ARL The University of Texas at Austin Information theory application to inversions of acoustic data from a continental shelf environment D. P. Knobles, J. D. Sagers, John Goff, and R. A. Koch Work Supported by the Office of Naval Research Code 321 OA 157TH MEETING OF THE ACOUSTICAL SOCIETY OF AMERICIA 18-22 May 2009 Portland, Oregon
ARL Outline The University of Texas at Austin • Inhomogeneous Oceans: Measurements and Inference • Maximum entropy versus Bayesian • Initial Computations • Example maximum likelihood range- dependent calculations • Summary
Example of acoustic measurements to infer ARL parameters for inhomogeneous seabed The University of Texas at Austin L-array Towed sources Small impulsive sources
Measurements and Inferences ARL Why is inversion by itself not sufficient? The University of Texas at Austin • Scientific objective: Interpret acoustic propagation in inhomogeneous ocean waveguides – Mode coupling mechanisms on shelf – Kramers Kronig relationships in seabed acoustics • A maximum likelihood or inversion solution for C min is useful but insufficient • Uncertainty in inferences is a natural consequence of – Environmental variability – Noise in data – Source-receiver motion and source level variability – Model errors • Thus the need for posterior probability ρ (W|D) that given data D the correct solution is W
ARL Two methods of inferring ρ (W|D) The University of Texas at Austin • Bayesian approach designed to solve this problem – Requires likelihood function • Alternative method is maximum entropy principle – Requires constraints
Maximum Entropy Approach to Uncertainty in ARL Ocean Acoustics The University of Texas at Austin Model Space n o i t a g a p M(W) o r P l e d o M Cost C(M(W), D) δ Entropy = 0 Canonical Processed Data Hypothesis Distribution D Space ρ (W|D) Feature Space W Features relative to observed moments of C Signal Constraints Processing Statistics of W Received Sensor Data
ARL A Canonical Distribution Approach (1) The University of Texas at Austin Claude Shannon Edwin Jaynes Shannon or Gibbs Entropy δ S = 0 subject to stated constraints S = - ∫ Ω dW ρ (W|D) ln [ ρ (W|D)/ ρ (W)] Analogy with statistical mechanics Constraints for a closed system in thermodynamic equilibrium with heat reservoir ∫ Ω dW ρ (W|D) = 1 ∫ Ω dW C(W) ρ (W|D) = <C> is global minimum determined from simulated annealing average value of cost function space = 1/N ∑ C(W i )
ARL A Canonical Distribution Approach (2) The University of Texas at Austin δ S = 0 subject to stated constraints ρ (W) exp(-C(W, D)/T) ρ (W|D) = Z Canonical Distribution Z = ∫ Ω dW ρ (W) exp(-C(W, D)/T) Average <C> constraint determines T ∫ Ω dW C(W) ρ (W|D) = <C>
Relationship to Bayes formula ARL The University of Texas at Austin ρ (W|D) = ρ (W) ρ (D|W) / ρ (D) ρ (D) = ∫ Ω dW ρ (W) ρ (D|W) ρ (W|D) = ρ (W) ρ (D|W) { ∫ Ω dW ρ (W) ρ (D|W) } -1 ρ (D|W) = exp(-C/T) Canonical distribution (not normalized) plays role of likelihood function
Average, standard deviation, and ARL marginal distributions The University of Texas at Austin Continuous formulation Monte-Carlo integration
Pros and cons of Bayesian versus ARL canonical distribution approach The University of Texas at Austin • When prior information on noise and model errors is available, Bayesian approach is well justified • Maximum entropy method appears well suited for problems with sparse data Does not require direct assumptions about model / data errors or noise Indirectly includes such information via constraints from observed features of cost Prior information on ρ (W) is included naturally via relative Shannon entropy Leads to most conservative distribution No restrictions on cost functions Posterior distribution depends on cost function Can include higher order moments of features, if available ,via constraints
Application to acoustic data ARL taken on continental shelf The University of Texas at Austin • Shallow Water 2006 experiment – Data set of interest because of large spatial and temporal inhomogenities on continental shelf Seabed Water column • Current Work – Maximum entropy principle applied to data with small range inhomogenities assuming range independence – Cross-slope range-dependent data using knowledge gained from MEP analysis, Goff geophysics characterization, and SSP measurements – Representation of range-dependent media - balance of representation versus number of parameters – Working to implement faster propagation model than PE
CSS 18 CSS 26 ARL Location of sources and receivers The University of Texas at Austin 39º 30´N 70 80 100 90 110 100 120 40 130 140 39º 20´N Array 2 70 39º 10´N 60 Array 1 70 70 Array 3 80 39º 00´N 38º 50´N 73º 20´W 73º 10´W 73º 00´W 72º 50´W 72º 40´W 72º 30´W 72º 10´W
ARL Chirp seismic reflection profiles The University of Texas at Austin 12 km Track 1 Weak variations in SSP along track Track 2 Strong variations 12 km in SSP along track
Marginal distributions from MEP for ARL short range data (1-4 km) taken on Array 2 The University of Texas at Austin Ratio(layer 1) Thickness(layer1) - m Water depth - m Measured Course sand Measured agreement with cores 2x10 6 samples
ARL Sound speed profile along Track 1 The University of Texas at Austin Sound speed - m/s 1480 1490 1500 1510 1520 1530 0 10 20 SSP had small Depth - m 30 variations along isobath 40 50 60 70 80
Geophysical structure of propagation ARL track 1 from chirp reflection sonar The University of Texas at Austin Weakly range-dependent track Array 1 68 Array 2 Water depth - m Array 1 Array 2 69.75 Outer shelf wedge Depth - m 78.00 Consolidated sands 86.25 SW NE Range - km Range - km
CSS event 26 model-data comparison for weakly range ARL dependent track range 26.3 km, 35-325 Hz band The University of Texas at Austin PE RAM Measured Pressure (arbitrary units) Mode 2, 35-50 Hz Requires structure below water-sediment interface 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time - sec Information from short-range data with additional information provided by direct geophysical and sound speed measurements is sufficient for acoustic prediction along ~ isobath
ARL Sound speed profile along Track 3 The University of Texas at Austin Sound speed - m/s 1480 1490 1500 1510 1520 1530 0 Stronger SSP variation across 10 shelf. Similar SSPs - but displacement 20 of thermocline Near Source Depth - m 30 At Array 3 40 Hypothesis: 50 Place measured SSPs at source and receiver and 60 interpolate for points between source and receiver 70 80
Geophysical structure of propagation ARL track 3 from chirp reflection sonar The University of Texas at Austin Moderately range-dependent track CSS 26 Array 3
Model Data Comparison of Received Time Series from CSS 26 on Track ARL 3 on Array 3, Range = 10.43 km, 50-325 Hz 24.75 m The University of Texas at Austin PE RAM Pressure (arbitrary units) Measured 39.75 m 62.25 m Cross-shelf track successfully modeled for weak SSP variations 77.25 m 0.0 0.1 0.2 0.3 0.4 0..5 Time - sec
ARL Sound speed profile along Track 2 The University of Texas at Austin Sound speed - m/s 1480 1490 1500 1510 1520 1530 0 Strong SSP variation across 10 shelf for track 2 20 Near Source Hypothesis 1: Depth - m 30 Place measured SSP At Array 3 at source and receiver and 40 interpolate for points between source and receiver 50 60 70 80
Geophysical structure of propagation ARL track 2 from chirp reflection sonar The University of Texas at Austin Moderately range-dependent track CSS 18 Array 3 70.0 78.25 Depth - m 86.50 94.75 103.00 Range - km
Model Data Comparison of Received Time Series on Track 2 SSP hypothesis 1 ARL 24.75 m The University of Texas at Austin PE RAM Pressure (arbitrary units) Measured 39.75 m 62.25 m Observed model-data phase shift 77.25 m 0.0 0.1 0.2 0.3 0.4 0..5 Time - sec
Model Data Comparison of Received Time Series on Track 2 SSP hypothesis 2 ARL 24.75 m The University of Texas at Austin PE RAM Pressure (arbitrary units) Measured 39.75 m Measured SSPs placed at source and 3 km from receiver. 62.25 m Observed model-data phase shift diminished with hypothesis 2 77.25 m 0.0 0.1 0.2 0.3 0.4 0..5 Time - sec
Recommend
More recommend
