; the four point estimate of uy,y is already too small.
Can this behavior be improved by using higher-order polynomial fits or splines, taking
additional data points? Again, no: because the arctan function flattens out to a horizontal
asymptote (a non-polynomial behavior), it cannot be approximated by a polynomial over
a range much greater than D. For example interpolation from sampling on an equal
spacing grid results in wild behavior between samples, due to a variant of the Runge
phenomenon (Dahlquist and Björk, 1974), which precludes obtaining valid derivatives of
deformation components at or between samples.
The critical problem with estimating strain from a sparse network of deformation
measurements is a cascade of errors. Since the sample density is not sufficiently dense
(less than about D/2 for presumed fault locking depth D), the first source of error is the
undersampling of a natural phenomenon, giving rise to aliasing. This implies information
at some or all the spatial wavelengths is corrupted by natural signal at higher spatial
wavelengths, erroneously mixed in with the lower spatial wavelengths. The second
source of error is using such aliased signal to estimate spatial derivatives: no matter what
3
method is used, estimation of derivatives is sensitive to the range of signal with highest
spatial frequency. The third source of error is computing combining functions of these
spatial derivatives to compute dilatation and strain magnitude. The dilatation, in
particular, is estimated as the difference of two comparatively large numbers, amplifying
any error present in the spatial derivative estimates.
References
Dahlquist, Germund ; Björk, Åke (1974), "4.3.4. Equidistant Interpolation and the Runge
Phenomenon", Numerical Methods, pp. 101–103, ISBN 0-13-627315-7
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