the effects of spatial field shifts in sensitivity...

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

The Effects of Spatial Field Shifts in

Sensitivity Measures

Dr. Stan E. Dosso and Mike MorleyUniversity of Victoria School of Earth and Ocean Sciences

Dr. Peter M. GilesGeneral Dynamics Canada Ltd.

Dr. Diana F. McCammonMcCammon Acoustical Consulting

Dr. Gary H. BrookeGeneral Dynamics Canada Ltd.

Prepared By:

General Dynamics Canada Ltd.11 Thornhill Dr, Suite 102Dartmouth, NS B3B 1R9

Project Manager: Peter M. Giles, 902-468-8068

Contract Number: W7707-063411

Contract Scientific Authority: Sean P. Pecknold, 902-426-3100 ext 222

Contract Report

DRDC Atlantic CR 2007-102

June 2007

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

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The Effects of Spatial Field Shifts in Sensitivity Measures Dr. Stan E. Dosso, Mike Morley University of Victoria School of Earth and Ocean Sciences Dr. Peter M. Giles General Dynamics Canada Dr. Diana F. McCammon McCammon Acoustical Consulting Dr. Gary H. Brooke General Dynamics Canada Prepared By: General Dynamics Canada 11 Thornhill Drive Dartmouth, NS Contract Project Manager: Peter M. Giles, 902-468-8068

Contract Number: W7707-063411

Contract Scientific Authority: Sean P. Pecknold, 902-426-3100 x222

Defence R&D Canada – Atlantic

Contract Report


June 2007

Author

Peter Giles

Approved by

Sean P. Pecknold

Contract Scientific Authority

Approved for release by

Kirk Foster

DRP Chair

The scientific or technical validity of this Contract Report is entirely the responsibility of the contractor and the contents do not necessarily have the approval or endorsement of Defence R & D Canada.

© Her Majesty the Queen as represented by the Minister of National Defence, 2007

© Sa majesté la reine, représentée par le ministre de la Défense nationale, 2007

Original signed by Peter Giles

Original signed by Sean P. Pecknold

Original signed by J.L. Kennedy

DRDC Atlantic CR 2007-102 i

Abstract

This report considers the effect of spatial shifts of acoustic-field structure on measures of acoustic sensitivity to variability and/or uncertainty in environmental parameters, particularly bathymetry. In general, acoustic-field perturbations due to an environmental perturbation can be considered to consist of two components: a spatial shift of the field structure, and a change to the field in addition to this shift. Quantifying the sensitivity at a fixed point without accounting for the field shift can result in high sensitivities that seem counter-intuitive. A robust numerical approach to determining field shifts is developed that maximizes the correlation between reference and perturbed acoustic fields defined over suitable range-depth windows. Fixed-point and field-shifted values can be computed for deterministic sensitivities (based on a specific environmental perturbation) or for stochastic sensitivities (based on Monte Carlo sampling of environmental uncertainty).

Sensitivity studies are carried out for an environmental model based on the Malta Plateau. The study indicates that for low frequencies and/or small perturbations to bathymetry, the field-shift component of sensitivity dominates, and field-shift correction substantially reduces the sensitivity. The effectiveness of field-shift corrections decreases with frequency, perturbation size, and overall complexity of the bathymetry. There appears to be little or no advantage to field-shift correction for sensitivities to geoacoustic parameters, with the possible exception of sediment-layer thickness.

Résumé

Le présent rapport porte sur les effets du déplacement spatial de la structure du champ acoustique sur les mesures de la sensibilité acoustique à la variabilité et aux incertitudes liées aux paramètres environnementaux, en particulier les caractéristiques bathymétriques. De façon générale, les perturbations du champ acoustique causées par un facteur environnemental comprennent deux composantes : le déplacement spatial de la structure du champ et une modification du champ en plus du déplacement. La quantification de la sensibilité en un point fixe sans tenir compte du déplacement du champ résultant risque de donner lieu à des sensibilités élevées, ce qui semble être contre-intuitif. Une méthode numérique robuste visant à déterminer les déplacements du champ qui maximise la corrélation entre les champs de référence et les champs perturbés pour des fenêtres de portée et de profondeur adéquates, a été élaborée. Les valeurs aux points fixes et les valeurs du champ déplacé peuvent être calculées en vue d’obtenir des sensibilités déterministes (pour une perturbation environnementale donnée) ou des sensibilités stochastiques (basées sur un échantillonnage d’incertitudes environnementales Monte Carlo).

Les études de sensibilité sont réalisées pour un modèle environnemental d’une région du plateau maltais. L’étude indique, que pour les basses fréquences et/ou les petites perturbations bathymétriques, la composante « déplacement du champ » de la sensibilité domine, et une correction du déplacement du champ permet de réduire considérablement la sensibilité. L’efficacité de la correction du champ diminue en fonction de la fréquence, de l’importance de la perturbation et de la complexité globale des caractéristiques bathymétriques. Il semble n’y avoir que peu ou pas d’avantages à corriger le déplacement du champ en vue d’influer sur la sensibilité aux paramètres géoacoustiques du fond marin, à une exception près : l’épaisseur de la couche de sédiments.

ii DRDC Atlantic CR 2007-102


DRDC Atlantic CR 2007-102 iii

Executive Summary

Introduction/Background

This work was undertaken as part of the Geoacoustic Parameter Sensitivity and Interaction Study. The Geoacoustic Parameter Sensitivity and Interaction Study builds on previous work performed as part of the Geoacoustic Sensitivity Study for DRDCAtlantic. The Geoacoustic Sensitivity Study investigated sensitivity of the acoustic pressure field to environmental variability in two different nominal environments and at two different frequencies. The work defined several different measures of sensitivity and quantitatively assessed the sensitivity to variations in various water column and sea sediment properties. In all cases the sensitivity was defined in terms of the acoustic pressure change at a fixed point in space.

Results

This report extends the work of the Geoacoustic Sensitivity Study in several ways. First, the measure of sensitivity is extended by recognizing that some environmental perturbations cause the acoustic field to shift in range and/or depth, and this can lead to a misleading assessment of sensitivity. The document develops a practical approach to account for the effects of spatial field-shifting in acoustic sensitivity analysis, thereby producing more meaningful measures of sensitivity. This approach is examined for a realistic environmental model based on the Malta Plateau. The study indicates that for low frequencies and/or small perturbations to bathymetry, the field-shift component of sensitivity dominates, and field-shift correction substantially reduces the sensitivity. The effectiveness of field-shift corrections decreases with increasing frequency, perturbation size, and overall complexity of the bathymetry. There appears to be little or no advantage to field-shift correction for sensitivities to seabed geoacoustic parameters, with the possible exception of sediment-layer thickness. Finally, interaction between geoacoustic parameters is considered and quantitatively assessed using a multidimensional Monte Carlo technique.

Significance

DRDC Atlantic has a practical interest in understanding the sensitivity of acoustic data to uncertainty in bathymetry and other environmental parameters in Canadian Forces operational areas. The field-shifting method developed and tested here produces a more meaningful sensitivity measure for such applications.

Future Plans

This work has developed a new measure of sensitivity, and implemented an algorithm for assessing this measure. The field-shifted sensitivity measure can now be applied to various environments of interest. Furthermore, the project developed the ability to assess

iv DRDC Atlantic CR 2007-102

sensitivity in fully range-dependent environments using PECan. This means that similar experiments can be conducted in quite realistic environments. It should not be difficult to adapt the results to other environments of interest, particularly environments where DRDCAtlantic has collected field trial measurements.

S.E. Dosso; M. Morley; P.M. Giles; D.F. McCammon; G.H. Brooke. 2007. The Effects of Spatial Field Shifts in Sensitivity Measures. DRDC Atlantic CR 2007-102. Defence R&D Canada – Atlantic.

DRDC Atlantic CR 2007-102 v

Sommaire

Introduction et contexte

Ces travaux ont été entrepris dans le cadre de l’Étude de sensibilité et d’interaction des paramètres géoacoustiques. L’Étude de sensibilité et d’interaction de paramètres géoacoustiques repose sur des travaux réalisés antérieurement dans le cadre de l’Étude de sensibilité géoacoustique réalisée pour RDDC Atlantique. L’Étude de sensibilité géoacoustique portait sur la sensibilité d’un champ de pression acoustique à la variabilité environnementale dans deux milieux distincts, et à deux fréquences différentes. Les travaux ont défini plusieurs mesures différentes de la sensibilité et ont permis d’évaluer quantitativement la sensibilité aux variations dans la colonne d’eau et aux propriétés des sédiments du fond marin. Dans tous les cas, la sensibilité a été définie en termes de changements de la pression acoustique en un point fixe dans l’espace.

Résultats

Le présent rapport complète de plusieurs manières les travaux réalisés dans le cadre de l’Étude de sensibilité géoacoustique. D’abord, la mesure de la sensibilité est élargie, compte tenu du fait que certaines perturbations environnementales peuvent causer un déplacement modifiant la portée et/ou la profondeur du champ acoustique et du fait que cela risque de mener à une fausse évaluation de la sensibilité. Le document présente une méthode pratique qui tient compte des effets du déplacement spatial du champ dans l’analyse de la sensibilité acoustique, ce qui permet d’obtenir des mesures plus représentatives de la sensibilité. Cette méthode est appliquée à un modèle environnemental réaliste situé sur le plateau maltais. L’étude indique que pour les basses fréquences et/ou petites perturbations bathymétriques, la composante « déplacement du champ » de la sensibilité domine, et une correction du déplacement du champ permet de réduire considérablement la sensibilité. L’efficacité de la correction du champ diminue en fonction de l’augmentation de la fréquence, de l’importance de la perturbation et de la complexité globale des caractéristiques bathymétriques. Il semble n’y avoir que peu ou pas d’avantages à corriger le déplacement du champ en vue d’influer sur la sensibilité aux paramètres géoacoustiques du fond marin, à une exception près : l’épaisseur de la couche de sédiments. Enfin, l’interaction entre les paramètres géoacoustiques est examinée et évaluée quantitativement à l’aide d’une technique Monte Carlo mutidimensionnelle.

Portée

RDDC Atlantique peut tirer parti d’une meilleure connaissance de la sensibilité des données acoustiques aux caractéristiques bathymétriques et à d’autres paramètres environnementaux dans les zones opérationnelles des Forces Canadiennes. La méthode de déplacement du champ élaborée et sa mise à l’essai présentée ici permettent d’obtenir des mesures de la sensibilité plus représentatives, pour de telles applications.

vi DRDC Atlantic CR 2007-102

Recherches futures

Les travaux ont permis d’élaborer une nouvelle mesure de la sensibilité et de mettre en œuvre un algorithme servant à évaluer cette mesure. La mesure de la sensibilité en champ déplacé peut maintenant être appliquée à différents environnements qui présentent de l’intérêt. En outre, dans le cadre du projet, on a développé la capacité d’évaluer la sensibilité dans des environnements définis par la portée à l’aide de PECan. Cela signifie que des expériences semblables peuvent être réalisées dans des environnements plutôt réalistes. L’adaptation des résultats à d’autres environnements d’intérêt ne devrait pas poser problème, en particulier dans les environnements où RDDC Atlantique a effectué des mesures de terrain.

Dosso, S.E., M. Morley, P.M. Giles, D.F. McCammon, G.H. Brooke. 2007. Effets du déplacement de champ spatial sur les mesures de la sensibilité. DRDC Atlantic CR 2007-102. Defence R&D Canada – Atlantic.

DRDC Atlantic CR 2007-102 vii

Table of contents

Abstract.............................................................................................................................................. i

Executive Summary ........................................................................................................................ iii

Sommaire.......................................................................................................................................... v

Table of contents ............................................................................................................................vii

List of figures ................................................................................................................................viii

1 Introduction......................................................................................................................... 1

1.1 Geoacoustic Parameter Sensitivity and Interaction Study ................................. 1

1.2 Document Objectives and Organization ............................................................. 1

2 Theory ................................................................................................................................. 2

2.1 Sensitivity.............................................................................................................. 2

2.2 Spatial Field Shifting............................................................................................ 3

3 Implementation................................................................................................................... 7

3.1 Propagation Modelling ......................................................................................... 7

3.2 Field Shifting and Shift Corrections.................................................................... 8

3.3 Plotting .................................................................................................................. 8

4 Sensitivity Study................................................................................................................. 9

4.1 Malta Plateau Environment.................................................................................. 9

4.2 Spatial Field Shifting for Range-independent Environments .......................... 10

4.3 Spatial Field Shifting for Range-dependent Environments ............................. 27

4.4 Multi-parameter Sensitivities............................................................................. 34

5 Summary and Future Work ............................................................................................. 36

References ...................................................................................................................................... 37

List of Symbols/Abbreviations/Acronyms/Initialisms ................................................................ 38

Distribution List ............................................................................................................................. 39

viii DRDC Atlantic CR 2007-102

List of figures

Figure 1. Schematic diagram of the Malta Plateau environment, including assumed parameter values and standard deviations. The asterisk indicates the source depth of 65 m. ................................. 9

Figure 2. Acoustic fields (magnitude) at 100 Hz for Malta Plateau environment with water depths of (a) 131 m, and (b) 130.5 m (i.e., a –0.5 m perturbation to water depth). ................................ 10

Figure 3. Deterministic sensitivity for a –0.5 m perturbation to water depth at 100 Hz (no spatial averaging). Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.................................................................................................................................................. 11

Figure 4. Same as Figure 3, but with spatial averaging (smoothing). ........................................ 11

Figure 5. Spatial field shifts for –0.5 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values. ....................................... 12

Figure 6. Stochastic sensitivity for a 0.5 m standard deviation in water depth at 100 Hz, computed from 100 Monte Carlo samples. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. ...................................................................................................... 13

Figure 7. Deterministic sensitivity for a –1 m perturbation to water depth at 100 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively............................ 14

Figure 8. Spatial field shifts for –1 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values............................................. 14


Figure 10. Spatial field shifts for –2 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values. ....................................... 16

Figure 11. Same as Figure 10, but without correction for anomalous field shifts. .................... 17

Figure 12. Deterministic sensitivity for a +2 m perturbation to water depth at 100 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively............................ 18

Figure 13. Spatial field shifts for +2 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values. ....................................... 18

Figure 14, Acoustic fields (magnitude) at 1200 Hz for Malta Plateau environment with water depths of (a) 131 m, and (b) 130.5 m (i.e., a –0.5 m perturbation to water depth). ................................. 19

Figure 15. Deterministic sensitivity for a –0.5 m perturbation to water depth at 1200 Hz (no spatial averaging). Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.................................................................................................................................................. 20

Figure 16. Same as Figure 15, but with spatial averaging (smoothing). .................................... 21

DRDC Atlantic CR 2007-102 ix

Figure 17. Spatial field shifts for –0.5 m water-depth perturbation at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. .................................... 21

Figure 18. Stochastic sensitivity for a –0.5 m perturbation to water depth at 1200 Hz, computed from 100 Monte Carlo samples. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. ............................................................................................................................ 22





Figure 23. Stochastic sensitivities for parameters v1, v2, h1, v0, and D, as indicated, computed from 100 Monte Carlo samples at 1200 Hz. Left and right columns show results for fixed-point and field-shifted sensitivities, respectively........................................................................................... 26

Figure 24. Deterministic sensitivity for a range-dependent perturbation consisting a 0.05° upslope (4 m over 5 km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. ............................................................................................................................ 27

Figure 25. Spatial field shifts for a range-dependent perturbation consisting a 0.05° upslope at 1200 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.................................................................................................................................................. 28

Figure 26. Deterministic sensitivity for a range-dependent perturbation consisting a 0.1° upslope (8 m over 5 km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. ............................................................................................................................ 29

Figure 27. Spatial field shifts for a range-dependent perturbation consisting a 0.1° upslope at 1200 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.................................................................................................................................................. 29

Figure 28. Deterministic sensitivity for a range-dependent perturbation consisting a 0.1° downslope (8 m over 5 km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively. ............................................................................................................................ 30

Figure 29. Spatial field shifts for a range-dependent perturbation consisting a 0.1° downslope at 1200 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.................................................................................................................................................. 30

Figure 30. Deterministic sensitivity for a range-dependent perturbation consisting of a single 5-km long wedge from 2.5–7.5 km that extends 4.36 m above the surround seafloor at its apex. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively...................... 31

x DRDC Atlantic CR 2007-102

Figure 31. Spatial field shifts for single-wedge perturbation at 1200 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values............................................. 32

Figure 32. Deterministic sensitivity for a range-dependent perturbation consisting of two 2.5 km long wedge from 2.5–5 km and 5–7.5 that extend 4.36 m above the surround seafloor. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively............................ 33

Figure 33. Spatial field shifts for two-wedge perturbation at 1200 Hz. Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values............................................. 33

Figure 34. Deterministic sensitivity contours for various combinations of environmental parameters for the Malta Plateau environment at 1200 Hz. Crosses indicate the true (unperturbed) parameter values.................................................................................................................................................. 34

Figure 35. Histograms for stochastic sensitivity from Monte Carlo sampling over the indicated environmental parameters for the Malta Plateau environment at 1200 Hz. ....................... 35

DRDC Atlantic CR 2007-102 1

1 Introduction

1.1 Geoacoustic Parameter Sensitivity and Interaction Study

This document is one of the deliverables of the Geoacoustic Parameter Sensitivity and Interaction Study. Some background on that project, and a description of the deliverables, is provided in a companion document titled Geoacoustic Parameter Sensitivity and Interaction Study: Summary of Work and Results. The scope of this document, and its relationship to other deliverable reports, is described in the following section.

1.2 Document Objectives and Organization

This report considers new approaches to quantifying the sensitivity of ocean acoustic fields to variability and/or uncertainty in physical parameters of the ocean environment. Previous work in [1] and [2] defined several useful sensitivity measures and investigated the issue of nonlinearity in sensitivity estimation. That work considered range-independent environmental variations, with a focus on the water-column sound-speed profile (SSP) and seabed geoacoustic parameters. However, in many practical applications sensitivity to variability/uncertainty in water depth is also an important issue, in both range-independent and range-dependent environments.

It has long been recognized in matched-field processing (MFP) that errors in environmental parameters (referred to as environmental mismatch) can lead to systematic errors in source localization [3]-[6]. This results from the fact that a change in some environmental parameters, particularly water depth, can lead to a systematic spatial shift in the acoustic field (i.e., the field structure remains essentially coherent but shifts in range and/or depth). The concept of spatial field shifting is also relevant to sensitivity studies, which are typically based on quantifying the perturbation in the acoustic field at a fixed point in space due to a perturbation to one or more environmental parameters. The crux of the issue involves the definition of what constitutes a significant change in the acoustic field. If a spatial shift in otherwise similar fields is not considered a significant change, then it makes sense to attempt to account for the shift in quantifying the sensitivity at a point. In practice, it may be useful to examine both fixed-point and field-shift corrected measures to properly understand the acoustic sensitivity in a particular environment.

This report develops a practical procedure to account for the spatial-shift of acoustic fields due to arbitrary environmental perturbations. To our knowledge, this represents the first systematic study of the effect of spatial field shifts in acoustic sensitivity studies. The remainder of this report is organized as follows. Section 2 provides background on the definitions of sensitivity used here, and develops the new spatial field-shift corrected sensitivity measure. Section 3 briefly describes our implementation of these sensitivity measures as practical computer codes. Section 4 provides a systematic study of sensitivity and shift corrections for Malta Plateau environment, which ties in directly to our earlier work [1], [2]. Finally, Section 5 summarizes this work.

2 DRDC Atlantic CR 2007-102

2 Theory

2.1 Sensitivity

Ocean acoustic sensitivity studies are generally based on examining the differences in modelled acoustic pressure fields (or transmission loss) that result from a given change in the environment (i.e., comparing fields computed for a reference environmental model and a perturbed model). This approach is referred to here as a deterministic sensitivity measure, since it applies to a single, specific environmental change. A useful deterministic sensitivity measure that is independent of the data and model scale and units was defined in [1] and [2] as

,||

||

||

|)(||)(|

P

P

P

mmPmPS

jjj !!=

+ = (1)

where mj represents the jth parameter of the model of environmental parameters m, P is the acoustic pressure, and mj and P are model and pressure perturbations. Deterministic sensitivities are generally highly specific to the particular environmental perturbation, but are efficient to compute and can be considered in studies based on a systematic approach to varying the environmental parameters.

In [1] and [2] the above concepts were generalized to consider acoustic sensitivities to uncertainties in environmental parameters. Environmental uncertainties are assumed to represent a Gaussian probability distribution centred on the assumed parameter value with the estimated parameter uncertainty representing the standard deviation of the distribution. Determining the acoustic sensitivity to environmental uncertainties then requires transferring these uncertainties through the forward problem (propagation modelling), and quantifying the resulting variability in the acoustic field. Monte Carlo methods provide a general approach to this problem, based on applying a large number of random Gaussian-distributed perturbations to the environmental parameters, and computing the corresponding perturbed acoustic fields. A stochastic sensitivity measure can then be defined based on quantifying the statistical variability of the ensemble of perturbed acoustic fields. In particular, [1] and [2] defined the sensitivity in terms of the coefficient of variation,

{ },

||

||2/12

P

PS

><=

(2)

where P is the reference acoustic pressure, P a realization of the perturbed pressure, and represents

an ensemble average.


The Monte Carlo approach to stochastic sensitivity estimation characterizes the full nonlinearity of the problem. However, for sufficiently small perturbations or perturbations to relatively insensitive parameters, a linear approximation to the nonlinear stochastic sensitivity can be a reasonable approximation [1], [2]

,||

|)(|

P

mPS

jj ! == (3)

where P( mj = !j) represents the acoustic pressure perturbation resulting from a perturbation to model parameter mj equal to its assumed uncertainty (standard deviation) of this parameter, !j. Hence, for a perturbation to an environmental parameter equivalent to its assumed uncertainty, the deterministic sensitivity represents a linear approximation to the nonlinear stochastic uncertainty (cf. Eqs. (1) and (2)).

The sensitivities given by equations (1)–(3) quantify the variability of the acoustic field at a point. However, these sensitivities may be to too specific for the purpose of evaluating the relative importance of environmental parameters in some applications. Propagating acoustic fields represent complicated interference patterns that are highly variable functions of source-receiver geometry, and hence the sensitivity for a particular point in space may not be representative of the overall sensitivity of the local field. To address this, data sensitivities can be spatially averaged (smoothed) to obtain a stable, representative sensitivity measure [1], [2], [7]. In particular, effective sensitivity stabilization is achieved here by appropriate spatial averaging of the numerator and denominator of the measures given by Eqs. (1)–(3). Details are given in Section 3.3.

2.2 Spatial Field Shifting

The sensitivity measures defined in the previous section consider changes in the acoustic field at a fixed point in space. However, the acoustic-field perturbation due to an environmental perturbation can, in general, include a component representing a spatial shift of the field in addition to a field-change component. In some applications it may be desirable to examine these components of sensitivity separately, since failing to account for even small spatial field shifts can result in high sensitivities at a point, which may conflict with an intuitive understanding of sensitivity.

To date, the effects of spatial field shifts have been investigated primarily for MFP source localization in cases where the water depth is not known exactly [3]-[6]. In this setting the result has been dubbed the “mirage” effect, in reference to the fact that water-depth mismatch typically manifests itself as systematic errors in the estimated source range and depth [5]. Spatial shifts have recently been recognized as relevant to sensitivity studies as well [8]. To our knowledge, this report represents the first systematic study of the effect of spatial field shifts in acoustic sensitivity studies.


To motivate our treatment of spatial field shifting, it is instructive to consider the simplest model for acoustic propagation due to a source at depth zs in a homogeneous ocean waveguide of depth D and sound speed c, with a pressure-release surface and perfectly-reflecting hard seabed. The far-field acoustic pressure at angular frequency ! is given in terms of the propagating normal modes by

,e

)sin()sin(2

e),(

1

4/

rkzkzk

hzrP

rj

rik

zjs

M

j

zj

i rj

=

=!

!

(4)

where Djkzj /)2/1( != are the vertical wave-numbers, 2/122 ][ zjrj kkk =

are the radial wave-

numbers, ck / = , and M < D!/"c + . Now consider the reference pressure field defined for an environment with water depth D and the perturbed field defined for a perturbed environment of depth D+ D. To estimate the approximate depth shift in the acoustic field due to the perturbed water depth requires finding the value of z such that the modal depth dependence is unaltered by the change in water depth, i.e., that satisfies

].)[()( zzDDkzDk zjzj ++= (5)

Substituting for the vertical wave-number leads to the relation

.zD

Dz

= (6)

According to Eq. (6) the field depth shift z is of the same sign as the water-depth perturbation D,

indicating that an increase (decrease) in water depth leads to a downward (upward) field shift. Further, the field shift is independent of frequency, increases linearly with D and z, and decreases with D. The maximum field shift of z = D occurs at the seabed, z = D.

The range evolution of the acoustic field is determined largely by modal interference, which is dominated by the interference between adjacent modes. Hence, to estimate the approximate range shift requires finding the value for r such that the phase difference between adjacent modes is unaltered by the depth perturbation, i.e., that satisfies

).()]()([)]()([ 11 rrDDkDDkrDkDk rjrjrjrj !!! ++ += ++

(7)

Substituting for the radial wave-number, expanding the square roots and neglecting second-order terms in D and r leads to

.2 rD

Dr

= (8)


According to Eq. (8), the range shift r is of the same sign as the perturbation D, indicating that an increase (decrease) in water depth leads to a outward (inward) field shift that is independent of frequency, increases linearly with D and r, and decreases with D. The fractional range shift of Eq. (8) is twice as large as the fractional depth shift of Eq. (6).

The above analysis is based on the simplest model for acoustic propagation, and is not expected to apply in detail to more realistic propagation conditions. Further, the analysis is approximate, in that it neglects possible changes in the number of modes resulting from the environmental perturbation, interference between non-adjacent modes, and second-order terms. The effects of these errors would be expected to increase with the size of the water-depth perturbation, and to increase with frequency (since increasing frequency increases the number of modes).

More advanced analyses (leading to similar results) have been developed for MFP based on penetrable seabeds and adiabatic mode theory [3]-[6]. Analytic field-shifting results for MFP have also been developed for perturbations to the SSP and to seabed geoacoustic parameters (through the concept of equivalent water depths) [4]. SSP field-shift results are mode dependent, and hence are more difficult to apply in practice. Geoacoustic field shifting results are frequency dependent, with effects that decrease with frequency.

It would seem to be clear that spatial field shifting is relevant not only to MFP localization, but also to acoustic sensitivity studies. However, there appears to be little existing work on this topic. Further, none of the existing analytic approaches to field shifting are sufficiently general to encompass the scope of the present sensitivity study, which involves sensitivity to all environmental parameters and arbitrary range dependence.

We have developed a straightforward and general numerical approach to estimate and account for the spatial field shift component of sensitivity. Consider a particular range-depth point on the reference acoustic field which is to be compared to the perturbed field. A two-dimensional window of pre-selected size is defined centred on this point. The correlation is then computed between the reference field (magnitude) over this window and the perturbed field over a series of windows of identical size which are centred at a grid of points about the point of interest. The window-centre for the perturbed field that produces the highest correlation with the reference field is taken to define the range and depth shift resulting from the environmental perturbation. The shift-corrected sensitivity can then be computed using any of Eqs. (1)–(3) by defining P to be the difference between the reference acoustic field and the perturbed field at the optimally shifted point. Note that the field-shifting procedure is based explicitly on maximizing the correlation of the local field structure over a suitable search region, not on minimizing the sensitivity over this region. Comparing sensitivity values computed both with and without the field-shifting procedure (referred to here as fixed-point and field-shifted sensitivities) can indicate the relative importance of the two components of sensitivity.

Due to the interplay between the field-shift and field-change components, it is possible for the numerical search procedure to find the largest correlation between reference and perturbed fields at a point that does not correspond to the appropriate spatial shift. This leads to erroneous field-shifted sensitivity values. For the many cases we have considered, these occasional failures are usually (although not always) relatively isolated events, and we have addressed this by applying a secondary correction step. This step is based on the fact that the field shifts are expected to vary slowly and systematically with depth and range. The computed spatial shift for each point on the range-depth surface is compared to the average shift


computed for the adjacent points. If a significant discrepancy is detected, the anomalous shift values are replaced by an average of shifts for the adjacent points (see Section 3.2 for details).

Finally, it should be noted the theoretical development in this section in terms of acoustic pressure magnitude can also be applied to transmission loss (TL). Since TL is already a relative measure, there is no need to consider relative sensitivities, i.e., if TL replaces |P| in Eqs. (1)–(3) the division by the reference TL can be omitted.


3 Implementation

The theory developed in the Section 2 has been implemented in a series of computer codes designed to calculate deterministic or stochastic acoustic sensitivities in terms of either acoustic pressure or transmission loss, account for spatial shifts, and plot the results. This section briefly outlines our implementation.

3.1 Propagation Modelling

Acoustic modelling is carried out using the PECan numerical propagation model [9], a split-step Padé-expansion parabolic equation model, which has been modified to run as a subroutine (i.e., the standard PECan input/out procedures based on reading/writing disc files is replaced by common blocks which pass values from/to a main calling program). The main program defines the environment, numerical grid, and control parameters, and calls the PECan subroutine to compute reference and perturbed acoustic fields, which are written to disc for subsequent processing. The main program can compute acoustic fields for multiple perturbed environments in a single run (e.g., for Monte Carlo sensitivity analysis), and handles memory allocation and de-allocation for each call to PECan.

For consistency, all cases considered in this report were based on PECan modelling with a numerical grid consisting of a range step r = 5 m and depth step z = 0.03125 m, although the output grid used for subsequent sensitivity calculations and plotting used an increased depth step of z = 1.0 m. Four Padé terms were included in the computations and a matched boundary condition was applied at the base of the numerical grid, just below the deepest seabed layer. Acoustic field computations to 20 km range for a 100 m deep water column at 1000 Hz required about 20 s on a 2.4 GHz PC running a Linux operating system. Acoustic fields computed with the PECan subroutine were compared to those of the stand-alone version of PECan for both range-independent and range-dependent cases, with identical results. PECan subroutine results were also compared to those of the wave-number integral code SAFARI for range-independent cases, with excellent agreement. Finally, PECan subroutine results were compared with the Gaussian-beam ray-theory code Bellhop for both range-independent and range-dependent cases. These comparisons ultimately identified deficiencies in the ray-theory model for range-dependent cases involving variations in bottom slope. The issues arising are discussed at length in an accompanying report titled Comparison Between Bellhop and PECan for Range-Dependent Bathymetry: Errors Arising

in Bellhop.


3.2 Field Shifting and Shift Corrections

Appropriate spatial shifts for the perturbed acoustic fields are computed using a secondary processing code that reads in the raw acoustic-field files written by the PECan modelling main program. As described in Section 2.2, the field shift for a particular point in space is based on determining the maximum correlation value between the reference acoustic field in a window about the point and similar windows of the perturbed field centred at a grid of points about the point of interest. The optimal range and depth shifts for each point of interest are computed using a straightforward grid search technique. After examining a variety of possibilities, an appropriate window size was determined to extend ±50 m in range (10 grid points) and ±5 m in depth (5 grid points) about the point of interest. Field shifts in increments of 5 m in range and 1 m in depth were tested, extending to ±500 m in range and ±10 m in depth. In some cases it may make sense to use larger field windows at lower frequencies and smaller windows at higher frequencies, but we did not find an appreciable advantage for the cases considered here.

As described in Section2.2, anomalous field shifts are occasionally obtained. To check and correct for these, the computed shift for each point on a range-depth surface is compared to the average shift for the adjacent points. If a significant discrepancy is detected, the anomalous shifts are replaced by the average shifts for the adjacent points. A number of variations of this procedure were considered, and the approach adopted here compares the field-shift for a particular point to the average shift for the adjacent points of equal or lesser depth and range, with the processing proceeding over a range-depth section from left-to-right, top-to-bottom. This approach is not failure proof, but was found to reduce the number of anomalous shift values to very low levels in cases where the field-shifting concept applies reasonably well. After this shift-correction procedure is carried out, the corrected spatial shifts are written to disc files for use by plotting programs.

3.3 Plotting

For computational efficiency, the propagation-modelling driver code and the spatial-shift code described in the previous two sections were written in FORTRAN. However, for convenience, the codes which read in the acoustic-field and spatial-shift files and plot the resulting sensitivities were written in MATLAB. As mentioned in Section 2.2, spatial averaging (smoothing) is generally applied in plotting to produce stable sensitivities with interpretable structure. The smoothing applied here was based on spatial averaging the computed sensitivities over ±5 m in depth and ±100 m in range (same as in [1], [2]).


4 Sensitivity Study

4.1 Malta Plateau Environment

The environmental parameters and uncertainties considered in this section are based on the Malta Plateau, a well-studied shallow-water region of the Mediterranean Sea which was a focus of our previous work [1], [2]. The environmental model, illustrated in Figure 1, is comprised of a water column of depth D = 131 m over-top a seabed consisting of three homogeneous layers. The ocean sound-speed profile (SSP), measured at the site, includes a strong negative gradient in the near-surface waters and a weak sound channel with its axis near mid-water depth. The three seabed layers are characterized by sound velocities v1, v2, v3, densities !1, !2, !3, and attenuation coefficients 1, 2, 3. The upper two sediment layers are of thicknesses h1 and h2 (the basement is semi-infinite). Environmental parameter values and uncertainties that are representative of the Malta Plateau region are given in Figure 1. The SSP profile uncertainty is taken to represent oceanographic variability due to surface heating/cooling and wind mixing, with the effects decaying exponentially with depth over the top 30 m (expressed in terms of the standard deviation of the surface sound-speed value, v0). Parameter sensitivities for the Malta Plateau environment were studied in [1], [2] using fixed-point sensitivity measures. The conclusions of that study were that the most important geoacoustic parameters were v0, v1, and h, and that the parameters of the second and third seabed layers (with the exception of v2) were essentially insensitive. Water depth D was studied in [1] and sensitivity to typical variations was found to be quite high. It was not considered in [2].

Figure 1. Schematic diagram of the Malta Plateau environment, including assumed parameter values and

standard deviations. The asterisk indicates the source depth of 65 m.


4.2 Spatial Field Shifting for Range-independent Environments

Since spatial field-shifting effects are most pronounced for variations in water depth, this parameter will be considered first and in the most detail. Further, since field-shift corrections are expected to be most straightforward for low frequencies and small perturbations, the study will begin with these cases and proceed to larger perturbations and higher frequencies.

Figure 2. Acoustic fields (magnitude) at 100 Hz for Malta Plateau environment with water depths of (a) 131

m, and (b) 130.5 m (i.e., a –0.5 m perturbation to water depth).

Figure 2(a) shows 100 Hz acoustic fields (magnitude) computed for the Malta Plateau environment for a source at 65 m depth. Figure 2(b) shows the same, except that the water depth is reduced by 0.5 m (one standard deviation) to D = 130.5 m. As would be expected for this small environmental change, the overall field structure in the two figures appears to be very similar. Figure 3(a) shows the deterministic sensitivity for this water-depth perturbation, computed according to Eq. (1) as the difference between the fields divided by the reference field at each point. According to this plot, the acoustic fields are highly sensitive to the small perturbation in D, with many localized regions of sensitivity S > 0.5, and a root-mean-square (RMS) average sensitivity computed over the range-depth surface of 0.3. However, the major component of this fixed-point sensitivity is due to small spatial shifts in the field structure. Accounting for the spatial shift before computing the sensitivity, as described in Section 2.2, leads to the field-shifted sensitivity results shown in Figure 3(b). Very few field-shifted sensitivity values are higher than 0.1 and the RMS average sensitivity is 0.03, representing a tenfold decrease over the fixed-point sensitivity. This could represent the difference between classifying the acoustic fields as sensitive or insensitive to this water-depth perturbation.


Figure 3. Deterministic sensitivity for a –0.5 m perturbation to water depth at 100 Hz (no spatial averaging).

Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.

Figure 3 shows raw sensitivity surfaces without the application of spatial averaging (smoothing). At higher frequencies, spatial averaging is generally required to produce stable, interpretable sensitivity results, and hence will be adopted as the standard in this report, even at low frequencies. Figure 4 shows the same sensitivity results as Figure 3, but with the spatial averaging described in Section 3.2 applied. Unless stated otherwise, spatial averaging is applied in all sensitivity plots in the remainder of this section.

Figure 4. Same as Figure 3, but with spatial averaging (smoothing).


The spatial field shifts that were computed in calculating the shift-corrected sensitivities of Figure 3 and Figure 4 are shown in Figure 5. Figure 5(a) shows that negative range shifts result from a decrease in water depth, and that the range shift is essentially independent of depth and increases uniformly with range. This general behaviour agrees with the simplistic model of Eq. (8). Further, the shift of about –0.15 km at 20 km range, shown in Figure 5(a), agrees closely with the prediction of Eq. (8). The depth shifts shown in Figure 5(b) consist of small values that fluctuate about zero. This is consistent with the simplistic model of Eq. (6), since the maximum depth shift predicted is –0.5 m (at the seabed) while the grid employed for the depth-shift correlation search had a depth increment of 1 m. Figure 5(c) shows that high correlation values (> 0.975) were generally achieved in the spatial-shifting procedure.

Figure 5. Spatial field shifts for –0.5 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in

range and depth; (c) shows corresponding correlation values.

Figure 6 shows stochastic sensitivities computed via Monte Carlo analysis for 100 Gaussian-distributed perturbations to the water depth with a standard deviation of 0.5 m. Comparisons (not shown) between several Monte Carlo runs with different random-number generator seeds indicated that 100 perturbations were sufficient to achieve convergence. The generally good overall agreement between the results in Figure 4 (deterministic sensitivity for –0.5 m perturbation) and Figure 6 (stochastic sensitivity for 0.5 m standard deviation) indicates that the water-depth sensitivity acts approximately linear at this scale. The RMS averages for the fixed-point and field-shifted sensitivities in Figure 6(a) and (b) are 0.3 and 0.04, respectively.


Figure 6. Stochastic sensitivity for a 0.5 m standard deviation in water depth at 100 Hz, computed from 100

Monte Carlo samples. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities,

respectively.

Deterministic sensitivity and field-shift results for a water-depth perturbation of D = –1 m are shown in Figure 7 and Figure 8. Figure 7 shows a significant sensitivity increase relative to the -0.5 m perturbation of Figure 4. Figure 7 also shows that accounting for spatial shifts substantially reduces the sensitivity, with RMS averages of 0.5 and 0.06 obtained for fixed-point and field-shifted sensitivities. The range shifts shown in Figure 8 (a) are approximately twice as large as those in Figure 5(a) where the depth perturbation was half as large, in agreement with the predictions of Eq. (8). Figure 8(b) shows negative depth shifts of approximately 1 m at larger depths, in general agreement with Eq. (6).


Figure 7. Deterministic sensitivity for a –1 m perturbation to water depth at 100 Hz. Panels (a) and (b) show

results for fixed-point and field-shifted sensitivities, respectively.

Figure 8. Spatial field shifts for –1 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in



Figure 9 and Figure 10 show sensitivities and field shifts for a water-depth perturbation of D = -2 m. Figure 9 shows that although field-shift correction again substantially reduces the sensitivity, it is not as effective as in the previous cases involving smaller water-depth perturbations, and relatively large field-shifted sensitivity values occur (RMS averages are 0.9 and 0.2 for fixed-point and field-shifted sensitivities). The range shifts shown in Figure 10 (a) are essentially depth-independent and increase uniformly with range, with values roughly consistent with predictions of Eq. (8). However, although the depth shifts shown in Figure 10(b) generally increase with depth, the range-dependent structure of this plot represents a significant departure from the predictions of Eq. (6). This indicates that that simplistic model of Eq. (6) is breaking down, while the numerical approach to estimating field shifts remains robust.

Figure 9. Deterministic sensitivity for a –2 m perturbation to water depth at 100 Hz. Panels (a) and (b) show





In all sensitivity results presented so far, corrections for anomalous shifts, described in Section 3.2, have been applied. To illustrate the results of this correction procedure, Figure 11 shows the raw range and depth shifts (i.e., without the correction procedure) for the same case as shown in Figure 10 for the corrected shifts. Comparing these figures indicates that a number of anomalous shifts (particularly in range), evident in Figure 11, have been corrected in Figure 10. (The number of anomalous shifts tends to increase with environmental perturbation size; hence, the correction procedure was not illustrated earlier for smaller perturbations).


Figure 11. Same as Figure 10, but without correction for anomalous field shifts.

The simple field-shift models of Eqs. (6) and (8) are anti-symmetric about zero perturbations (i.e., depth and range shifts of equal magnitude but opposite sign are predicted for perturbations of opposite sign). Sensitivities and field shifts for an increase in water depth of D = +2 m are shown in Figure 12 and Figure 13. The sensitivities in Figure 12 are qualitatively similar to those obtained for D = -2 m in Figure 9 (RMS averages are 1.0 and 0.3 for fixed-point and field-shifted sensitivities in Figure 12). The range and depth field shifts shown in Figure 13 are also similar to those obtained for the equivalent water-depth decrease (Figure 10), but of opposite sign. Note in Figure 13 that depth shifting fails near the seabed for a positive water-depth perturbation since the acoustic field cannot be shifted downward into the seabed.


Figure 12. Deterministic sensitivity for a +2 m perturbation to water depth at 100 Hz. Panels (a) and (b) show


Figure 13. Spatial field shifts for +2 m water-depth perturbation at 100 Hz. Panels (a) and (b) show shifts in



Figure 14 through Figure 22 consider a similar analysis to that described above for a frequency of 100 Hz, but for a frequency of 1200 Hz (same as considered in [1] and [2]). The modelled acoustic fields for water depths of 131 and 130.5 m are shown in Figure 14, i.e., D = -0.5 m. Note that at this higher frequency, refracted propagation paths within the sound channel become more significant (cf. Figure 2 and Figure 14).

Figure 14, Acoustic fields (magnitude) at 1200 Hz for Malta Plateau environment with water depths of (a) 131

m, and (b) 130.5 m (i.e., a –0.5 m perturbation to water depth).

Figure 15 and Figure 16 show the deterministic sensitivity (fixed-point and field-shifted), both with and without spatial averaging applied. At this higher frequency, the benefit of spatial smoothing in producing simpler, more robust structure is clearly evident, and only Figure 16 will be considered further.


Figure 15. Deterministic sensitivity for a –0.5 m perturbation to water depth at 1200 Hz (no spatial

averaging). Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.

Figure 16 shows a low sensitivity region mid-water column due to refracted sound-channel propagation which has limited interaction with the seabed. Outside of the sound channel the sensitivity is higher at 1200 Hz than for the same water-depth perturbation at 100 Hz (Figure 4). The sensitivity in Figure 16 is considerably reduced by the field-shifting procedure; however, the reduction is not as great as at the lower frequency (Figure 4). RMS averages for the fixed-point and field-shifted sensitivities in Figure 16 are 0.5 and 0.2, respectively. The range and depth field shifts are shown in Figure 17. The range shifts, shown in Figure 17(a), are no longer depth independent (cf. Figure 5), and the effects of sound-channel propagation are evident as reduced shifts in the mid-water region. The depth shifts, shown in Figure 17(b) are near zero, as expected for a depth perturbation of -0.5 m and a correlation search grid with 1 m depth increments. The correlation values obtained for each point by the field-shifting procedure, shown in Figure 17(c), are considerably smaller than those obtained at 100 Hz (Figure 4c). All of the above indicates that the field-shift component plays a reduced role in the total sensitivity at 1200 Hz compared to 100 Hz.


Figure 16. Same as Figure 15, but with spatial averaging (smoothing).

Figure 17. Spatial field shifts for –0.5 m water-depth perturbation at 1200 Hz. Panels (a) and (b) show results

for fixed-point and field-shifted sensitivities, respectively.


Figure 18 shows stochastic sensitivities computed via Monte Carlo analysis for 100 Gaussian-distributed perturbations to the water depth with a standard deviation of 0.5 m. The generally good overall agreement between the results in Figure 16 (deterministic sensitivity for -0.5 m perturbation) and Figure 18 (stochastic sensitivity for 0.5 m standard deviation) indicates that the water-depth sensitivity acts approximately linear for this perturbation and frequency. The RMS average sensitivities are 0.5 and 0.2 for Figure 18(a) and (b), respectively.

Figure 18. Stochastic sensitivity for a –0.5 m perturbation to water depth at 1200 Hz, computed from 100

Monte Carlo samples. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities,

respectively.

Figure 19 and Figure 20, and Figure 21 and Figure 22, consider deterministic sensitivities at 1200 Hz to water-depth perturbations of D = -1 m and -2 m, respectively. Figure 19 and Figure 21 indicate that both the fixed-point and field-shifted sensitivities increase with increasing water-depth perturbations, and that these sensitivities are considerably higher than those for the same perturbation at 100 Hz (Figure 7 and Figure 9). The RMS averages for fixed-point and field-shifted sensitivities are 0.7 and 0.4 for Figure 19, and 0.9 and 0.7 for Figure 21. The range and depth shifts, shown in Figure 20 and Figure 22, exhibit increasingly complex structure. The correlation values obtained in the field-shifting procedure, also shown in Figure 20 and Figure 22, are much lower than the corresponding values at 100 Hz (Figure 8 and Figure 9), and the correlation values decrease with increasing water-depth perturbation size.


Figure 19. Deterministic sensitivity for a –1 m perturbation to water depth at 1200 Hz. Panels (a) and (b)

show results for fixed-point and field-shifted sensitivities, respectively.




Figure 21. Deterministic sensitivity for a –2 m perturbation to water depth at 1200 Hz. Panels (a) and (b)

show results for fixed-point and field-shifted sensitivities, respectively.




So far, the effects of spatial field shifts on environmental sensitivity have been considered only for perturbations to the water depth, which is expected to be the dominant cause of field shifts. However, it is possible for spatial field shifts to result from perturbations to SSP parameters and even seabed geoacoustic parameters (through the concept of equivalent water depth). Hence, it is of interest to assess the importance of field shifts for other parameters of the Malta Plateau environment. Figure 23 shows fixed-point and shift-corrected stochastic sensitivities for parameters v1, v2, h1, v0, and D at 1200 Hz based on 100 Monte Carlo samples for each parameter with standard deviations equal to the parameter uncertainty (Figure 1). This figure represents an extension of a similar figure in [2], which considered fixed-point sensitivities for v1, v2, h1, v0, but not D. RMS average fixed-point and field-shifted (in brackets) sensitivity values are: v1 6.0 (5.8) , v2 0.1 (0.1), h1 6.0 (5.2), v0, 1.0 (1.0), and D 0.5 (0.2). The RMS average sensitivities and the results in Figure 23 indicate that, for this environment, spatial field shifting represents a negligible contribution to the sensitivity for parameters v1, v2, and v0, and is, at best, a minor contribution for h1. Only for the water depth D is field shifting an important component of sensitivity.


Figure 23. Stochastic sensitivities for parameters v1, v2, h1, v0, and D, as indicated, computed from 100 Monte

Carlo samples at 1200 Hz. Left and right columns show results for fixed-point and field-shifted sensitivities,

respectively.


4.3 Spatial Field Shifting for Range-dependent Environments

This section considers the effects of spatial field shifts on acoustic sensitivity for range-dependent environmental perturbations. Since the Section 4.2 indicated that water-depth perturbations are the dominate cause of field shifting, variability/uncertainty in bathymetry is considered. In all cases, the reference acoustic fields are based on 1200 Hz propagation to 5 km range in the reference Malta Plateau environment (Figure 1) with a flat seafloor at 131 m depth.

The perturbed environment for the first range-dependent example involves a seafloor that slopes upward at a 0.05° angle (a 4 m decrease in water depth over 5 km range). Figure 24(a) shows relatively high fixed-point sensitivity to this perturbation, with an RMS average sensitivity of 0.7. Figure 24(b) shows that accounting for the spatial field shift due to the sloping seafloor significantly reduces the sensitivity, with an RMS average field-shifted sensitivity of 0.3. Figure 25(a) shows negative (inward) range shifts that increase in magnitude with range but are largely depth independent (with slightly decreased values near the sound-channel axis). Figure 25(b) shows negative (upward) depth shifts that increase in magnitude with both depth and range. Figure 25(c) indicates correlations values that are not particularly high, suggesting a significant component of field change in addition to field shift.

Figure 24. Deterministic sensitivity for a range-dependent perturbation consisting a 0.05° upslope (4 m over 5

km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.


Figure 25. Spatial field shifts for a range-dependent perturbation consisting a 0.05° upslope at 1200 Hz.

Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.

Figure 26 shows fixed-point sensitivities for a perturbed environment in which the seafloor slopes upward at a 0.1° angle (8 m over 5 km). Figure 26(a) shows high fixed-point sensitivities, with an RMS average sensitivity of 0.9. Figure 26(b) shows somewhat reduced field-shifted sensitivities, with an RMS average of 0.5. In particular, high field-shifted sensitivities are obtained beyond about 4 km range. The calculated range shifts, shown in Figure 27(a), are similar to those for the 0.05° slope (Figure 25a), but are approximately twice as large. The negative depth shifts, shown in Figure 27(b), increase uniformly in magnitude with range and depth similar to those for the 0.05° slope (Figure 25b) out to about 4 km range. However, beyond this range, the regular pattern in depth shifting is no longer observed. In addition, Figure 27(c) indicates unusually low correlation values beyond about 4 km range. Thus, it appears that beyond 4 km range the environmental change due to the sloping seafloor is such that the concept of field shifting begins to break down. This could be due, in part, to a significant change in the number of modes between the reference and perturbed environments which cannot be accounted for by a simple spatial shift.

Figure 28 and Figure 29 are similar to Figure 26 and Figure 27 except that they involve an environmental perturbation consisting of a 0.1° downslope (water depth increase of 8 m over 5 km range). The sensitivities shown in Figure 28 are similar to those in Figure 26, and the range and depth shifts in Figure 29 are similar to those in Figure 27, except of opposite sign. A region of high field-shifted sensitivity, with a concomitant breakdown in the structure of the depth shifts and low correlation values are again obtained beyond about 4 km range.


Figure 26. Deterministic sensitivity for a range-dependent perturbation consisting a 0.1° upslope (8 m over 5

km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.

Figure 27. Spatial field shifts for a range-dependent perturbation consisting a 0.1° upslope at 1200 Hz. Panels

(a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.


Figure 28. Deterministic sensitivity for a range-dependent perturbation consisting a 0.1° downslope (8 m over

5 km) at 1200 Hz. Panels (a) and (b) show results for fixed-point and field-shifted sensitivities, respectively.

Figure 29. Spatial field shifts for a range-dependent perturbation consisting a 0.1° downslope at 1200 Hz.

Panels (a) and (b) show shifts in range and depth; (c) shows corresponding correlation values.

Figure 30 and Figure 31 consider the acoustic sensitivity to an environmental perturbation consisting of a bathymetric wedge in which the water depth decreases uniformly from 131 to 129 m over 2.5–5 km range, then increases to 131 m again from 5–7.5 km range (i.e., 0.05° upslope and downslope). Figure 30


shows zero sensitivity to just beyond 2.5 km range since the reference and perturbed environments are identical for these ranges. Beyond 2.5 km range Figure 30(a) shows reasonably large fixed-point sensitivity values, particularly outside of the sound channel. However, there is no clear relationship between the sensitivity and the upslope, downslope or flat bathymetry. The field-shifted sensitivities, shown in Figure 30(b), are significantly lower, with RMS sensitivities for fixed-point and field-shifted sensitivities of 0.4 and 0.1, respectively. Figure 31(a) shows that the range shifts are zero to about 3 km range, become increasingly negative from 3–6 km range, then remain approximately constant. Figure 31(b) shows negative depth shifts that increase in magnitude with range and depth from about 3–5 km range over the upslope section of the wedge, then decrease in magnitude from 5–7 km range over the downslope section. Beyond about 7 km, the depth shifts are generally zero, but with a numerous small regions of nonzero values (positive and negative). This suggests that the shift is not particularly well determined after the field has traversed the upslope and downslope sections of the wedge, although reasonably high correlation values are indicated in Figure 31(c).

Figure 30. Deterministic sensitivity for a range-dependent perturbation consisting of a single 5-km long

wedge from 2.5–7.5 km that extends 4.36 m above the surround seafloor at its apex. Panels (a) and (b) show



Figure 31. Spatial field shifts for single-wedge perturbation at 1200 Hz. Panels (a) and (b) show shifts in range

and depth; (c) shows corresponding correlation values.

Figure 32 and Figure 33 consider the acoustic sensitivity to an environmental perturbation consisting of two bathymetric wedges of the same height as the previous example (2 m) but half the length (2.5 km), extending from 2.5–5 km and 5 –7.5 km. Figure 32(a) shows that the fixed-point sensitivity is quite similar to that for the single wedge (Figure 30a). However, the field-shifted sensitivity, shown in Figure 32(b), is considerably larger than for the single wedge (Figure 30b). RMS averages are 0.4 and 0.3 for the fixed-point and field-point sensitivities, respectively. The range shifts, shown in Figure 33(a), are negative and depth independent, and increase in magnitude out to about 6 km range, similar to the single-wedge case (Figure 31a). However, beyond this range the pattern breaks down and the range shifts appear to become unstable, including regions of large positive shifts. The depth shifts, shown in Figure 33(b), follow the same pattern over the first wedge as was observed for the single-wedge case (Figure 31b). However, beyond about 6 km range the pattern breaks down and the depth shifts also appear unstable. Relatively low correlation values are obtained beyond 6 km range (Figure 33c). This example suggests that the cumulative effect of propagating over complicated bathymetric structure (such as two wedges) is that the acoustic-field perturbation becomes increasingly less amenable to representation as a simple spatial shift.


Figure 32. Deterministic sensitivity for a range-dependent perturbation consisting of two 2.5 km long wedge

from 2.5–5 km and 5–7.5 that extend 4.36 m above the surround seafloor. Panels (a) and (b) show results for

fixed-point and field-shifted sensitivities, respectively.

Figure 33. Spatial field shifts for two-wedge perturbation at 1200 Hz. Panels (a) and (b) show shifts in range

and depth; (c) shows corresponding correlation values.


4.4 Multi-parameter Sensitivities

Finally, another aspect of acoustic sensitivity analysis that was not considered in our previous work [1], [2] is that of the simultaneous dependence on perturbations to multiple environmental parameters (sound speed, density, attenuation, etc.). A practical approach to this problem is to specify self-consistent sets of parameter values that define distinct seabed types (e.g., sand, silt, clay) applicable to the particular environment of interest, and then consider the sensitivity to seabed type. This approach is used for the Emerald Basin environment in the companion report titled The Sensitivity of Transmission Loss Modeling

to Environmental Resolution. Here we briefly consider more general approaches to quantify multi-parameter sensitivities, applied to the Malta Plateau environment.

A straightforward way to examine sensitivity interactions between pairs of environmental parameters is to compute deterministic sensitivities over a two-dimensional grid of perturbed parameter values, as illustrated in Figure 34 for various combinations of geoacoustic parameters v1, 1, !1, h1, and v2 for the Malta Plateau environment at 1200 Hz. The sensitivity contours in Figure 34 indicate that the sensitivity to v1 dominates 1, !1, and v2 with little interaction. However, the v1- h1 sensitivity contours are more interesting. The positive slope to the contour lines over much of the plot indicates that similar sensitivities are obtained by increasing both v1 and h1. This likely occurs because an appropriate increase in both the sound speed and the thickness of the sediment layer keeps a constant acoustic transit time through the layer.

Figure 34. Deterministic sensitivity contours for various combinations of environmental parameters for the

Malta Plateau environment at 1200 Hz. Crosses indicate the true (unperturbed) parameter values.


Two-dimensional plots such as Figure 34 are not possible when considering the dependence of sensitivity on more than two parameters. One approach that can be generalized to any number of parameters is to plot histograms of the stochastic sensitivity computed from multi-dimensional environmental perturbations, as illustrated in Figure 35. The five histograms in Figure 35, from bottom to top, represent the relative sensitivity distributions obtained by applying a large number (5000) of random Gaussian perturbations to parameters v1; v1 and !1; v1, !1 and 1; v1, !1, 1 and v2; and v1, !1, 1, v2 and h1. In each case, the standard deviation for the Gaussian distribution for each parameter perturbation is equal to that parameter’s assumed uncertainty, as given in Figure 1. The histogram widths become wider as the random variability of more parameters are included. However, the histogram width does not increase significantly by including !1, 1 and v2 in addition to v1, indicating that the sensitivity to v1 dominates the sensitivity to these other parameters. A more substantial increase in width occurs when h1 is included, indicating a significant effect of this parameter on multi-dimensional sensitivity.

Figure 35. Histograms for stochastic sensitivity from Monte Carlo sampling over the indicated environmental

parameters for the Malta Plateau environment at 1200 Hz.


5 Summary and Future Work

This report considered new approaches to quantifying acoustic sensitivity to ocean environmental parameters. In particular, sensitivity to water depth in both range-independent and range-dependent environments was considered in detail. It has been previously recognized in MFP that uncertainty/variability in water depth can result in a spatial shift of the acoustic field structure. This also has consequences for sensitivity studies in that the acoustic-field perturbation due to a water-depth perturbation may be considered to consist of two components: a spatial shift of the field structure and a change to the field in addition to this shift. Quantifying the sensitivity at a fixed point in space does not account for the field shift, and may result in a much higher sensitivity measure that can seem counter-intuitive. Existing analytic expressions for acoustic field shifts, while of interest, are not general enough for application to sensitivity analyses in range-dependent environments. Hence, a robust numerical approach was devised here based on determining the shift that maximizes the correlation between reference and perturbed acoustic fields defined over suitable range-depth windows. This approach allows both fixed-point and field-shift corrected sensitivities to be considered; in addition, the optimal range and depth shifts and correlation values can be examined. Fixed-point and field-shifted values can be computed for deterministic sensitivities based on a single, specific environmental perturbation) or for stochastic sensitivities (based on Monte Carlo sampling to environmental uncertainty).

Sensitivity studies were carried out for an environmental model based on the Malta Plateau. The study indicated that for low frequencies (100 Hz) and/or small perturbations to water depth (range-independent and range dependent), the field-shift component of sensitivity dominated, and accounting for this field shift could substantially reduce the measure of sensitivity. The importance of field-shift corrections was decreased for higher frequencies, but was still significant at 1200 Hz for most water-depth perturbations considered. There appeared to be a cumulative reduction in the effectiveness of field-shift corrections in propagating over variable bathymetry (e.g., multiple upslopes and downslopes). There appeared to be little or no advantage to field-shift corrections for sensitivities to geoacoustic parameters, except possibly sediment-layer thickness.

Finally, two approaches to examining sensitivity to uncertainties in multiple environmental parameters were briefly considered.

This work has developed a new measure of sensitivity, and implemented an algorithm for assessing this measure. The field-shifted sensitivity measure can now be applied to various environments of interest. Furthermore, the project developed the ability to assess sensitivity in fully range-dependent environments using PECan. PECan, in fact, supports fully three-dimensional environments. This means that future sensitivity experiments can be conducted in quite realistic environments. It should not be difficult to adapt the results to other environments of interest, particularly environments where DRDC Atlantic has collected field trial measurements.


References

[1] P. M. Giles, D. F. McCammon, and S. E. Dosso, 2006. Geoacoustic sensitivity study: Phases

II and III, DRDC Atlantic Contract Report CR 2006-066.

[2] S. E. Dosso, P. M. Giles, G. H. Brooke, D. F. McCammon, S. Pecknold, and P. C. Hines, 2007. “Linear and nonlinear measures of ocean acoustic environmental sensitivity,” J.

Acoust. Soc. Am., 121, 42–45.

[3] D. R. Del Balzo, C. Feuillade, and M. M. Rowe, 1988, “Effects of water-depth mismatch on matched-field localization in shallow water,” J. Acoust. Soc. Am., 83, 2180–2185.

[4] E. C. Shang and Y. Y. Wang, 1991. “Environmental mismatching effects on source localization in mode space,” J. Acoust. Soc. Am., 89, 2285–2290.

[5] G. L. D’Spain, J. J. Murray, W. S. Hodgkiss, N.O. Booth, and P. W. Schey, 1999. “Mirages in shallow water matched field processing,” J. Acoust. Soc. Am., 105, 3245–2165.

[6] C. H. Harrison and M. Siderius, 2003. “Effective parameters for matched-field geoacoustic inversion in range-dependent environments,” IEEE J. Ocean. Eng., 28, 432–445.

[7] R. T. Kessel, 1999. “A mode-based measure of field sensitivity to geoacoustic parameters in weakly range-dependent environments,” J. Acoust. Soc. Am., 105, 122–129.

[8] K. R. James and D. R. Dowling, 2006. “Approximating acoustic field uncertainty in underwater sound channels,” J. Acoust. Soc. Am., 119, 3353 (Abstract).

[9] G. H. Brooke, D. J. Thompson and G. R. Ebbeson, 2000. “PECan: A Canadian parabolic equation model for underwater sound propagation,” J. Comp. Acoust., 9, 69–100.


List of Symbols/Abbreviations/Acronyms/Initialisms

DND Department of National Defence

OPI Office of Primary Interest

R&D Research & Development


Distribution List

Document No.: DRDC Atlantic CR 2007-102

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The Effects of Spatial Field Shifts in Sensitivity Measures

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Dosso, Stan E.; Morley, Michael; Giles, Peter M.; McCammon, Diana F.; Brooke, Gary H.

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This report considers the effect of spatial shifts of acoustic-field structure on measures of acoustic sensitivity to variability and/or uncertainty in environmental parameters, particularly bathymetry. In general, acoustic-field perturbations due to an environmental perturbation can be considered to consist of two components: a spatial shift of the field structure, and a change to the field in addition to this shift. Quantifying the sensitivity at a fixed point without accounting for the field shift can result in high sensitivities that seem counter-intuitive. A robust numerical approach to determining field shifts is developed that maximizes the correlation between reference and perturbed acoustic fields defined over suitable range-depth windows. Fixed-point and field-shifted values can be computed for deterministic sensitivities (based on a specific environmental perturbation) or for stochastic sensitivities (based on Monte Carlo sampling of environmental uncertainty).

Sensitivity studies are carried out for an environmental model based on the Malta Plateau. The study indicates that for low frequencies and/or small perturbations to bathymetry, the field-shift component of sensitivity dominates, and field-shift correction substantially reduces the sensitivity. The effectiveness of field-shift corrections decreases with frequency, perturbation size, and overall complexity of the bathymetry. There appears to be little or no advantage to field-shift correction for sensitivities to geoacoustic parameters, with the possible exception of sediment-layer thickness.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a

document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title). Environmental variability Spatial sensitivity Acoustic propagation modeling

the effects of spatial field shifts in sensitivity...

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