#### 2.1. Acoustic Fields with Normal Mode

In the stratified shallow water like pekeris waveguide, the acoustic field can be described as follows [

5]:

where,

According to Equation (1), the fields at each frequency can be expressed as a sum of individual modal contributions where p(r,z) is the acoustic pressure field at the location (r,z), and ρ is the density of the water, ${z}_{s}$ is the depth of source, ${\phi}_{m}$ is the mode depth function of the mth mode, ${\xi}_{m}$ is the horizontal wave number of the mth mode, M is the number of normal mode.

Derived from Equation (1), the acoustic field can be written simply as:

where

${d}_{m}(r,{z}_{s})$ is the mode amplitude of the

mth mode:

For a receiver array of

N hydrophones which covers the whole water column, assuming that the depth of

nth hydrophone is

${z}_{n}$ , the fields of all the receivers as Equation (3) can be expressed as follows [

12]:

Substitute Equation (3) to Equation (5), the fields

${\mathit{p}}_{N}$ can be written as:

And

${\mathit{p}}_{N}$ can be described by a matrix as follows:

where

${\mathit{p}}_{N}$ represents the pressure fields of

N hydrophones;

${\mathit{\phi}}_{NM}$ is the sample matrix of mode depth function, and the dimension is

$N\times M$;

${\mathit{d}}_{M}$ is the mode amplitudes vector, and its dimension is

$M\times 1$. When the environmental parameters are well known,

${\mathit{\phi}}_{NM}$ can be obtained accurately by the normal mode without requiring any other information. In this paper, we use the KRAKEN normal mode program [

14].

In this way, the received acoustic fields are decomposed into depth function and amplitudes of modes. In particular, we found that no matter whichever receiver arrays are adopted, the amplitudes of normal mode in a range-independent ocean waveguide are all the same. If we can derive the mode amplitudes from the received fields, let

${\widehat{\mathit{d}}}_{M}$ be the estimation, then the received acoustic field of any array can be reconstructed as follows:

Assuming a vertical linear array which can sample all the modes sufficiently needs at least

N hydrophones, now only

L elements of this array near the water surface can work well. So the array becomes a small aperture vertical linear array, and the received fields of this small array can be expressed as:

Assuming we can get the estimation of

${\mathit{d}}_{M}$ from the received fields of small array in Equation (7), then the fields of

$N-L$ remainder hydrophones can be written as:

where

${\widehat{\mathit{d}}}_{M}$ is also the estimation of

${\mathit{d}}_{M}$. As we known,

${\mathit{p}}_{L}$ and

${\mathit{p}}_{N-L}$ are subsets of

${\mathit{p}}_{N}$,

${\mathit{\phi}}_{L}$ and

${\mathit{\phi}}_{N-L}$ are subsets of

${\mathit{\phi}}_{N}$, so the fields of the whole water derived from the small aperture array can be described as:

Substitute Equation (10) into Equation (11), we can get that:

Now the problem that we consider is how to obtain the estimation of mode amplitudes

${\mathit{d}}_{M}$ from the fields of the small array. When

L is larger than

M, the mode amplitudes

${\mathit{d}}_{M}$ can be estimated by computing the pseudo inverse directly using the least squares [

12]:

This method of computing the mode amplitudes works well for overdetermined systems. However, when $L<M$, Equation (9) is underdetermined and the least square does not work, some new method must be adopted to obtain the estimation of mode amplitudes.

#### 2.2. Estimation of Mode Amplitudes by Least Squares

According the Nyquist sample theorem, a small aperture array with

L elements can only sample

L modes at the most, so the mode amplitudes estimated from Equation (7) can only include

L elements which have contribution to the fields. Additionally, the sample matrix of mode depth function

$\mathit{\phi}$ that is used must have some errors, because there is a mismatch between the model and real environment, meanwhile, there are also some system errors and noise. Therefore, the real received fields of the small array can be expressed as follows:

where

${\mathit{p}}_{L}$ is the received fields without errors,

${\mathit{\phi}}_{L}$ is the mode depth function without errors,

$\Delta {\mathbf{p}}_{L}$ and

$\Delta \mathit{\phi}$ are the perturbations which represent all the errors and mismatch.

For estimating the mode amplitudes accurately, the errors or perturbations must be as small as possible, and the optimum problem can be described like this:

where

${\u2225\xb7\u2225}_{2}$ denotes the 2-norm, and sometimes the constraint condition can be written as

${\mathit{p}}_{L}+\Delta {\mathbf{p}}_{L}\in Range\left({\mathit{\phi}}_{L}+\Delta {\mathit{\phi}}_{L}\right)$. For simplification, Equation (14) is expressed as:

Here,

$\mathit{B}=[{\mathit{\phi}}_{L},{\mathit{p}}_{L}]$ is the data augmented matrix,

$\mathit{D}=[\Delta {\mathit{\phi}}_{L},\Delta {\mathbf{P}}_{L}]$ is the augmented matrix correction or the perturbation matrix. It is obvious that whether

$\mathit{B}$ or

$\mathit{D}$ is

$L\times M+1$ matrix,

$\mathit{x}={\left[{\mathit{d}}^{T},-1\right]}^{T}$ is

$(M+1)\times 1$ vector. The problem that Equation (15) described is to find a perturbation matrix

$\mathit{D}\in {\mathit{C}}^{L\times M+1}$ whose norm square is minimized, which involves

$\mathit{B}+\mathit{D}$ in rank deficiency. So the optimization problem of Equation (15) can be written to be a standard least squares form:

Now, we decompose the augmented matrix

$\mathit{B}$ by the singular value decomposition (SVD) method:

where

$\Sigma =diag({\sigma}_{1},{\sigma}_{2},\cdots ,{\sigma}_{M+1}),{\sigma}_{1}\ge {\sigma}_{2}\ge \cdots \ge {\sigma}_{M+1}$ are the singular values,

${\mathit{v}}_{1},{\mathit{v}}_{2},\cdots {\mathit{v}}_{M+1}$ are the right singular vectors, and they all have specific physical meanings in our problem: the former

M singular vectors correspond to the

M modes of acoustic fields, and the last one is least square solution, usually it is called total least square solution [

15,

16,

17,

18]

where

$\mathit{v}(i,M+1)$ is the

ith element of the

$(M+1)$th column in the right singular matrix

$\mathit{V}$.

However, as previously shown, an array with L elements can only sample L modes at the most. Assuming a small array with L hydrophones can sample k modes completely, so we can only know that k singular values satisfying ${\sigma}_{1}\ge {\sigma}_{2}\ge \cdots \ge {\sigma}_{k+1}$, in this sense the meaning of singular values and vectors have changed, and Equation (19) is not the optimum total least square solution.

Let

${\mathit{v}}_{i}$ be a vector in the subspace

$\mathit{S}$ defined by:

Then every right singular vector corresponds to a total least square solution:

where

${\alpha}_{i}$ is the first element of

${\mathit{v}}_{i}$, and

${\mathit{y}}_{i}$ consists of the other elements of

${\mathit{v}}_{i}$, that is

${\mathit{v}}_{i}={\left[\alpha ,{\mathit{y}}_{i}^{T}\right]}^{T}$. So there are

$M+1-k$ TLS solutions, we must find an only TLS solution in some special meaning. Golub and Vanloan proposed a TLS solution in the sense of minimum norm [

19], which can solve this problem perfectly. Their algorithm is summarized as follows:

Decompose the augmented matrix $\mathit{B}$ by SVD: $\mathit{B}={\mathit{U}\mathbf{\Sigma}\mathit{V}}^{H}$.

Let L is the elements of small array, M is the mode number in a shallow waveguide, k is the number of main singular value. If L is less than M, then let k equal to L, else let k equal to M.

Let ${\mathit{V}}_{1}={\mathit{v}}_{k+1},{\mathit{v}}_{k+2},\cdots ,{\mathit{v}}_{M+1}$, it is obviously that ${\mathit{V}}_{1}$ consists of the latter $M+1-k$ column of $\mathit{V}$, so the dimension of ${\mathit{V}}_{1}$ is $(M+1)\times (M+1-k)$.

Let

${\overline{\mathit{v}}}_{1}$ denotes the first row of

${\mathit{V}}_{1}$,

${\mathit{V}}_{2}$ consists of the last rows of

${\mathit{V}}_{1}$, viz.

${\mathit{V}}_{1}=\left[{\overline{\mathit{v}}}_{1};{\mathit{V}}_{2}\right]$, so the dimension of

${\mathit{V}}_{2}$ is

$M\times (M+1-k)$, and

${\mathit{V}}_{2}$ is a row less than

${\mathit{V}}_{1}$. Then the TLS solution in the sense of minimum norm can be expressed as:

As previously mentioned, we can only sample k modes completely; however, given the solution estimated in Equation (22) suppose that the number of modes is equal to the columns of $\mathit{\phi}$ in Equation (14), then it is consistent with the physical scene.

#### 2.3. Matched Field Processing Based on Least Squares

The conventional matched field processor (CMFP) is usually also called the Bartlett processor, and its ambiguity surface is as follows [

4,

20]

where

$\mathit{R}$ is the cross spectral density matrix of received fields:

In Equation (24) ${\mathit{p}}_{c}(r,z)$, which are called replica fields, are the acoustic fields on the assumption that the virtual source is located at $(r,z)$ by the normal mode or other propagation model, ${\mathit{\omega}}_{c}$ is the normalized weight vector. In Equation (25) $\mathit{p}({r}_{s},{z}_{s})$, which are called the measured fields, are the received acoustic fields of the real source at $({r}_{s},{z}_{s})$. The essence of MFP is to obtain the square of correlation coefficient between measured fields and replica forecasted by propagation models. The dimensions of ${\mathit{p}}_{c}(r,z)$, ${\mathit{\omega}}_{c}(r,z)$, and $\mathit{p}({r}_{s},{z}_{s})$ are all $L\times 1$ for small array with L elements.

In this paper, one method in which the acoustic fields received by small array are used directly to match with replica is called CMFP, and the other method in which the fields used is recalculated with Equation (12) is called the least squares matched field processor (LSMFP):

where

${\mathit{p}}_{cls}(r,z)$ is also the replica acoustic fields on the assumption that the virtual source is located at

$(r,z)$,

${\mathit{\omega}}_{cls}(r,z)$ is the normalized weight vector,

${\mathit{p}}_{cls}({r}_{s},{z}_{s})$ is the reconstructed acoustic fields using the received fields of small array by LS at the location

$({r}_{s},{z}_{s})$ for real source. In Equation (27), the dimensions of

${\mathit{p}}_{cls}(r,z)$,

${\mathit{\omega}}_{cls}(r,z)$, and

${\mathit{p}}_{r}({r}_{s},{z}_{s})$ are all

$N\times 1$ even that the receiver array is a small vertical linear array with

L elements.

Comparing Equations (23) and (26), we can see that the performance of the two methods is depends on the similarity between the received or reconstructed received fields and the replica fields. To evaluate the similarity between the reconstructed ones and the real pressure fields, the generalized cosine-squared between two vectors is defined as:

When the whole acoustic fields are reconstructed, the performance of matched field processing or other algorithm with small aperture array is improved.