The Wolf Ramp: Reﬂection Characteristics of a Transition Layer · transition zone yields the...

The Wolf Ramp: Reflection Characteristics of a Transition

Layer

Christopher L. Liner and Bernhard G. Bodmann∗

∗The University of Houston,

4800 Calhoun Road,

Houston, Texas 77004

(March 19, 2010)

GEOPHYSICS LETTERS

Running head: Wolf Ramp

ABSTRACT

The modern use of spectral decomposition has shown that reflection events in practice are

always frequency dependent, a phenomenon we call reflectivity dispersion. This can often

be attributed to strong interference effects from neighboring reflection coefficients of the

classical type (i.e., parameter discontinuities or jumps). However, an intrinsic frequency

dependence from a single layer is possible if the contact is not a jump discontinuity but

a gradual transition. Here we revisit and expand the normal incidence theory of a linear

velocity transition zone (termed a Wolf ramp) and show how it leads to frequency-dependent

reflectivity. The development of waveform forward modeling leads to a ramp detection

method that is demonstrated on migrated field data.

1

of 24 Geophysics Manuscript, Accepted Pending: For Review Not Production

INTRODUCTION

The development of seismic spectral decomposition (Gridley and Partyka, 1997), and the

more general field of time-frequency representations (Goupillaud et al., 1983; Chakraborty

and Okaya, 1995), has invited close scrutiny of seismic reflection events for frequency-

dependent behavior. This has effectively added an information-rich dimension to data

interpretation, analogous to the development of amplitude versus offset analysis (AVO).

In this paper, ‘reflectivity dispersion’ (or simply dispersion) refers to frequency depen-

dence of a normal incidence reflection event. Intrinsic dispersion occurs when the effect

is present for a single reflection event (two half-spaces in contact). Apparent dispersion is

spectral modification due to interference between several events. Mixed dispersion is some

combination of intrinsic and apparent type.

The known causes of intrinsic reflectivity dispersion are few: rough surface scatter-

ing (Clay and Medwin, 1977), Biot reflection (Geertsma and Smit, 1985), and vertical

transition zones (Wolf, 1937). For the development of future quantitative seismic interpre-

tation methods, it is necessary to model these effects and identify any characteristics that

can be used to distinguish them. We note in passing that normal incidence reflection from

an interface separating attenuating media also leads to dispersion as shown experimentally

by Wuenschel (1965).

The problem of normal incidence P-wave reflection from a linear velocity transition

ramp was first studied by Wolf (1937) who derived an exact solution. His analysis is limited

to constant density, but illustrates the nature of reflection from a linear transition zone.

Bortfeld (1960) considered the same problem from the viewpoint of internal multiples to

find a closed form solution. The general acoustic reflection problem (variable density and

2

of 24Geophysics Manuscript, Accepted Pending: For Review Not Production

oblique incidence) for a Wolf ramp has been treated by Gupta (1965). Transition zones have

also been studied in relation to synthetic seismograms (Berryman et al., 1958a,b; Sherwood,

1962; Wuenschel, 1960)

In the next section we review Wolf’s exact solution, extend his result for zero crossing

of the reflection coefficient as a function of frequency, and develop waveform simulation and

detection methods.

THEORY

Our analysis focusses on reflection due to a transition zone for the special case of constant

density and normal incidence (Wolf, 1937). We term this a Wolf ramp. Here we briefly

recount the theoretical development.

The earth model consists of an upper layer of velocity v, a linear transition zone of

thickness h, and a lower layer of velocity v2 = k v. The differential equation to be solved is

the one-dimensional constant-density elastic wave equation

∂

∂zV 2(z)

∂u

∂z=

∂2u

∂t2, (1)

where the velocity function is

V = v z < 0 (2)

V = v

(1 +

k − 1h

z

)0 < z < h (3)

V = k v z > 0 . (4)

Extending the linear part of the velocity function, we define the variable

s(z) = v

(1 +

k − 1h

z

), (5)

3


and its derivatives,

∂s

∂z= v

k − 1h

(6)

∂

∂z=

∂s

∂z

∂

∂s= v

k − 1h

∂

∂s. (7)

We take the Fourier transform with respect to time and express the wave equation in the

new variable as

v2(

1 +k − 1

h

)2 ∂

∂ss2 ∂u

∂s= ω2 u , (8)

where u = u(s, ω) and ω is angular frequency. Absorbing constants into the frequency

variable,

ω̄ =ω

v(1 + k−1

h

) , (9)

the transformed wave equation simplifies to

2s∂u

∂s+ s2 ∂2u

∂s2+ ω̄2 u = 0 ; u = u(s, ω̄) . (10)

The form of this differential equation suggests power-law solutions, so we make the following

ansatz

u = c sm , (11)

where c is a constant with respect to s. Substitution of this trial solution yields the poly-

nomial equation

2 m u + m(m− 1) u + ω̄2 u = 0 , (12)

that can be written in compact form as

[m(m + 1) + ω̄2

]u = 0 , (13)

identical to Equation 6 in Wolf (1937). The values of m are given by the roots of the brack-

eted polynomial. Applying the continuity of the field and its z-derivative across the velocity

4


transition zone yields the frequency-dependent normal incidence reflection coefficient for a

Wolf ramp

Rw(f) =1

2σ + 2γ coth(γ log(k)), (14)

where log(k) is the natural logarithm of k, frequency f is in Hz, and σ and γ depend on

frequency by

σ(f) =i 2π f h

(k − 1)v; γ(f) =

√1/4 + σ2 . (15)

A useful approximation to Rw(f) is given in Appendix A.

In the limit of zero frequency or thickness, σ = 0 and γ = 1/2, leading to

lim(f,h)→0

Rw(f) =1

coth(12 log(k))

=k − 1k + 1

= R0 (16)

where R0 is the classical constant density normal incidence reflection coefficient. To arrive

at this result, the definition

coth(x) ≡ ex − e−x

ex + e−x(17)

has been utilized along with some simplifying algebra.

Discussion of exact Rw

In general, Rw(f) is a complex function of frequency. As noted above, at zero frequency

the Wolf reflection coefficient reduces to the well-known normal incidence acoustic reflection

coefficient because any transition zone is negligible for the corresponding infinite wavelength.

The function Rw(f) has an absolute maximum at f = 0 and decreases in magnitude as

frequency increases. The reflection coefficient is zero if either σ or coth(γ log k) are infinite,

the latter occuring when γ log(k) is a non-zero purely imaginary integer multiple of π. In

5


other words, |Rw(f)| = 0 when

γ log(k) = i n π ; n = ±1,±2,±3, · · · (18)

This clarifies Wolf’s zero-crossing result by avoiding use of the ambiguous expression |γ|.

Furthermore, it accounts for all zero crossings, not just the first as given by Wolf. Solving

equation 18 for the nth zero crossing frequency we obtain

fn =v (k − 1)

2 h

√(n

log(k)

)2

+1

4 π2(19)

which yields a positive frequency for all k > 0 and all n.

In Figure 1 we illustrate behavior of the Wolf reflection coefficient by plotting Rw(f)

from 0-100 Hz for a transition thickness of h = 10 m, upper velocity v = 3500 m/s, and

velocity contrast k = 0.8. This case could represent, for example, a high-velocity cap rock

overlying and grading into low velocity reservoir rock. In this case, the real part of Rw (A)

decays slowly away from the static value of R0 = −0.1, with a zero crossing and polarity

reversal at 80 Hz. The imaginary part (B) is zero at zero frequency (as it should be, since R0

is real) and slowly builds to a maximum at about 55 Hz. These combine to give a phase (D)

that is linear from 180 degrees at zero frequency (consistent with negative R0) and finishing

at 60 degrees near 100 Hz. Note the amplitude (C) does not exhibit a zero crossing in the

0-100 Hz frequency range since |Rw(f)| = 0 requires both real and imaginary parts to be

zero. Equation 19 indicates a first zero crossing of |Rw(f)| for this parameter set is 157 Hz.

Figure 2 shows the same model, except the transition zone is now 50 m thick. All

significant change occurs in this case below 20 Hz, above which Rw oscillates around zero,

meaning these waves do not see a reflecting interface but a smoother, linear v(z) medium.

For this thicker ramp, there are six real part zero crossings and three amplitude ones.

6


Waveform Simulation and Detection

Convolutional modeling allows us to examine the characteristics of a Wolf ramp reflection in

a different way. By convolving the complex reflection coefficient with a time-domain delta

function and multiplying by e−iωt, we build the integrand of the inverse Fourier Transform.

Symmetry of this complex reflection coefficient (the real part is even, the imaginary part

is odd) assures the inverse Fourier transform will be a real-valued time function, and vice

versa. The integrand of the inverse Fourier transform can be viewed and analyzed for

features characteristic of the Wolf ramp reflection. Since the integrand is complex, we can

view the real, imaginary, amplitude, or phase attributes. In this paper we limit analysis to

the real part while acknowledging that additional information my be encoded in the other

attributes.

Figure 3 displays an example Wolf ramp reflection 1-100 Hz waveform simulation. Note

the time scale is in seconds with t = 0 located at the top of second layer (base of the Wolf

ramp). The real part of the zero crossings of Rw(f) seen in Figure 3 occur at t = 0, while

the amplitude zero crossings (not shown) are vertical bands at one frequency for all time.

The waveform simulation method leads naturally to a detection algorithm. Let g(t) be

a windowed seismic trace containing a reflection suspected of being due to a Wolf ramp,

and let g(f) be its Fourier transform. The inverse Fourier transform of this data is

g(t) =∫

g(f) e−i2πft dt (20)

which allows interrogation of a time-frequency version of the windowed data

g(t, f) = Re[g(f) e−i2πft] (21)

for characteristics similar to those related to Wolf ramp reflection. Numerical tests (not

7


shown) confirm that the algorithm decomposes Wolf ramp waveform traces to accurately

reproduce known time-frequency variations like those shown in Figure 3B. Again, we note

that other complex attributes could be studied in addition to the real part.

FIELD DATA EXAMPLE

To illustrate various aspects of Wolf ramp theory applied to real data, we consider a 2D

migrated seismic line from the Andaman Sea, Thailand. Due to a strong terrigenous sed-

iment flux, this basin is known to have a soft sea floor. Such a situation may reasonably

be expected to exhibit a gradational boundary at the sea floor from mud to poorly consoli-

dated sediments and, finally, to lithified material. Our goal here is to examine the sea floor

reflection for evidence of a Wolf-type transition layer.

Our field example uses migrated seismic data to make a tentative, plausible case for a

transition zone reflection at the seafloor. This is offered as an illustration of the decomposi-

tion algorithm and concept of discriminating transition zone effects from other possibilities.

Reviewers have correctly pointed out that prestack field data that is better suited to this

kind of analysis. While we concede this point in general, the fact is that spectral decompo-

sition is routinely applied to migrated seismic data, and this is the domain where principle

interpretation is done. Furthermore, the authors do not currently have access to suitable

prestack data. Therefore, our field example should be considered tentative.

Figure 4A shows a portion of the migrated data from 1.0-1.5 seconds centered on the

sea floor reflection, 4B is a detailed plot of trace number 1950 over the time range 1.2-1.35

sec, and 4B is the time-frequency decomposition of TR1950 as described in equation 21.

Comparing with a Wolf ramp TF plot, Figure 3, we see two compelling characteristics in

8


common. First, the presence of a notch at about 55 Hz of the amplitude type discussed

above, and, second, decay of energy above the notch till the bandwidth limit is reached

around 80 Hz.

We understand that a migrated seismic trace is a highly-processed object, representing

various prestack processes and summation over offset. But insofar as migrated data is used

routinely to map geologic horizons and estimate rock properties, it may also be suitable for

detection of Wolf-type transition zones.

If Figure 4A is the expression of a Wolf ramp, we can ask what set of parameters would

give a reasonable fit to the observed data. Layer one in this case is water so we take v = 1500

m/s (with the understanding that this could be incorrect by up to 5%). In the absence of

any hard data on sea floor velocities in the area, we make the assumption that velocity

below the transition zone is 2100 m/s (k=1.4). With these parameters set it is possible to

test various ramp thickness values and we find a good fit for h = 15 m, as illustrated in

Figure 5B.

Another possible explanation of the observed data is interference of two reflection co-

efficients in the shallow sea floor. Figure 5C is an interference simulation that preserves

the notch frequency with two equal reflection coefficients 9 ms apart. The key difference

is quick recovery of energy beyond the notch frequency. This is always true of interference

in which frequencies are not lost or attenuated, but temporally phased out. Compare this

to the Wolf ramp simulation that shows clear decay above the notch in keeping with the

exponential term in equation 22 of Appendix A. This is an effect that cannot be simulated,

for example, by interference of two unequal reflection coefficients (Figure 5D).

Two final comments about possible causes for the time-frequency behavior of the field

9


data (Figure 5A). The amplitude notch may also be due to a source or receiver ghost.

Usual practice is to design the acquisition system so that ghost notches lie outside the data

bandwidth, although a 60 Hz notch corresponds to a reasonable source and/or receiver

depth of 6.25 m (vertical travel time). Without prestack data (unavailable to the authors)

this has to remain an open question.

Finally, we note that rough surface scattering (Clay and Medwin, 1977) is another kind

of dispersive reflection that shows exponential frequency decay, but it does not involve

development of a notch and was therefore discounted from our analysis.

CONCLUSIONS

The tools of modern spectral analysis make it possible to interpret seismic data in new

ways and at new levels of interrogation. We have examined one example of early work

on reflection from a vertical velocity transition zone (Wolf, 1937), with an eye toward the

usefulness of such analytic solutions in light of modern spectral decomposition tools. Even in

the simple case of normal incidence and constant density, frequency dependent reflectivity is

predicted. We have developed time-frequency waveform simulation and detection algorithms

that show features of complex, dispersive Wolf ramp reflectivity, including characteristic

notches, sinc-like oscillations of the real part of the reflection coefficient, and exponential

decay with increasing frequency. This pattern of behavior distinguishes Wolf ramp reflection

from pure interference effects due to closely spaced nondispersive reflection coefficients.

This is demonstrated by synthetic tests and decomposition of field migrated seismic data.

Our analysis of a soft seafloor reflection in the Andaman Basin shows evidence for the

presence of a Wolf-type transition zone, results that must be considered tentative since

prestack data may be better suited for such analysis. We see the work reported here as

10


a step toward developing quantitative interpretation tools aimed at unraveling competing

frequency-dependent effects in seismic data at a local scale.

APPENDIX A: REFLECTION COEFFICIENT APPROXIMATION

The exact form of Rw is ideal for numerical experiments like those shown above, but analytic

approximations are useful for interpretation applications. Wolf (1937) has thoughtfully

provided an accurate approximate form valid in the usual situation of k < 2. In this case

the difference between γ and σ can be ignored, and the reflection coefficient reduces to

Rw(f) ≈ sin(|σ| log k)2|σ|

e−σ log k (22)

The approximate form shows exponential decay in frequency because σ is proportional to

f through equation 15. Examination of the real and imaginary plots in Figure 2 clearly

shows sinc-like behavior as expressed in the approximation.

ACKNOWLEDGMENTS

The authors would like to thank Assoc. Editor J. Blanch and reviewer John Stockwell for

constructive criticism. This work is a contribution of the Allied Geophysical Lab at the

University of Houston.

11


REFERENCES

Backus, G. E., 1962, Long-wave elastic anisotropy produced by horizontal layering: J.

Geophys. Res., 67, 4427–4440.

Berryman, L. H., P. L. Goupillaud, and K. H. Waters, 1958a, Reflections from multiple

transition layers part I: Theoretical results: Geophysics, 23, 223–243.

——–, 1958b, Reflections from multiple transition layers part Ii: Experimental investigation:

Geophysics, 23, 244–252.

Bortfeld, R., 1960, Seismic waves in transition layers: Geophys. Prosp., 08, 178–217.

Chakraborty, A., and D. Okaya, 1995, Frequency-time decomposition of seismic data using

wavelet-based methods: Geophysics, 60, 1906–1916.

Clay, S. C., and H. Medwin, 1977, Acoustical oceanography: Principles and applications:

John Wiley and Sons.

Geertsma, J., and D. C. Smit, 1985, Some aspects of elastic wave propagation in fluid-

saturated porous solids: Geophysics, 50, 1797–1809.

Goupillaud, P., A. Grossmann, and J. Morlet, 1983, Cycle-octave representation for in-

stantaneous frequency spectra: 53rd Ann. Internat. Mtg, Soc. of Expl. Geophys., Ses-

sion:S24.5.

Gridley, J., and G. Partyka, 1997, Processing and interpretational aspects of spectral de-

composition: 67th Ann. Internat. Mtg, Soc. of Expl. Geophys., 1055–1058.

Gupta, R. N., 1965, Reflection of plane-waves from a linear transition layer in liquid media:


Sherwood, J. W. C., 1962, The seismoline an analog computer of theoretical seismograms:


Wolf, A., 1937, The reflection of elastic waves from transition layers of variable velocity:

12



Wuenschel, P. C., 1960, Seismogram synthesis including multiples and transmission coeffi-

cients: Geophysics, 25, 106–129.

Wuenschel, P. C., 1965, Dispersive body waves an experimental study: Geophysics, 04,

552–570.

Figures

13


LIST OF FIGURES

1 Exact normal incidence reflection coefficient, Rw(f) , for a Wolf ramp (linear ve-

locity transition zone). Parameters for this case are h = 10 m, v = 3500 m/s, and k = 0.8.

(A) Real part of the Wolf reflection coefficient as a function of frequency. If no ramp were

present, the standard reflection coefficient would be R0 = −0.1 for all frequencies. (B)

Imaginary part. (C) Amplitude, or complex absolute value. (D) Phase.

2 Exact Rw(f) as previous figure, except that the ramp is now 50 m thick.

3 Convolutional 1-100 Hz waveform modeling of a Wolf ramp (h = 50 m, v = 3500

m/s, k = 0.8). (A) Time domain response formed by summation of (B) over frequency. (B)

Time-frequency representation formed by convolving the complex Wolf reflection coefficient

with a time-domain unit spike. Only the real part is shown.

4 Offshore migrated field data from Anadman Basin, Thailand. (A) View of sea floor

reflector. (B) Detail plot of trace 1950 from 1.20-1.35 sec. (C) Time-Frequency decomposi-

tion from 0-100 Hz, showing characteristic notch and polarity reversal at 55 Hz as well as

diminished amplitude above the notch frequency.

5 Time-Frequency comparisons (real part only). (A) Decomposition of field data

trace 1950, (B) Wolf ramp simulation (v = 1500, k = 1.4, h = 15 m), (C) Interference effect

simulation (RC2 = RC1, δt = −9 ms), (C) Interference effect simulation (RC2 = 0.8 RC1,

δt = −9 ms).

14


0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"

Value

!A" Real

0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"Value

!B" Imaginary

0 20 40 60 80 1000.00

0.05

0.10

0.15

0.20

Frequeny !Hz"

Value

!C" Amplitude

0 20 40 60 80 1000

50

100

150

Frequeny !Hz"

Degrees

!D" Phase

Figure 1: Exact normal incidence reflection coefficient, Rw(f) , for a Wolf ramp (linear

velocity transition zone). Parameters for this case are h = 10 m, v = 3500 m/s, and

k = 0.8. (A) Real part of the Wolf reflection coefficient as a function of frequency. If no

ramp were present, the standard reflection coefficient would be R0 = −0.1 for all frequencies.

(B) Imaginary part. (C) Amplitude, or complex absolute value. (D) Phase.

Liner and Bodmann – GEOPHYSICS LETTERS

15


0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"

Value

!A" Real

0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"Value

!B" Imaginary

0 20 40 60 80 1000.00

0.05

0.10

0.15

0.20

Frequeny !Hz"

Value

!C" Amplitude

0 20 40 60 80 1000

50

100

150

Frequeny !Hz"

Degrees

!D" Phase

Figure 2: Exact Rw(f) as previous figure, except that the ramp is now 50 m thick.


16


-0.05

0

0.05

0.10

Sec

20 40 60 80 100Hz

(B) Re[ Rw(t,f) ]

-0.05

0

0.05

0.10

Sec

0Amp

(A) Summed

Figure 3: Convolutional 1-100 Hz waveform modeling of a Wolf ramp (h = 50 m, v = 3500

m/s, k = 0.8). (A) Time domain response formed by summation of (B) over frequency. (B)

Time-frequency representation formed by convolving the complex Wolf reflection coefficient

with a time-domain unit spike. Only the real part is shown.


17


1.0

1.2

1.4

Sec

1800 1850 1900 1950 2000 2050 2100 2150Trace

(A) Migration Data

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Sec

0Amp

(B) TR1950

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Sec

20 40 60 80 100Hz

(C) TF of TR1950 (Real)

Figure 4: Offshore migrated field data from Anadman Basin, Thailand. (A) View of sea

floor reflector. (B) Detail plot of trace 1950 from 1.20-1.35 sec. (C) Time-Frequency

decomposition from 0-100 Hz, showing characteristic notch and polarity reversal at 55 Hz

as well as diminished amplitude above the notch frequency.


18


1.22

1.24

1.26

1.28

1.30

1.32

1.34

Sec

20 40 60 80 100Hz

(A) TR1950 TF

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (m

s)

20 40 60 80 100Hz

(B) Wolf Ramp TF

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (m

s)

20 40 60 80 100Hz

(C) Interference TF (rc1=rc2)

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (m

s)

20 40 60 80 100Hz

(D) Interference TF (rc1>rc2)

Figure 5: Time-Frequency comparisons (real part only). (A) Decomposition of field data

trace 1950, (B) Wolf ramp simulation (v = 1500, k = 1.4, h = 15 m), (C) Interference effect

simulation (RC2 = RC1, δt = −9 ms), (C) Interference effect simulation (RC2 = 0.8 RC1,

δt = −9 ms).


19


0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"

Value

!A" Real

0 20 40 60 80 100!0.2

!0.1

0.0

0.1

0.2

Frequeny !Hz"

Value

!B" Imaginary

0 20 40 60 80 1000.00

0.05

0.10

0.15

0.20

Frequeny !Hz"

Value

!C" Amplitude

0 20 40 60 80 1000

50

100

150

Frequeny !Hz"

Degrees

!D" Phase


-0.05

0

0.05

0.10

Tim

e (s

)

20 40 60 80 100Frequency (Hz)

b) Re[ Rw(t,f) ]

-0.05

0

0.05

0.10

Tim

e (s

)0

Amp

a) Summed


1.0

1.2

1.4

Tim

e (s

)1800 1850 1900 1950 2000 2050 2100 2150

Trace

a) Migration Data

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (s

)

0Amp

b) TR1950

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (s

)

20 40 60 80 100Frequency (Hz)

c) TF of TR1950 (Real)


1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (s

)20 40 60 80 100

Frequency (Hz)

a) TR1950 TF

1.22

1.24

1.26

1.28

1.30

1.32

1.34Ti

me

(s)

20 40 60 80 100Frequency (Hz)

b) Wolf Ramp TF

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (s

)

20 40 60 80 100Frequency (Hz)

c) Interference TF (rc1=rc2)

1.22

1.24

1.26

1.28

1.30

1.32

1.34

Tim

e (s

)

20 40 60 80 100Frequency (Hz)

d) Interference TF (rc2=0.7*rc1)


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