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GEOPHYSICAL PROSPECTING

Geophysical prospecting

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GEOPHYSICAL PROSPECTING

Average Velocity This is simply the depth z of a reflecting surface below a datumdivided by the observed one – way reflection time t from the datum to the surface sothat.

( 7 – 22 )

If z represents the sum of the thicknesses of layers z1, z2, z3, …, zn, the average velocityis defined as

( 7 – 23 )

The average velocity is used for time – to – depth conversions and for migration.

Interval Velocity if two reflectors at depths z1 and z2 give reflections having respectiveone – way time of t1 and t2, the interval velocity Vint between z1 and z2 is definedsimply as (z2 – z1) / (t2 – t1) .

Instantaneous Velocity. If the velocity varies continuously with depth, its value at a particulardepth z is obtained from the formula for interval velocity by contracting the interval z2 – z1 until itbe comes an infinite simally thin layer having a thickness dz. The interval velocity computed bythe formula above becomes the derivative of z with respect to t, and we designate it as theinstantaneous velocity Vinst, defined as

( 7 – 24 )

1 2 3 1

1 2 3

1

av

n

k

nav n

nk

zV

t

zz z z z

Vt t t t

t

inst

dzV

dt

Root – Mean – Square Velocity. If the section consists of horizontal layers withrespective interval velocities of V1, V2, V3, . . . , Vn, and one – way interval travel timet1, t2, t3, . . . , tn, the root – mean – square (rms) velocity is obtained from the relation

( 7 – 25 )

The slope of this line in t2, x2 space, using the reflection from a flat bed, is shown inFig. 7.22. Kleyn15 offers a further discussion of velocity is obtained from the relation

Stacking Velocity Stacking velocity, Vst , is based on the relation

( 7 – 26 )

The best fit over all offsets of hyperbolic move outs derived from Eq. (7.26) to theactual reflection events. In general.

2

2 2 2 2

1 1 2 2 3 3 1

1 2 3

1

n

k k

n nrms n

nk

V tV t V t V t V t

Vt t t t

t

22 2

0 2

st

av rms st

xT T

V

V V V

Mirror or fold plane, so that

the times ( T, T0) are two

way time.

T

½T½T½T0

½T0 ½T

T0

X

Offset

Source Reseive

r

2

2 2

0

XT T

V

Figure 7 . 22. A simple derivation of the normal-move out equation uses the zero offset time T0,the time T tc a given offset s, a fold plane at thereflector, the Pythagorean theorem Vst, thestacking velocity, is the processing parameterthat achieves the best time alignment of areflection at an offset with the zero – offsetreflection time.

x

Sd

Shot ZQ

D

Well

Detector

1

cos SD

Z d

T

Recording Truck

16000

14000

12000

10000

8000

6000

4000

2000

20001000 3000 4000 5000 6000 7000

Depth ft

Vel

oci

ty f

t/se

c

Interval Velocity

Average overall velocity, V̅

Detector positions

Figure 7 – 23 Well shootingarrangement with typical interval –velocity and average – velocity curvesthus obtained.

t

A B

Z True locations of reflecting

points

True locations of reflecting

points

Apparent reflecting points positions

when plotted vertically below

reflection spreads

(a) (b)

Figure 7 – 39 (a). Tight syncline showing the true and apparent spatial locations of thereflected doing energy. (b) The “bow- tie effect” that is the time response over thesyncline. A further discussion of this effect is found in Secs. 8-3 and 8-5.

The need for migration has been recognized since the first reflection surveys [Jakosky30 (pp. 670 – 696)]. Figure 7 -40 traces the development of migrationtechniques

Graphical MethodsStraight Ray Curved Ray path – wave front ChartsDiffraction Overlays

Digital Computation Ray Tracing Diffraction Summation Wave front Interference

Wave Equation (CDP Data)Finite Differences Frequency Domain Kirchhoff (Summation)

Imaging in depth Before Sack

Figure 7 – 40 Development of migration techniques. ( From Johnson and French. 32)

When discussing migration topics, certain variables have widespread preassingnedmeanings :

z = deptht = one – way timeω = frequencyx = spatial location (the midpoint axis)y = the second spatial coordinate, orthogonal to x, used in 3 – d situationsv = velocity

Manual and Graphical MethodsStraight – Ray (Constant – Velocity) Manual Method . Manual approximate migrationcan be performed by use of the correct average velocity overlying the dipping event,on the assumption of stratified media. Using Claerbout ‘S14 formulation , we canassociatiate an apparent location (t0, x0) and an apparent time dip θa = dt/dx, witheach reflection on the CDP stack. We wish to find its migrated position (tm, xmo) anddip, θm. From Fig. 7 – 41 we see that

( 7 – 34 )

Where v is the overlying velocity. The basic geometric equation linking the unmigrated and the migrated dips is

( 7 – 35 )

sin m

dtv

dx

sin tanm m

0 xm0 xm1 x0 x1

θmθaθm

θa

VtM

VtD’ VtD

D

Vdt

Figure 7 – 41 . Diagram of terms used in development of hand – migration equations.θm = migrated dip θa = apparent dip. The basic geometric equation linking theunmigrated and migrated dips is sin θm = tan θa .

Which can be observed from triangles OX1D’ and OX1D.

( 7 – 36 )

The migrated time is

( 7 – 37 )The lateral location after migration is

( 7 – 38 )

Where v sin θa is horizontal component of velocity. The migrated dip of the reflectionsegment, ρm, is given by

( 7 – 39 )

2 2

0

2

0 0 0 0

2 2

tan

cos 1

sin

tan

1

a

dtm a a dx

dtmo a dx

a am

a

dtv

dx

t t t v

x x t v x t v

pv v

0

1

2

3

4-4 -2 2 4 km sec

( a )

-4 -2 2 4 km0

0

1

2

3

4sec

0

( b )

Figure 7 – 42(a). Diffraction overlay, also

known as a surface ofmaximum convexity.

(b). Wave front chart,showing the position ofseismic wave front at100 – ms increments oftime, for a givenvelocity function. (fromKleyn. 15)

S1 S2

R2

R1

R’2

R’1

D

X

T

MAC curve

Figure 7 43 . The use of the maximum convexity curve and a wave front curve tomigrate a dipping event, R1 to its correct location, R2, and determine its true dip. (fromKlen.15)