GEOL 335.3

Refraction Seismic Method

Intercept times and apparent velocities;Critical and crossover distances;Hidden layers;Determination of the refractor velocity and depth;The case of dipping refractorInversion methods:

Hagedoorn plus-minus method;Generalized Reciprocal Method;Travel-time continuation.

Reading:➢ Reynolds, Chapter 5➢ Shearer, Chapter 4➢ Telford et al., Sections 4.7.9, 4.9

GEOL 335.3

Refraction Seismic Method

Uses travel times of refracted arrivals to derive:1) Depths to velocity contrasts (“refractors”);2) Shapes of refracting boundaries;3) Seismic velocities.

GEOL 335.3

Apparent VelocityRelation to wavefronts

Apparent velocity, Vapp,

is the velocity at which the wavefront sweeps across the geophone spread.

Because the wavefront also propagates upward, V

app, ≥ V

true:

A Cθ

B

wavefront

Prop

agat

ion

dire

ctio

n

Apparent propagation direction

AC=BC

sinV app=

Vsin

.

2 extreme cases:➢ θ = 0: V

app = ∞;

➢ θ = 90°: Vapp

= Vtrue

.

GEOL 335.3

Two-layer problemOne reflection and one refraction

xcritical

S

ic

i V1

V2>V

1

Direct

HeadwaveRefracted

Refle

cted

x

t

xcrossover

Direct

Reflected: Head

pre-criticalpost-critical

t=t 0x

V 2=t0x p2

t= xV 1

=x p1

t0

At pre-critical offsets,record direct wave and reflection

In post-critical domain,record direct wave, refraction, and reflection

h1

GEOL 335.3

Travel-time relationsTwo horizontal layers

For a head wave (“often called refraction”):

For a reflection (we'll use this later):

this also equals sin i

intercept time

sin ic= pV 1 cos i c=1− pV 12p= 1

V 2

t=2h1

V 1 cos icp x−2h1 tan ic = t0px

t 0=2h1

V 1 cos ic1−pV 1sin i c=

2h1

V 1cos ic

t=2 h12 x

2V 1

=4 h12 x2

V 1

pV 1=sin i

tan i= x2h1 here, p is variable and controlled

By arbitrary angle i

“linear moveout” term

hyperbolicmoveout”

ic

i V1

V2>V

1

h1

S

S'

R

A B R'

x

intercept time

GEOL 335.3

Critical and cross-over distances

Critical distance:

Cross-over distance:

xcritical=2 h1 tan ic=2 h1

V 1/V 2

1−V 1/V 22=

2 h1V 1

V 22−V 1

2

ic

V1

V2>V

1

h1

S

A B

xcritical x

crossover

t direct xcrossover=t headwave xcrossover

xcrossover

V 1=t 0

xcrossover

V 2

xcrossover=t 0

1/V 1−1/V 2

“slownesses”

GEOL 335.3

Multiple-layer case(Horizontal layering)

p is the same critical ray parameter;t0 is

accumulated across the layers:

p= 1V refractor

t=∑k=1

n 2hk

V kcos i kpx

Critical p determined here

sin i k= pV k in any layer

GEOL 335.3

Dipping Refractor Caseshooting down-dip

S

ic

V1

V2>V

1

hd

Rx

α (dip)

ic

huAB

R'xsinα

x(cosa-sinα tanic)

xsinα/cosic

sinic

would change to '-' for up-dip recording

t=2hd

V 1cos ic

1V 2

x cos−sin tan i c1

V 1

x sincos ic

t=2hd

V 1cos ic

xV 1 cos ic [V 1

V 2cos cos i c−sinsin i csin]

t=2hd

V 1cos ic

xV 1

cossin i csin cos ic

t=2hd

V 1cos ic

xV 1

sin ic

GEOL 335.3

Refraction InterpretationReversed travel times

One needs reversed recording (in opposite directions) for resolution of dips.

The reciprocal times, TR, must be the the same for

reversed shots.

Dipping refractor is indicated by:Different apparent velocities (=1/p, TTC slopes) in the two directions;

➢ determine V2 and α (refractor velocity and dip).

Different intercept times.➢ determine h

d and h

u (interface depths).

Sx

t

R

TR

2zd cosic

V 1

2zu cosic

V 1

slope=p1=1V 1

pu=sin ic−

V 1

pd=sin ic

V 1

GEOL 335.3

Determination of Refractor Velocity and Dip

Apparent velocity is Vapp

= 1/p, where p is the ray parameter (i.e., slope of the travel-time curve).

Apparent velocities are measured directly from the observed TTCs;V

app = V

refractor only in the case of a horizontal layering.

For a dipping refractor: ➢ Down dip: (slower than V

1);

➢ Up-dip: (faster).

From the two reversed apparent velocities, ic and α

are determined:

V d=V 1

sin ic

V u=V 1

sinic−

ic=12sin−1 V 1

V dsin−1 V 1

V u ,

ic−=sin−1 V 1

V u.ic=sin−1 V 1

V d,

=12sin−1 V 1

V d−sin−1 V 1

V u .

From ic, the refractor velocity is: V 2=

V 1

sin i c.

GEOL 335.3

Determination of Refractor Depth

From the intercept times, td and t

u, refractor depth is

determined:

hd=V 1 t d

2cos ic,

hu=V 1 tu

2 cos ic.

S

ic

V1

V2>V

1

hd

Rx

α (dip)

ic

huAB

GEOL 335.3

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: t

SR = t

SX + t

XR.

Relate the time tSX

to a time along the refractor, tBX

: tSX

= tSA

– tBA

+ tBX

= tS Delay

+x/V2.

S

ic

V1

V2>V

1

hs

x

BA X

Rh/cosic

htanic

t S Delay=SAV 1

− BAV 2

=hs

V 1cos i c−

hs tan i c

V 2=

hs

V 1 cos ic1−sin2i c=

hs cosic

V 1.

Note that V2=V

1/sini

c

Thus, source and receiver delay times are:

t S , R Delay=hs ,r cosi c

V 1.

t SR=t S DelaytR DelaySRV 2.

and

hr

GEOL 335.3

Plus-Minus Method(Weathering correction; Hagedoorn)

Assume that we have recorded two headwaves in opposite directions, and have estimated the velocity of overburden, V

1.

How can we map the refracting boundary?

S1

V1

D(x) S2

S1

x

t

S2

TR

tS1 D

tS2 D

DSolution:

➢ Profile S1 → S

2:

➢ Profile S2 → S

1:

Form PLUS travel-time:

t S 1 D=x

V 2t S1

t D ;

t S 2 D=S1 S 2−x

V 2t S2

tD.

tPLUS= t S1 Dt S 2 D=S1 S 2

V 2t S1

t S 22tD=t S 1 S 2

2tD.

tD=12tPLUS−t S 1 S 2

.Hence:

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

2.

GEOL 335.3

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:

tMINUS=t S 1 D−t S 2 D=2 xV 2

−S 1 S 2

V 2t s1

−t s2.

slope[ tMINUS x ]=2

V 2.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!

GEOL 335.3

Generalized Reciprocal Method (GRM)

S1

V1

D S2

S1

x

t

S2

TR

tS1 D

tS2 D

DThe 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:1) Corresponding to the most linear velocity analysis function;

2) Corresponding to the most detail of the refractor.XY

XY

t D=12t S 1 Dt S 2 D−t S 1 S 2

−XYV 2 .

t V=12 t S 1 D−t S 2 Dt S 1 S 2 should be linear, slope = 1/V

2;

The time-depth function:

this is related to the desired depth: hD=tD V 1 V 2

V 22−V 1

2

GEOL 335.3

Head-wave “migration”(travel-time continuation) method

S1

S2

“Migration” refers to transforming the space-time picture (travel-time curves here) into a depth image (position of refractor). Refraction (head-wave) migration:

Using the observed travel times, draw the head-wave wavefronts in depth;Identify the surface on which:

This surface is the position of the refractor.

P

t forward x , z t reversed x , z =T R

Constant traveltimes from S

1

Constant traveltimes from S

2

x

z

GEOL 335.3

Phantoming

Refraction imaging methods work within the region sampled by head waves, that is, beyond critical distances from the shots;In order to extend this coverage to the shot points, phantoming can be used:

Head wave arrivals are extended using time-shifted picks from other shots;However, this can be done only when horizontal structural variations are small.

x

t

Phantom arrivals

GEOL 335.3

Hidden-Layer Problem

Velocity contrasts may not be visible in refraction (first-arrival) travel times. Three typical cases:

Low-velocity layers;

Relatively thin layers on top of a strong velocity contrast;

Short travel-time branch may be missed with sparse geophone coverage.