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Geophysics Manuscript, Accepted Pending: For Review Not Production
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.
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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 frequencydependent 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 dependence 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 scattering (Clay and Medwin, 1977), Biot reflection (Geertsma and Smit, 1985), and vertical
transition zones (Wolf, 1937). For the development of future quantitative seismic interpretation 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
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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
∂ 2 ∂u
∂2u
V (z)
= 2
∂z
∂z
∂t
,
(1)
z<0
(2)
0<z<h
(3)
z>0 .
(4)
where the velocity function is
V
= v
V
= v 1+
V
= kv
k−1
z
h
Extending the linear part of the velocity function, we define the variable
s(z) = v 1 +
3
k−1
z
h
,
(5)
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and its derivatives,
∂s
∂z
∂
∂z
k−1
h
∂s ∂
k−1 ∂
=v
∂z ∂s
h ∂s
= v
=
(6)
.
(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
∂ 2 ∂u
s
= ω2 u ,
∂s
∂s
(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
∂2u
+ s2
+ ω̄ 2 u = 0
∂s
∂s2
;
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 polynomial equation
2 m u + m(m − 1) u + ω̄ 2 u = 0 ,
(12)
that can be written in compact form as
h
i
m(m + 1) + ω̄ 2 u = 0 ,
(13)
identical to Equation 6 in Wolf (1937). The values of m are given by the roots of the bracketed polynomial. Applying the continuity of the field and its z-derivative across the velocity
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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
i 2π f h
σ(f ) =
(k − 1)v
;
γ(f ) =
q
1/4 + σ 2 .
(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 ) =
coth( 12
1
k−1
=
= R0
k+1
log(k))
(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
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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
v (k − 1)
fn =
2h
s
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.
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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
Z
g(t) =
g(f ) e−i2πf t dt
(20)
which allows interrogation of a time-frequency version of the windowed data
g(t, f ) = Re[g(f ) e−i2πf t ]
(21)
for characteristics similar to those related to Wolf ramp reflection. Numerical tests (not
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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 sediment 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 consolidated 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 decomposition 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 decomposition 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
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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 coefficients 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
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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
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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) −σ
e
2|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.
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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 fluidsaturated porous solids: Geophysics, 50, 1797–1809.
Goupillaud, P., A. Grossmann, and J. Morlet, 1983, Cycle-octave representation for instantaneous frequency spectra: 53rd Ann. Internat. Mtg, Soc. of Expl. Geophys., Session:S24.5.
Gridley, J., and G. Partyka, 1997, Processing and interpretational aspects of spectral decomposition: 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:
Geophysics, 30, 122–132.
Sherwood, J. W. C., 1962, The seismoline an analog computer of theoretical seismograms:
Geophysics, 27, 19–34.
Wolf, A., 1937, The reflection of elastic waves from transition layers of variable velocity:
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Geophysics, 02, 357–363.
Wuenschel, P. C., 1960, Seismogram synthesis including multiples and transmission coefficients: Geophysics, 25, 106–129.
Wuenschel, P. C., 1965, Dispersive body waves an experimental study: Geophysics, 04,
552–570.
Figures
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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 decomposition 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
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!A" Real
0.2
0.1
Value
Value
0.1
0.0
!0.1
!0.2
!B" Imaginary
0.2
0.0
!0.1
0
20
40
60
Frequeny !Hz"
80
!0.2
100
0
20
!C" Amplitude
0.20
40
60
Frequeny !Hz"
80
100
40
60
Frequeny !Hz"
80
100
!D" Phase
150
Degrees
Value
0.15
0.10
50
0.05
0.00
100
0
20
40
60
Frequeny !Hz"
80
100
0
0
20
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
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Geophysics Manuscript, Accepted Pending: For Review Not Production
!A" Real
0.2
Value
Value
0.1
0.0
!0.1
!0.2
!B" Imaginary
0.2
0.1
Page 16 of 24
0.0
!0.1
0
20
40
60
Frequeny !Hz"
80
!0.2
100
0
20
!C" Amplitude
0.20
40
60
Frequeny !Hz"
80
100
40
60
Frequeny !Hz"
80
100
!D" Phase
150
Degrees
Value
0.15
0.10
50
0.05
0.00
100
0
20
40
60
Frequeny !Hz"
80
100
0
0
20
Figure 2: Exact Rw (f ) as previous figure, except that the ramp is now 50 m thick.
Liner and Bodmann – GEOPHYSICS LETTERS
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Amp
0
20
0
60
80
100
0
0.05
0.05
0.10
Hz
-0.05
Sec
Sec
-0.05
40
(A) Summed
0.10
(B) Re[ Rw(t,f) ]
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.
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1.0
Trace
1800 1850 1900 1950 2000 2050 2100 2150
Sec
1.2
1.4
(A) Migration Data
0
Amp
20
1.22
1.24
1.24
1.26
1.26
Sec
Sec
1.22
1.28
40
Hz
60
80
100
1.28
1.30
1.30
1.32
1.32
1.34
1.34
(B) TR1950
(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.
Liner and Bodmann – GEOPHYSICS LETTERS
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20
40
Hz
60
80
100
20
1.22
1.24
1.24
1.26
1.26
40
Hz
60
80
100
80
100
Sec
Time (ms)
1.22
1.28
1.28
1.30
1.30
1.32
1.32
1.34
1.34
(A) TR1950 TF
20
40
Hz
60
(B) Wolf Ramp TF
80
100
20
1.22
1.24
1.24
1.26
1.26
40
Hz
60
Time (ms)
1.22
Time (ms)
Page 19 of 24
1.28
1.28
1.30
1.30
1.32
1.32
1.34
1.34
(C) Interference TF (rc1=rc2)
(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).
Liner and Bodmann – GEOPHYSICS LETTERS
19
!A" Real
0.2
!B" Imaginary
0.2
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0.1
Value
Value
0.1
0.0
!0.1
!0.2
0.0
!0.1
0
20
40
60
Frequeny !Hz"
80
!0.2
100
0
20
!C" Amplitude
0.20
40
60
Frequeny !Hz"
80
100
40
60
Frequeny !Hz"
80
100
!D" Phase
150
Degrees
Value
0.15
0.10
50
0.05
0.00
100
0
20
40
60
Frequeny !Hz"
80
100
0
0
20
!A" Real
!B" Imaginary
0.2 21
0.2
Page
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of 24
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0.1
Value
Value
0.1
0.0
!0.1
!0.2
0.0
!0.1
0
20
40
60
Frequeny !Hz"
80
!0.2
100
0
20
!C" Amplitude
0.20
40
60
Frequeny !Hz"
80
100
40
60
Frequeny !Hz"
80
100
!D" Phase
150
Degrees
Value
0.15
0.10
50
0.05
0.00
100
0
20
40
60
Frequeny !Hz"
80
100
0
0
20
Geophysics Manuscript, Accepted Pending: For Review Not Production
Amp
0
20
-0.05
Time (s)
Time (s)
-0.05
0
0
0.05
0.05
0.10
Frequency (Hz)
40
60
a) Summed
0.10
b) Re[ Rw(t,f) ]
80
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100
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1.0
Trace
1800 1850 1900 1950 2000 2050 2100 2150
Time (s)
1.2
1.4
a) Migration Data
0
Amp
20
1.22
1.24
1.24
1.26
1.26
80
Time (s)
Time (s)
1.22
Frequency (Hz)
40
60
1.28
1.28
1.30
1.30
1.32
1.32
1.34
1.34
b) TR1950
c) TF of TR1950 (Real)
100
Geophysics Manuscript, Accepted Pending: For Review Not Production
20
Frequency (Hz)
40
60
80
100
20
1.24
1.24
1.26
1.26
80
100
80
100
Time (s)
1.22
Time (s)
1.22
Frequency (Hz)
40
60
Page 24 of 24
1.28
1.28
1.30
1.30
1.32
1.32
1.34
1.34
a) TR1950 TF
20
Frequency (Hz)
40
60
b) Wolf Ramp TF
80
100
20
1.22
1.24
1.24
1.26
1.26
Time (s)
Time (s)
1.22
Frequency (Hz)
40
60
1.28
1.28
1.30
1.30
1.32
1.32
1.34
1.34
c) Interference TF (rc1=rc2)
d) Interference TF (rc2=0.7*rc1)