Refraction Statics

Plus-Minus method;

Generalized Linear Inverse (GLI, tomographic) method;

Delay-time ('time-term') method;

L1 and L2 norms.

Reading:

Yilmaz (refraction statics section and Appendix C in the new edition)

Delay time

Consider a nearly horizontal, shallow interface with strong velocity contrast (a typical case for weathering layer).

In this case, we can separate the times associated with the source and receiver vicinities: tSR = tSX + tXR.

Relate the time tSX to a time along the refractor, tBX: tSX = tSA – tBA + tBX = tS Delay+x/V2.

S

ic

V1

V2>V1

hs

x

B

A

X

R

h/cosic

htanic

Note that V2=V1/sinic

Thus, source and receiver delay times are:

and

hr

Plus-Minus Method
(Weathering correction; Hagedoorn)

Assume that we have recorded two head-waves in opposite directions, and have estimated the velocity of overburden, V1.

How can we map the refracting interface?

S1

V1

D(x)

S2

S1

x

t

S2

TR

tS1 D

tS2 D

D

Solution:

Profile S1  S2:

Profile S2  S1:

Form PLUS travel-time:

Hence:

To determine ic (and depth), still need to find V2.

Plus-Minus Method
(Continued)

S1

V1

D(x)

S2

S1

x

t

S2

TR

tS1 D

tS2 D

D

To determine V2:

Form MINUS travel-time:

Hence:

The slope is usually estimated by using the Least Squares method.

Drawback of this method – averaging over the pre-critical region.

this is a constant!

Generalized Reciprocal Method (GRM)

S1

V1

D

S2

S1

x

t

S2

TR

tS1 D

tS2 D

D

The velocity analysis function:

Introduces offsets ('XY') in travel-time readings in the forward and reverse shots;

so that the imaging is targeted on a compact interface region.

Proceeds as the plus-minus method;

Determines the 'optimal' XY:

Corresponding to the most linear velocity analysis function;

Corresponding to the most detail of the refractor.

XY

XY

should be linear, slope = 1/V2;

The time-depth function:

this is related to the desired image:

Example:
Refraction survey over a proposed waste disposal site
(Lankston, 1990), recorded travel times

Reverse-direction travel times

Forward first-arrival travel times

Refraction survey over a proposed waste disposal site
(Lankston, 1990)

Phantomed forward- and reverse-direction first-arrival travel times

Refraction survey over a proposed waste disposal site
(Lankston, 1990), velocity analysis

Time-distance function

Choose the smoothest one

Choose the one with best detail

Velocity analysis function

Refraction survey over a proposed waste disposal site
(Lankston, 1990), results

Velocity variation

Depth variation

(note vertical exaggeration)

Two changes in velocity

The interface is an envelope

of circular arcs Dt=const

Generalized Linear Inverse

Usually overdetermined travel-time inversion problem

Impossible to fit all times at once;

Trying to minimize (tij - tij observed)2

Solved using damped least squares method.

In matrix form:

Regularized using damping parameter, m:

Generalized Linear Inverse Method
(Hampson and Russel (1984), Amorim et al. (1987))

Tomographic model

Parameterized in terms of slownesses within the weathering (LVZ) layer

s

s'

These equations can be viewed in two ways:

fixed H, linear equations in s and s' (fixed-depth scheme);

fixed s, linear equations in H and s' (variable- depth scheme);

GLI Refraction Statics

Iterations to reduce dt:

start

finish

Delay-time (time-term) method

Linear inverse model for delay times (DtS and DtR) and retractor slowness s = 1/V:

DtS and DtR are usually surface consistent (dependent on common locations only), and s can be spatially variant

Spatial variability described using a grid, similarly to GLI method.

Effects of statics in shot gathers

no statics

conventional

statics

GLI

(tomography)

statics

Effects of statics in stacked image

conventional

statics

GLI

statics

L1 vs. L2 Norms

Most tomographic methods use the so-called L2 norm: (tij - tij observed)2

However, L2 is not the best norm:

sensitive to outliers (wild mis-picks).

Other used norms are:

L1 = |tij - tij observed| (more stable in respect to outliers);

L = max|tij - tij observed| (extremely sensitive to outliers, the most stringent norm).

L1-norm is inverted using linear programming (simplex) techniques.

For good picks (not plagued by outliers) the results of L2 and L1-norm inversion are very close.

Refraction_Statics

GEOL882

Igor Morozov

2003-01-21T21:24:09

Igor Morozov