Three dimensional geoacoustic inversion on the New Jersey shelf - - PowerPoint PPT Presentation
Three dimensional geoacoustic inversion on the New Jersey shelf Megan S. Ballard and Kyle M. Becker Applied Research Laboratory and Graduate Program in Acoustics The Pennsylvania State University PO Box 30, State College, PA 16804 Applied
Applied Research Laboratory and Graduate Program in Acoustics The Pennsylvania State University PO Box 30, State College, PA 16804
Applied Research Laboratory
The Hankel Transform Pair using the far field approximation
4
( ; , ) ( ; , ) 2
r
i ik r r r
e g k z z p r z z re dr k
π
π
∞ − −∞
=
4
( ; , ) ( ; , ) 2
r
i ik r r r r
e p r z z g k z z k e dk r
π
π
∞ − −∞
=
1
r
k r >>
The Short-Time Fourier Transform (STFT) Auto Regression (AR)
1 p n k n k k
x a x −
=
=∑
2 2 2 1
1
AR p i fkT k k
T P a e
π
σ
− =
= +∑
4
ˆ ˆ ( ; , , ) ( ; ) ( ; , ) 2
r
i ik r r L r
e g k r z z w r r p r z z re dr k
π
π
∞ − −∞
=
Hankel Transform Estimate Auto Regressive Spectrum
Auto regressive (AR) techniques were used to estimate wave numbers from pressure field
Window length of the AR estimator is 2000m. k5 k4 k3 k2 k1 k6 k8 k7
k1 k4 k3 k9 k8 k5 k6 k7 k2
k9
1 2 2
n n n
∞ −
This equation can be written in the form of a Fredholm integral of the first kind:
D i i
Which can be written in matrix form as: is a vector representing the data is a matrix representing the forward model is a vector representing the model parameter x A y
A relation between a perturbation to sound speed in sediment and a perturbation to horizontal wavenumbers is formulated from the depth separated normal mode equation:
where is a discrete version of the differential operator Solve the ill-conditioned problem: by choosing the smoothest solution.
y = Ax
n n
d dx
1 n = 2 n = favors the flattest solution; favors the smoothest solution. L-Curve Criterion: The Lagrange multiplier is chosen such that it both the residual and the semi-norm are minimized simultaneously.
λ
The Regularization solution is given by:
y A L L A A x
T T T 1 2
) ( ˆ
−
+ = λ
2 2 2 2 2
min Lx y Ax λ + −
2
y Ax −
2
Lx
where the set is on orthogonal basis for .
where is created by the user.
q
1
( )
r T q i i i
I
=
= −∑ L L q q
1
{ }r
i i=
q Q
q
is given by: Solve the ill-conditioned problem: by choosing the solution that best fits some prior knowledge. y = Ax The Qualitative Regularization solution is given by:
y A L L A A x
T q T q T 1 2
) ( ˆ
−
+ = λ
2 2 2 2 2
min x L y Ax
q
λ + −
More accurate estimation of the bottom parameters can be made by allowing the solution to be a layered medium instead of a smooth profile.
Ship tracks oriented along, across, and oblique to the shelf break on radials with respect to the Shark VLA. All ship tracks are about 5km long.
chirp seismic reflection data Chirp data provided by John Goff
Input data to the inversion scheme: wavenumber estimates from 125 and 175 Hz data. Over lapping regions are inconsistent due to noise on the wavenumber estimates.
Wavenumber estimates could not be obtained for ranges less than 1.5 km because ship speed could not be approximated as constant along a radial with respect to the VLA.
Longer apertures were required for wavenumber estimation to account for closely spaced wavenumbers and average over the effects of the range dependent water column sound speed profile.
Upper Unit: 1639m/s, 1624m/s, 1657m/s Lower Unit: 1554m/s, 1652m/s (single values) Below R: 1850m/s (Upper range of very erratic measurements)
( ) ˆ( ) P r P r
min
min
1 ( ) ( )
R r
PQ P r Q r dr R r = −
* 12 * *
( ) ˆ ( ) ( ) ˆ ˆ ( ) ( ) ( ) ( ) cor R P r P r P r P r P r P r =
Calculated Field Measured Field 50 75 125 175
0.7 0.8 0.9
Along Shelf Correlation Depth [m] 0.7 0.8 0.9
Across Shelf Correlation 0.7 0.8 0.9
Oblique Shelf Correlation
50 75 125 175
0.7 0.8 0.9
Along Shelf Correlation Depth [m] 0.7 0.8 0.9
Across Shelf Correlation 0.7 0.8 0.9
Oblique Shelf Correlation 0.7 0.8 0.9
Along Shelf Correlation 0.7 0.8 0.9
Across Shelf Correlation 0.7 0.8 0.9
Oblique Shelf Correlation
Model to Data Correlation Data to Data Correlation
The complete solution of the inverse problem requires not only the estimates
estimates.
Monte Carlo Error Propagation:
T
N = D D Cov
T T i i
− D = x x 1,2,...., i N = where The empirical estimate of the covariance matrix
i i
Ax n y = + ( ) ( )
m z
diag σ = Cov
Error bars: standard deviation of the model estimates
For the Oblique Shelf Track
2 1 2
( )
M ij j ii
R rl i R
=
= ∑
T
T m v m v
R = (G C G + C ) G C G d + v = Gm
m v m
C = (G C G + C )
For the nonlinear problem examined here, the solution is arrived at iteratively. Assuming the final solution to the problem is linear, it is valid to use linear theory to obtain the resolution and covariance matrices. Beginning with the linear problem: The resolution matrix is given by: The posterior model covariance matrix is given by:
v
C
m
C
Data covariance matrix; Model covariance matrix Resolution length is given by:
For the Oblique Shelf Track