Comparison of seafloor sound-speed profile measurement
between the image source and reflection seismology
techniques
Samuel Pinson
To cite this version:
Samuel Pinson. Comparison of seafloor sound-speed profile measurement between the image
source and reflection seismology techniques. 2015. <hal-01254106>
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Comparison of seaoor sound-speed prole measurement between the
image source and reection seismology techniques
S. Pinson
a)
Laboratório de Vibração e Acústica, Universidade Federal de Santa Catarina, Floriaónopolis, Brazil
(Dated: July 13, 2015)
In the context of sediment characterization, the image source method provides a fast and
automated sound-speed prole measurement of the seaoor. In this letter, this method is
compared to reection seismology methods. The comparison highlights similarities in the
signal processing and a dierence in the basic equations of the wave reection travel time analysis. Understanding the links between the two methods helps sheds additional light on each method.
PACS numbers: 43.30 Ma, 43.30 Pc, 43.60 Fg, 43.60 Rw
(a)
1. Introduction
A technique has recently been developed for rapidly
estimating sub-seabed sound-speed proles using the
concept of image sources.
Hydrophones
Source
The image source method
(ISM) has been used for measurements from a vertical
4,5 which yields a point estimate of sound
receive array
speed vs depth in the seabed.
More recently it has
4,6 yielding the
been applied to horizontal array data
layer 0 (water)
layer 1
layer 2
two-dimensional (depth vs range) sound-speed structure
(b)
of the seabed to 50 m in sub-bottom and 14 km in range
with resolutions roughly 1 m in depth and 5 m in range.
It turns out that the fundamental equations of the
40
image source method can be understood and compared
with those from classical seismic reection methods.
underlying connections between the two methods.
The
secondary purpose is to briey consider, in the light of
the connections, relative strengths and possibilities of
extension for the ISM.
The letter is organized as follows.
of both methods are presented.
First, the principles
This is followed by a
short discussion to explore some basic dierences and
similarities between them.
2. Basic ideas of the methods
35
Offset (m)
This is the purpose of this letter, rst to explain the
30
25
20
0.025
0.03
0.035
0.04
0.045
Time (s)
0.05
0.055
Figure 1: (a): Sketch of the experiment. (b): Simulated
seismogram of the seaoor reections.
lated by a numerical evaluation of the Sommerfeld integral (see appendix of Ref.
?
) using the following geome-
A typical measurement conguration for seaoor char-
try: the source and 10 hydrophones are 20 m above the
acterization by reection seismology is a source and a
seaoor, the hydrophones are linearly spaced between 20
linear array of hydrophones (Fig. 1a). The source emits
and 38 m from the source. The environment parameters
a low frequency, short, powerful pulse such that the
are presented in the Table I. A Mexican hat centered at
wave goes deep into the seaoor.
Then the wave is re-
1500 Hz has been used for the emitted pulse shape and
ected from geological interfaces and recorded by the hy-
0.2% amplitude Gaussian noise related to the maximum
drophone array.
amplitude reection is added to the simulated signals.
For illustration purposes, a set of data has been simu-
The low noise is added to avoid division by zero in the
semblance functions described in the next sections. The
simulated reected echoes are displayed in Fig. 1b. Multiple paths between the seaoor and the sea surface are
a)
Electronic address: [email protected]
not considered here.
Image source & reection seismology
1
If one assumes that
Layer n◦ Thickness (m) Sound speed (m/s) Density (kg/m3 )
0 (water)
20
1500
1000
1
5
1600
1500
2
6
1700
2000
3
∞
1800
2500
Table I: Layered media parameters.
c(l)
rms
(l)
(l)
(l)
cnmo ≈ crms with:
v
u Pl
(i) 2 (i)
u
i=0 (c ) τ⊥
t
,
=
Pl
(i)
i=1 τ⊥
(2)
c(i) is
(i)
the
sound speed and τ⊥ is the travel time at
normal incidence in the layer i, then one can apply the
crms
layer i
where
is the root mean square sound speed,
2
Dix formula to obtain layer sound speeds :
c(l)
v
u (l)
2 (l−1)
u (crms )2 t(l) − (c(l−1)
rms ) t⊥
⊥
.
=t
(l)
(l−1)
t⊥ − t⊥
(3)
2.1. Reection seismology
There are dierent ways to determine the normal move
Obtaining the sound speed in reection seismology is
out sound speeds.
One of them consists in calculating
a rst step for imaging the earth's interior by migrat-
a velocity spectrum (velocity rather than sound speed is
ing the recorded signals at the correct reector's depths.
used in the seismic community)
For reection from interface
l
of the layered media, the
10 . The velocity spectrum
is calculated by summing the recorded signals along hy-
layered media is approximated by an equivalent homoge-
perbolas (colored curves on Fig. 1b) dened by Eq. (1):
(l)
neous medium in which the sound speed is cnmo . The subscript
nmo
s

X
2 2
1 N H
x
−
x
n
s
 ,
V S(t⊥ , cnmo ) = sn  t2⊥ +
N
c
nmo
n=1
is used in the seismic community and stands
10 . To determine c(l) , the
nmo
for Normal MoveOut (NMO)
three sides of the right triangle are considered: the o-
(4)
(xs − xn ), twice
(l)
(l)
the depth of the interface t⊥ × cnmo , and the hypotenuse
(l)
(l)
(l)
tn × cnmo (Fig. 2). t⊥ is the null oset travel time (i.e.
(l)
at normal incidence) of the interface l reection and tn
is the travel time of the interface l reection at the oset
set between the source and the receiver
(xs − xn ).
(l)
time tn of
the reection from interface
l
trum avoids that problem:
P
1 N
N n=1 sH
n
V Ssemb (t⊥ , cnmo ) =
PN H
1
n=1 sn
N
z
(l)
θnmo
(l)
θnmo
(l)
source
c (0
)
o
m
interface
xn −xs
cnmo
(5)
signals are identical along travel time hyperbolas and
l
close to 0 when dierent.
t (l)
×
t (l)
×
image
t2⊥
The semblance is a similarity measurement and is 1 when
(l)
cn (l)
(l)
!
2 2
+
.
!
r
2 2
x
−x
t2⊥ + cnnmos
r
x
θ0
t⊥ × cnmo
stands for Hilbert trans-
low amplitude reections. The semblance velocity spec-
is written:
= (xn , 0)
rc = (xc , 0)
H
plitude reected pulses thus making it dicult to detect
sensor
rn
= (xs , 0)
signal recorded at the hy-
form). The velocity spectrum is not sensitive to low am-
Using the Pythagorean theorem, the travel
source
rs
sH
n (t) is the analytic
drophone n (the superscript
where
local maxima of
It is then possible to select
V Ssemb (t⊥ , cnmo )
to calculate a sound-
speed prole using Eq. (3).
image source
located in a
2.2. Image source method
water sound-speed
medium
Figure 2: Ray diagram for the reection on the interface l in a
(l)
cnmo equivalent sound-speed medium. In gray is the ray diagram
for the image source l located in a water sound-speed c(0) medium.
The idea of the ISM is to consider wave reections from
geological interfaces as waves coming from image sources
located symmetrically to the real source relative to the interfaces. The image sources are located in a water sound
speed
c(0)
medium by migrating the recorded signals in
the space domain:
t(l)
n ≈
s
(l)
t⊥
2
+
xn − xs
(l)
cnmo
2
,
(1)
2
N
1 X
Im (r) = sH
n (tn (r)) ,
N
(6)
n=1
Image source & reection seismology
2
where
tn (r) = krn − rk /c(0)
and
r = (x, z)
is the spatial
3. Link between the methods
coordinate.
Im (r)
This
function is displayed in Fig. 3a where the 3
One can notice the strong similarity between Eq. (4)
z ≈
and Eq. (6), the velocity spectrum and the image source
images relative to the 3 interfaces are apparent at
−40,
-50 and -62 m.
But because their amplitudes are
a function of impedance contrast, this means that low
map respectively.
Both consist of summing the signals
along hyperbolic travel times which are driven by dier-
cnmo
ent parameters:
a semblance function of the migrated signals is used for
and,
an automatic detection:
The similarity is such that it is possible to change the
semblance
function
in
Fig.
3d
results
and
z
change of variables from
(7)
r = (x, z)
s
in
small
Applying a threshold on
this function, the image source search area is conned
to a neighborhood.
cnmo =
Im (r).
c(0) |xc − xs |
,
tc sin θ0
s
One can
t⊥ = tc cos θnmo = tc
1−
notice in Fig. 3d that image sources related to multiple
reections between interfaces also appear.
Even weak
noise on the simulated signals generally hides those
low
amplitudes
from
real
echoes
?
.
Also
environment,
the
for
signals
multiple
recorded
reection
be-
(cnmo , t⊥ )
using
(10)
and:
The image source locations are
found by taking the local maxima of
to
Eq. (9) and simple trigonometry (Fig. 2):
regions where the focused signals are coherent which
simplies image detection.
for the velocity spectrum
for the image source map.
image source map into a velocity spectrum through a
2
P
1 N
(t
(
r
))
N n=1 sH
n
n
Isemb (r) = 1 PN
.
2
H
n=1 |sn (tn (r))|
N
The
x
and
t⊥
impedance contrast layers will be dicult to detect. So
tc = krc − rk /c0
cnmo
sin θ0
c0
2
(11)
|xc −x|
|zc −z| .
The result of this variable change is shown on Fig. 3b for
where
the
Im
and
θ0 = tan−1
map and on Fig. 3e for the
Isemb
map. Fig. 3c
tween interfaces are assumed to be hidden by reections
and 3f are a zoom on the NMO sound speeds of Fig. 3b
from deeper interfaces through the Born approximation .
and 3e.
6
After image source localization, the receive array is collapsed to a point sensor at the array center
1
N
(0)
P
c
rc = (xc , zc ) =
n rn . With the assumption of a water sound speed
equivalent medium, the image sources are not on the
vertical source axis as they should be (gray drawing in
Fig. 2).
From the location of image sources, one can
(l)
between the image source
deduce the travel times tc
(l)
l and the array center rc and the angle of arrival θ0
(the subscript stands for the angle of incidence in water
i.e.
layer 0 and the upper index stands for the interface
number on which the wave is reected). With these parameters it is then possible to deduce layer sound speeds
and thicknesses using ray theory
(l)
time tc
and the angle of arrival
1,46 .
(l)
θ0
From the travel
4. Discussion
A comparison between Eq. (1) and Eq. (8) provides
the basic dierence between classical reection seismology and the image source method.
From the reection
seismology point of view, the problem is written considering the right triangle dened by the source, the receiver
and the image source in a
medium.
(l)
cnmo
equivalent sound-speed
From the ISM point of view, the problem is
written considering the source-receiver oset, the travel
time and the angle of arrival on the receiver in a water
sound speed
c(0)
medium.
In reection seismology, one
considers more realistic travel times by approximating
, it is also possible to
the wave front shape as being ellipsoidal. From a physics
obtain the sound speed of the homogeneous media, equiv-
point of view, reection seismology is better because ISM
alent in travel time, to the layered media. Eq. (1) of the
approximates the wave front as spherical. For this reason
echo travel time is rewritten using the angle of arrival on
the ISM has to use a limited aperture array to enable a
the sensor:
coherent echo summation in the image source localization
t(l)
c ≈
where
(l)
sin θnmo
1
(l)
×
sin θnmo
|xc − xs |
(l)
(8)
cnmo
can be deduced from
Snell-Descartes relation.
process.
,
ections on interfaces, it becomes useful to understand
(l)
sin θ0
through the
Then the equivalent medium
sound-speed is given by:
s
c(l)
nmo
=
As the ISM is an intuitive way of modeling the wave resome possibly unexpected phenomena associated with the
velocity spectrum semblance function. Using the migration function (Eq. (6)) to nd the image sources, the resolution of the image location is driven by two dierent
c(0) |xc − xs |
(l)
tc
(l)
sin θ0
experimental parameters: the array aperture that drives
.
(9)
the angle resolution of the images and the pulse shape
that drives the range resolution of the images to the array. Using the semblance function for the ISM, the angle
Finally, using the Eq. (3) one can obtain the layer sound
resolution remains the same but the range resolution is
speeds.
lost. Indeed, if the range resolution is driven by the pulse
Image source & reection seismology
3
Figure 3: (a) and (d): Image source maps using respectively Im (Eq. (6)) and Isemb (Eq. (7)). The red point corresponds to the real
source and the black points correspond to the hydrophones. (b) and (e): Im (Eq. (6)) and Isemb maps with coordinates changed from x
and z to cnmo and t⊥ using Eq. (10) and Eq. (11) in order to obtain a velocity spectrum. (c) and (f): Zoom of (b) and (e). The colorbars
on the right are related to the rows. The Im colormap is normalized by the maximum. Isemb is intrinsically between 0 and 1.
shape for the migration function, the semblance shows a
geologic
interfaces
are
coherent value of 1 for the whole pulse duration (in the
geologic
interfaces
and,
neglected),
case of a good signal to noise ratio).
isotropic
at
and
and
parallel
homogeneous
This interpreta-
layers. The reection seismology methods have far fewer
tion in terms of angle and range resolution is lost for the
assumptions, given many decades of progress since their
semblance velocity spectrum (Eq. (5)). It results in the
introduction.
impossibility of an accurate automatic detection of the
anisotropy
NMO sound speeds using this semblance velocity spec-
ters
trum.
8,9
For example, extension to include the
or
inhomogeneities/statistical
parame-
3,7 . It is possible, now with a clearer understanding
Because of that, using the ISM, the semblance
of the connections between the ISM and the classical
function is only used to window small areas of the migra-
methods, that ISM may be able to exploit some of these
tion function (Eq. (6)) in which signals are summed co-
advances.
herently and in which accurate localization is performed
using local maxima. The same process should be apply
when using velocity spectrum.
Using Eq. (1) to dene the travel times implies the use of
an horizontal array for seismic techniques.
6. Acknowledgements
By contrast
in ISM, since the sound-speed equations are a function
of the array center coordinate
rc , the array conguration
The
author
would
like
to
gratefully
acknowledge
Charles Holland for his help and stimulating discussions
is very exible as long as its aperture is not too large.
about that work.
Indeed, the rst results of the ISM was obtained using a
This work is funded by Direction Générale de l'Armement
vertical array of hydrophones moored on the seaoor .
(DGA) through the Laboratoire Domaine Océaniques
Originally the ISM did not employ the NMO approxi-
(LDO) of the Institut Universitaire Européen de la Mer
mation to give sound-speed proles.
(IUEM).
5
Nevertheless, this
formulation, borrowed from reection seismology, could
be useful for eliminating the parallel layering restriction.
1
2
5. Conclusion
3
In its present form, the method requires the Born
approximation
(the
multiple
reections
between
the
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The hydrophonepinger experiment.
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