Refraction- III -

Ali K. Abdelfattah

Geology Department

Collage of science

King Saud University

Delay Time Method

•Allows Calculation of Depth Beneath Each Geophone

• Requires refracted arrival at each geophone from opposite directions

• Requires offset shots

• Data redundancy is important

Delay Time Methodx

V1

V2

•Irregular travel time curves is due to bedrock topography or glacial fill, much analysis is based on delay times.

•Total Delay Time is the difference in travel time along actual ray path and projection of ray path along refracting interface.

Delay Time Methodx

V1

V2

2

)

21

) )tan((

)cos(

(

V

ihh

V

AB

iV

hhT

cBA

c

BAAB

Delay Time Methodx

2

)

21

) )tan((

)cos(

(

V

ihh

V

AP

iV

hhT

cPA

c

PAAP

V1

V2

2

)

21

) )tan((

)cos(

(

V

ihh

V

AB

iV

hhT

cBA

c

BAAB

Delay Time Methodx

2

)

21

) )tan((

)cos(

(

V

ihh

V

BP

iV

hhT

cPB

c

PBBP

V1

V2

2

)

21

) )tan((

)cos(

(

V

ihh

V

AB

iV

hhT

cBA

c

BAAB

2

)

21

) )tan((

)cos(

(

V

ihh

V

AP

iV

hhT

cPA

c

PAAP

Delay Time Methodx

t T T TAP BP AB0

Definition:

V1

V2

(1)

ABBPAP TTTt 0

)cos(

)tan()tan(

)cos( 122210

c

PcPcA

c

A

iV

h

V

ih

V

ih

V

AP

iV

ht

)cos(

)tan()tan(

)cos( 12221 c

PcPcB

c

B

iV

h

V

ih

V

ih

V

BP

iV

h

)cos(

)tan()tan(

)cos( 12221 c

BcBcA

c

A

iV

h

V

ih

V

ih

V

AB

iV

h

2120

)tan(2

)cos(

2

V

ih

iV

h

V

ABBPAPt

cP

c

p

But from figure above, BPAPAB . Substituting, we get

2120

)tan(2

)cos(

2

V

ih

iV

h

V

BPAPBPAPt

cP

c

p

or

210

)tan(2

)cos(

2

V

ih

iV

ht

cP

c

p

)cos(

)sin(

)cos(

12

210

c

c

cp

iV

i

iVht

)cos(

)sin(

)cos(2

21

1

21

20

c

c

cp

iVV

iV

iVV

Vht

)cos(

)sin(

)cos(2

2121

1

2

10c

c

cp

iVV

i

iVVVV

Vht

2

1sin

V

VicSubstituting from Snell’s Law,

)cos(

)sin(

)cos(sin

1

22121

10c

c

c

cp

iVV

i

iVViVht

)cos(

)sin(

)cos(sin

1

22121

10c

c

c

cp

iVV

i

iVViVht

Multiplying top and bottom by sin(ic)

)cos()sin(

)(sin

)cos()sin(

12

21

2

2110

cc

c

ccp

iiVV

i

iiVVVht

)cos()sin(

)(cos2

21

2

10cc

cp

iiVV

iVht

)sin(

)cos(2

20

c

cp

iV

iht

)sin(

)cos(2

20

c

cp

iV

iht

2

1sin

V

Vic

Substituting from Snell’s Law,

10

)cos(2

V

iht

cp (2)

We get

11

)cos(

2

)cos(2

2 Ppoint at Delay time

V

ih

V

ihtD

cpcpoTP (3)

Reduced Traveltimes

Definition:

T’AP = “Reduced Traveltime” at point P for a source at A

T’AP=TAP’

x

Reduced travel times are useful for determining V2. A plot of T’ vs. x will be roughly linear, mostly unaffected by changes in layer thickness, and the slope will be 1/V2.

Reduced Traveltimesx

From the above figure, T’AP is also equal to TAP minus the Delay Time. From equation 9, we then get

2'

oAPTAPAP

tTDTT P

Reduced Traveltimesx

Earlier, we defined to as

t T T TAP BP AB0 Substituting, we get

22'

ABBPAPAP

oAPAP

TTTT

tTT

(1)

(4)

Reduced Traveltimes

T

T T TAP

AB AP BP'

2 2

Finally, rearranging yields

The above equation allows a graphical determination of the T’ curve. TAB is called the reciprocal time.

(5)

Reduced Traveltimes

TT T T

APAB AP BP

'

2 2The first term is represented by the dotted line below:

Reduced Traveltimes

TT T T

APAB AP BP

'

2 2The numerator of the second term is just the difference in the traveltimes from points A to P and B to P.

Reduced Traveltimes

TT T T

APAB AP BP

'

2 2Important: The second term only applies to refracted arrivals. It does not apply outside the zone of “overlap”, shown in yellow below.

Reduced Traveltimes

TT T T

APAB AP BP

'

2 2The T’ (reduced traveltime) curve can now be determined graphically by adding (TAP-TBP)/2 to the TAB/2 line. The slope of the T’ curve is 1/V2.

We can now calculate the delay time at point P. From Equation 4, we see that

1

)cos(

2 V

iht cpo

According to equation 8

2'

oAPAP

tTT

1

0 )cos(

2'

V

ihT

tTT

cpAPAPAP

So

Now, referring back to the equation of refracted waves

212

)cos(2

V

x

V

ihT

c

(6)

(2)

(4)

It’s fair to say that

21

)cos(2

V

x

V

ihT

cpAP

Combining equations 12 and 13, we get

1211

)cos()cos(2)cos('

V

ih

V

x

V

ih

V

ihTT

cpcpcpAPAP

Or

21

)cos('

V

x

V

ihT

cpAP

(7)

(8)

1

)cos(

V

ihD

cpTp

Referring back to equation 3, we see that

Substituting into equation 8, we get

221

)cos('

V

xD

V

x

V

ihT pT

cpAP

Or

2'

V

xTD APTp

hD V

iP

T

c

P

1

co s( )

Solving equation 9 for hp, we get

(9)

(10)

(3)

We know that the incident angle i is critical when r is 90o. From Snell’s Law,

2

1

sin

sin

V

V

r

i

2

1

90sin

sin

V

Vic

2

1sin

V

Vic

2

11sinV

Vic

Substituting back into equation 16,

)cos(

1

c

Tp

i

VDh

p

2

11

1

sincosVV

VDh

pTp

(10)

(11)

we get

or

21

22

21

VV

VVDh PTp

In summary, to determine the depth to the refractor h at any given point p:

1.Measure V1 directly from the traveltime plot.

2.Measure the difference in traveltime to point P from opposing shots (zone of overlap only).

3.Measure the reciprocal time TAB.

4. From equation 5,

TT T T

APAB AP BP

'

2 2

divide the reciprocal time TAB by 2.

,

5. From equation 5,

TT T T

APAB AP BP

'

2 2add ½ the difference time at each point P to TAB/2 to get the reduced traveltime at P, T’AP.

,

6. Fit a line to the reduced traveltimes, compute V2 from slope.

2'

V

xTD APTp

7. Using equation 15,

Calculate the Delay Time DT at P1, P2, P3….PN

(9)

8. Using equation 17,

Calculate the Depth h at P1, P2,

P3….PN

2

11

1

sincosVV

VDh

pTp (10)

Faulted Planar Interface ( Diffraction )

• If refractor faulted, then there will be a sharp offset in the travel time curve.

• We can estimate throw on fault from offset in curves, i.e. difference between two intercept times, from simple formula:

21

22

21

VV

VVtZ

Blind layer problem

• Blind layers occur when there is a low velocity layer (LVL).

• Head waves only occur for a velocity increase. Thus, there will be no refraction from the top of the LVL.

• The LVL will not be detected on the time-distance plot.

• This is described below.

2

1

sin

sin

V

V

r

i

What if V2 < V1?

Snell’s Law

If V1>V2, then as i increases, r increases, but not as fast.

What if V2 < V1?

If V2<V1, the energy refracts toward the normal.

None of the refracted energy makes the rays back to the surface.

Seismic Refraction requires that velocities increase with depth.

A slower layer beneath a faster layer will not be detected by seismic refraction. This can lead to errors in depth calculations.

Hidden Layer Problem

• Layers may not be detected by first arrival analysis:

1- Travel time curve produces no critical refraction from layer 2

2- Insufficient velocity contrast makes refraction difficult to identify

3- Refraction from thin layer does not become first arrival

4- Geophone spacing too large to identify second refraction

Important: The Length of the Geophone Spread Should be 4-5 times the depth of interest.