Sensitivity Analysis in Ocean Acoustic Propagation

Abstract

The sensitivity of acoustic pressure to sound speed is investigated through the application of adjoint-based sensitivity analysis using an acoustic propagation model. The sensitivity analysis is extended to temperature and salinity, by deriving the adjoint of the sound polynomial function of temperature and salinity. Numerical experiments using a range dependent model are carried out in a deep and complex environment at the frequency of 300 Hz. It is shown that through the adjoint sensitivity analysis one can infer reasonable variations of sound speed, and thus temperature and salinity. Successful extension of the sensitivity of acoustic pressure to temperature and salinity implies that acoustic pressure observations in a given range-depth plane can be assimilated into an ocean model using the acoustic propagation model as the observation operator.

Appendix: Equation of Sound Speed with Its Tangent Linear and Adjoint

The equation for the speed of sound in seawater in m s⁻¹, given by Chen and Millero (Chen and Millero 1977) is:

$$ U(s,t,P) = C_{w} (t,P) + A(t,P)s + B(t,P)s^{{\frac{3}{4}}} + D(t,P)s^{2} $$

(A1)

where s is the salinity in PSS-78, t the temperaure in °C and P the water column pressure in decibars, not to be confused with the acoustic pressure p used in the text above. A, B, C and D are temperature- and pressure-dependent parameters. The term C_w is defined as:

$$ \begin{aligned} C_{w} (t,P) & = C_{{00}} + C_{{01}} t + C_{{02}} t^{2} + C_{{03}} t^{3} + C_{{04}} t^{4} + C_{{05}} t^{5} \\ {\text{ }} & {\text{ + }}\left( {C_{{10}} + C_{{11}} t + C_{{12}} t^{2} + C_{{13}} t^{3} + C_{{14}} t^{4} } \right)P \\ {\text{ }} & {\text{ + }}\left( {C_{{20}} + C_{{21}} t + C_{{22}} t^{2} + C_{{23}} t^{3} + C_{{24}} t^{4} } \right)P^{2} \\ {\text{ }} & {\text{ + }}\left( {C_{{30}} + C_{{31}} t + C_{{32}} t^{2} } \right)P^{3} \\ \end{aligned} $$

(A2)

The term A is defined as:

$$ \begin{aligned} A(t,P) & = A_{{00}} + A_{{01}} t + A_{{02}} t^{2} + A_{{03}} t^{3} + A_{{04}} t^{4} \\ {\text{ }} & {\text{ + }}\left( {A_{{10}} + A_{{11}} t + A_{{12}} t^{2} + A_{{13}} t^{3} + A_{{14}} t^{4} } \right)P \\ {\text{ }} & {\text{ + }}\left( {A_{{20}} + A_{{21}} t + A_{{22}} t^{2} + A_{{23}} t^{3} } \right)P^{2} \\ {\text{ }} & {\text{ + }}\left( {A_{{30}} + A_{{31}} t + A_{{32}} t^{2} } \right)P^{3} \\ \end{aligned} $$

(A3)

The term B is defined as:

$$ B(t,P) = B_{{00}} + B_{{01}} t + \left( {B_{{10}} + B_{{11}} t} \right)P $$

(A4)

The term D is defined as:

$$ D(t,P) = D_{{00}} + D_{{10}} P $$

(A5)

Linearization

Note that in the derivations that follow we have neglected the variations of the water column pressure (P) with temperature and salinity. According the first order Taylor’s approximation, the equations (A1)–(A5) above can be linearized as follows, with the prime symbol appended to the linearized variables:

$$ \begin{gathered} U'(s,t,P,s',t') = C_{w}^{'} (t,P,t') + A'(t,P,t')s + A(t,P)s' \hfill \\ {\text{ }} + B'(t,P,t')s^{{\frac{3}{4}}} + B(t,P)s^{{ - \frac{1}{4}}} s' + 2D(t,P)ss' \hfill \\ \end{gathered} $$

(A6)

$$ \begin{gathered} C_{w}^{'} (t,P,t') = \left[ {\left( {C_{{01}} + 2C_{{02}} t + 3C_{{03}} t^{2} + 4C_{{04}} t^{3} + 5C_{{05}} t^{4} } \right)} \right. \hfill \\ {\text{ + }}\left( {C_{{11}} + 2C_{{12}} t + 3C_{{13}} t^{2} + 4C_{{14}} t^{3} } \right)P \hfill \\ {\text{ + }}\left( {C_{{21}} + 2C_{{22}} t + 3C_{{23}} t^{2} + 4C_{{24}} t^{3} } \right)P^{2} \hfill \\ {\text{ }}\left. {{\text{ + }}\left( {C_{{31}} + 2C_{{32}} t} \right)P^{3} } \right]t' \hfill \\ \end{gathered} $$

(A7)

$$ \begin{gathered} A'(t,P,t') = \left[ {\left( {A_{{01}} + 2A_{{02}} t + 3A_{{03}} t^{2} + 4A_{{04}} t^{3} } \right)} \right. \hfill \\ {\text{ + }}\left( {A_{{11}} + 2A_{{12}} t + 3A_{{13}} t^{2} + 4A_{{14}} t^{3} } \right)P \hfill \\ {\text{ + }}\left( {A_{{21}} + 2A_{{22}} t + 3A_{{23}} t^{2} } \right)P^{2} \hfill \\ {\text{ + }}\left. {{\text{ }}\left( {A_{{31}} + 2A_{{32}} t} \right)P^{3} } \right]t' \hfill \\ \end{gathered} $$

(A8)

$$ B'(t,P,t') = \left( {B_{{01}} + B_{{11}} P} \right)t' $$

(A9)

The Adjoint

In the following equation the * symbol is appended to the adjoint variables. Given the adjoint of sound speed as resulting from the adjoint of the acoustic propagation model, the adjoint variables associated to both temperature and salinity are obtained from transposing the equations (A6)–(A9) according the L2 inner product

$$ \begin{gathered} s* = \left[ {A(t,P) + B(t,P)s^{{ - \frac{1}{4}}} + 2D(t,P)s} \right]U* \hfill \\ B* = s^{{\frac{3}{4}}} U* \hfill \\ A* = sU* \hfill \\ C_{w}^{*} = U* \hfill \\ \end{gathered} $$

(A10)

$$ \begin{gathered} t* = t* + \left[ {\left( {C_{{01}} + 2C_{{02}} t + 3C_{{03}} t^{2} + 4C_{{04}} t^{3} + 5C_{{05}} t^{4} } \right)} \right. \hfill \\ {\text{ + }}\left( {C_{{11}} + 2C_{{12}} t + 3C_{{13}} t^{2} + 4C_{{14}} t^{3} } \right)P \hfill \\ {\text{ + }}\left( {C_{{21}} + 2C_{{22}} t + 3C_{{23}} t^{2} + 4C_{{24}} t^{3} } \right)P^{2} \hfill \\ {\text{ }}\left. {{\text{ + }}\left( {C_{{31}} + 2C_{{32}} t} \right)P^{3} } \right]C_{w}^{*} (t,P,t') \hfill \\ \end{gathered} $$

(A11)

$$ \begin{gathered} t* = t* + \left[ {\left( {A_{{01}} + 2A_{{02}} t + 3A_{{03}} t^{2} + 4A_{{04}} t^{3} } \right)} \right. \hfill \\ {\text{ + }}\left( {A_{{11}} + 2A_{{12}} t + 3A_{{13}} t^{2} + 4A_{{14}} t^{3} } \right)P \hfill \\ {\text{ + }}\left( {A_{{21}} + 2A_{{22}} t + 3A_{{23}} t^{2} } \right)P^{2} \hfill \\ {\text{ + }}\left. {{\text{ }}\left( {A_{{31}} + 2A_{{32}} t} \right)P^{3} } \right]A*(t,P,t') \hfill \\ \end{gathered} $$

(A12)

$$ t* = t* + \left( {B_{{01}} + B_{{11}} P} \right)B*(t,P,t') $$

(A13)

The coefficients for the above terms are given in Table 1 below.

Table 1 Coefficients of the polynomials (A1)—(A5)