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 Research Laboratory
Outline • Methodology – Horizontal wavenumber estimation – Perturbative inversion – Qualitative Regularization • The Shallow Water 2006 Experiment – Acoustic and oceanographic measurements – Inversion results – Three dimensional model for the environment – Validation of results: comparison to core data and ability to predict the acoustic field
Horizontal Wavenumber Estimation Hankel Transform Estimate The Hankel Transform Pair using the far field k r >> 1 approximation r − π ∞ i e 4 ∫ = ik r p r z z g k z z k e dk ( ; , ) ( ; , ) r r r r 0 π 0 r 2 −∞ π ∞ i e 4 ∫ − = ik r g k z z p r z z re dr ( ; , ) ( ; , ) r r 0 π 0 k 2 −∞ r The Short-Time Fourier Transform (STFT) Auto Regressive Spectrum π ∞ i e 4 ∫ − = ik r g k r z z w r r p r z z re dr ˆ ˆ r ( ; , , ) ( ; ) ( ; , ) r L 0 π 0 k 2 −∞ r Auto Regression (AR) σ T 2 = p P = ∑ x a x − AR + ∑ 2 p − π i fkT a e n k n k 2 1 k = k k = 1 1
Horizontal Wavenumber Estimation Auto regressive (AR) techniques were used to estimate wave numbers from pressure field data. Mode shapes aided in identifying modes. Across shelf data for 125 Hz shown here. k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 Window length of the AR estimator is 2000m. k 9 k 8 k 7 k 6 k 5 k 4 k 3 k 2 k 1
Perturbative Inversion 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: ∞ ∆ c z 1 ( ) ∫ ∆ = ρ − k z Z z k z dz 1 2 2 ( ) ( ) ( ) n n k c z ( ) n 0 0 This equation can be written in the form of a Fredholm integral of the first kind: D = ∫ = i N y x z A z dz 1,..., ( ) ( ) i i 0 y = Ax Which can be written in matrix form as: y is a vector representing the data A is a matrix representing the forward model x is a vector representing the model parameter
Tikhonov Regularization y = Ax Solve the ill-conditioned problem: by choosing the smoothest solution. Ax − y 2 + λ Lx 2 2 min 2 2 n d L where is a discrete version of the differential operator d x n n = favors the flattest solution; 1 n = favors the smoothest solution. 2 λ L-Curve Criterion: The Lagrange multiplier is chosen such that it both the Ax − y Lx residual and the semi-norm are minimized simultaneously. 2 2 The Regularization solution is given by: x A A L L − A y = T + λ T T 2 1 ˆ ( )
Qualitative Regularization y = Ax Solve the ill-conditioned problem: by choosing the solution that best fits some prior knowledge. 2 Ax y L x − 2 + λ 2 min q 2 2 L where is created by the user. q r − ∑ L L L q q = T I ( ) is given by: q q i i = i 1 Q q { } r where the set is on orthogonal basis for . i i = 1 The Qualitative Regularization solution is given by: x A A L T L A y = T + λ − T 2 1 ˆ ( ) q q
Comparison: Tikhonov and QR 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 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.
The New Jersey Shelf chirp seismic reflection data Chirp data provided by John Goff
Layering Information from Chirp Data
Layering Information from Chirp Data
Water Column Sound Speed
Inversion Results: Oblique Shelf Track 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.
Inversion Results: Across Shelf Track 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.
Inversion Results: Along Shelf Track 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.
Simple Model
Model Agreement: SW06 Cores 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)
Evaluation of Results: TL Prediction 50 Hz
Evaluation of Results: TL Prediction 125 Hz
Evaluation of Results: Correlation Along Shelf Across Shelf Oblique Shelf Incoherent Correlation -10 -10 -10 of the pressure fields -20 -20 -20 averaged over all -30 -30 -30 tracks Depth [m] -40 -40 -40 P r ( ) Calculated Field -50 -50 -50 P r ˆ( ) Measured Field -60 -60 -60 = cor R ( ) P r P r ˆ * ( ) ( ) -70 -70 -70 12 P r P r P r P r ˆ ˆ * * ( ) ( ) ( ) ( ) -80 -80 -80 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 Correlation Correlation Correlation R 1 ∫ = PQ P r Q r dr ( ) ( ) − R r 50 75 125 175 r min min
Conclusions • Inversion Results – Range Dependent Inversion Results for three distinct tracks – Creation of a 3-D model by determining for sound speed for each layer • Validation of Results – Comparison shows agreement between inversion result and core data – Ability to predict the acoustic field at all depths and frequencies
Back Up Slides
Evaluation of Results: Correlation Model to Data Correlation Data to Data Correlation Along Shelf Across Shelf Oblique Shelf Along Shelf Across Shelf Oblique Shelf -10 -10 -10 -10 -10 -10 -20 -20 -20 -20 -20 -20 -30 -30 -30 -30 -30 -30 Depth [m] -40 -40 -40 -40 -40 -40 -50 -50 -50 -50 -50 -50 -60 -60 -60 -60 -60 -60 -70 -70 -70 -70 -70 -70 -80 -80 -80 -80 -80 -80 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 Correlation Correlation Correlation Correlation Correlation Correlation 50 75 125 175
Monte Carlo Error Estimates The complete solution of the inverse problem requires not only the estimates of the model parameter values, but also a measure of the uncertainty of the estimates. Monte Carlo Error Propagation: y n Ax + = = i N 1,2,...., i i The empirical estimate of the covariance matrix = D D T Cov D = x − x T T where i i N Error bars: standard deviation of the model estimates σ = Cov m z diag ( ) ( )
Inversion Results: Error Bars For the Oblique Shelf Track
Calculated Error Bars 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. d + v = Gm Beginning with the linear problem: C C Data covariance matrix; Model covariance matrix v m R = (G C G + C ) G C G T -1 -1 T The resolution matrix is given by: m v m v = ∑ M R 2 ij j = rl i Resolution length is given by: 1 ( ) R 2 ii � The posterior model covariance C = (G C G + C ) T -1 -1 -1 m v m matrix is given by:
Calculated Error Bars For the Oblique Shelf Track
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