ISSN 1063-7710, Acoustical Physics, 2018, Vol. 64, No. 1, pp. 91–98. © Pleiades Publishing, Ltd., 2018. Original Russian Text © A.G. Sazontov, I.P. Smirnov, 2018, published in Akusticheskii Zhurnal, 2018, Vol. 64, No. 1, pp. 86–93.
ACOUSTIC SIGNAL PROCESSING AND COMPUTER SIMULATION
Source Localization by a Partially Calibrated Array in an Uncertain Transmission Channel A. G. Sazontova, b, * and I. P. Smirnova a
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603600 Russia b Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603022 Russia *e-mail:
[email protected] Received January 23, 2017
Abstract⎯A robust Capon-type algorithm is constructed for source localization by a partially calibrated array operating in an uncertain environment. Results of statistical modeling are presented to determine the accuracy of source localization and the probability of correct source detection. Experimental testing of the proposed method is carried out to demonstrate its performance in Ladoga Lake. Keywords: source localization, partially calibrated subarray-based array, robust algorithm, statistical modeling, experimental testing DOI: 10.1134/S1063771018010153
acoustics to partially compensate the mismatch effects (see, e.g., [9–13]). With these methods, the source reconstruction quality can be improved to a certain extent. However, most of the algorithms include timeconsuming simultaneous estimation of both the desired coordinates and the unknown parameters of the waveguide. Therefore, it is necessary to develop stable methods of solving the inverse problem with lesser computation cost.
1. INTRODUCTION Source localization in an underwater sound channel is an important applied problem of underwater acoustics [1–3]. However, its solution based on the matched field method encounters some fundamental difficulties. This is caused by the fact that the aforementioned approach is highly sensitive to the data on both the geometry of the receiving array and the parameters of ocean environment, and, hence, the inevitable discrepancy (mismatch) between the received sound field and its calculated model may lead to incorrect solution of the inverse problem in fullscale conditions. To improve the accuracy of coordinate estimation and to achieve high resolution, it is desirable to use spatially developed arrays composed of subarrays. However, practical realization of such large-aperture arrays encounters natural technical difficulties related to total array calibration. Therefore, in the literature, one can find a considerable number of publications devoted the methods that allow source localization without knowing the relative positions of subarrays and their mutual amplitude–phase calibration [4–8]. At the same time, in actual conditions, not only the array geometry but also the true parameters of the medium are a priori unknown and, hence, a calculated signal replica always differs from the received one by an error caused by the uncertainty in data on the channel characteristics (the sound velocity profile, the sea depth, and the parameters of the bottom). Several adaptive approaches have been developed in ocean
It should be noted that in the last decade, in the general theory of adaptive arrays, rapid progress had been observed in the field of research related to the development of robust algorithms taking into account the deviation of the expected replica from the true one and minimizing the mismatch effects (see, e.g., [14– 17]). The approach uses the worst-case principle; it assumes the norm bounded vector mismath, and, the adaptation to the conditions of a priori uncertainty of the channel parameters consists in determining the optimal signal vector, which ensures maximum output power of the processor and satisfies the constraint imposed upon it. In this paper, using the worst-case principle, we construct a robust Capon-type algorithm intended for source localization by a partially calibrated array operating in an uncertain propagation channel. We present the results of statistical modeling and experimental testing of the proposed method to characterize its efficiency in an actual shallow water channel. 91
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2. PROBLEM STATEMENT. INITIAL RELATIONS
M
G (0, z (jk )
ϕ n(z 0 )ϕ n(z (jk ) ) exp[(i κ n − α n )r0 θ) = 8πκ nr0 n =1 + i π 4], j = 1,2, , N k .
∑
Let a narrow-band sound source be positioned in a waveguide at the point with coordinates θ = (r0, z 0 )T and let the source emit a noise signal with envelope s(t ), which represents a statistically stationary random process with zero mean and variance σ 2s . (Superscript T denotes transposition.) The signal is received by an N-element vertical array consisting of K linear nonoverlapping subarrays containing N k elements each and positioned at depths {z (jk )} Nj =k1. (The range origin is at the array site.)
Here, ϕ n(z (jk ) ) are eigenfunctions of a regular waveguide at the site of jth hydrophone of kth subarray and κ n and α n are the propagation constant and the bottom attenuation factor of nth mode, respectively. In its turn, for the bottom model in the form of a homogeneous absorbing fluid half-space with density ρ b and longitudinal sound velocity cl = cb (1 − i α), mode coefficients α n can be calculated by formula [18]
In the narrow-band approximation, the total field at the array input is characterized by N-dimensional sample observation vector x l :
ρ k n ϕ (H ) αn = w 0 b n α. ρ b 2κ n κ 2n − k 02nb2
x l = s l a(θ) + nl , l = 1,2,… L,
(1)
where l is the time snapshot index, a(θ) is the Ndimensional spatial array signal vector, nl is the Ndimensional sample vector of additive noise, and L is the sample size. In the following analysis, we assume that individual subarrays are calibrated but their mutual amplitude– phase calibration is a priori unknown. In this case, the array signal vector can be represented as [4, 5]
a(θ) = V(θ)h.
(2)
Here, V(θ) is an N × K , matrix:
⎛ a1(θ) 0 0 ⎞ ⎜ 0 a (θ) 0 ⎟ 2 ⎟, V(θ) = ⎜ ⎜ ⎟ ⎜ 0 0 a K (θ) ⎟⎠ ⎝
(
)
T
where a k (θ) = G (0, z1(k ) θ), ,G (0, z N(kk) θ) is the signal vector of kth subarray with dimension N k × 1, and G (0, z (jk ) θ) is Green’s function of the Helmholtz equa-
2
2 2
Here, H is the depth of the channel, k 0 is the reference wave number of the sound wave, ρ w is the density of water, nb is the acoustic index of refraction in the bottom, and parameter α characterizes the absorbing properties of bottom sediments and is related to measured attenuation coefficient β by the formula β = 40π log e α dB/λ. The problem consists in constructing an adaptive processing algorithm that makes it possible to estimate the desired source position from received sample {x l }lL=1 without knowing both the amplitude–phase calibration of the array and the true parameters of the acoustic waveguide. One of the most popular spectral estimation methods is the Capon algorithm according to which the desired source position can be determined from the condition [19]
θˆ = arg max P (θ), P (θ) = θ
1 , ˆ a (θ)Γ −x 1a(θ)
(3)
+
where Γˆ x is the N × N sample covariance matrix determined as L
Γˆ x = 1 L
∑x x , l
+ l
tion. Unknown vector h = (h1, h2, , hK ) determines the mutual amplitude–phase calibration of subarrays with respect to the first one. Evidently, h1 = 1, and the kth element of vector h can be represented as hk = g k e iϕ k , where g k and ϕ k are the unknown gain factor and the unknown phase difference between kth and first subarrays, respectively.
and symbol (⋅) + denotes Hermitian conjugate. Using representation (2), criterion (3) can be reformulated as follows:
Below, in calculating the expected signal replica, we use the wave approach in terms of which Green’s function G (0, z (jk ) θ) can be represented as superposition of a finite number M of propagating normal modes:
where C(θ) = V + (θ)Γˆ −x 1V(θ) is a K × K matrix. Complex vector h, appearing on the right-hand side of Eq. (4) is a priori unknown and should be estimated (together with the desired source coordinates) from the condition that objective function h +Ch is mini-
T
l =1
θˆ = arg min{min{h +C(θ)h}, θ
(4)
h
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SOURCE LOCALIZATION BY A PARTIALLY CALIBRATED ARRAY
mum under additional constraint h1 ≡ h +e1 = 1, where e1 = (1,0, ,0)T is a K-dimensional vector. The problem of determining conditional extremum +
1
which is valid for an arbitrary positive definite matrix C. Then, assuming that C = V +Γˆ −x1V, we obtain
e1T (V +Γˆ −x 1V) −1e1 ≥
+
min h C(θ)h subject to h e1 = 1 h
93
1
eT1 (V +Γˆ 1x V)e1
.
As a result, instead of expression (6), we obtain
has well-known solution
T + −1 min e1 (V Γˆ x V)e1 subject to
C −1(θ)e h(θ) = T −1 1 , e1 C (θ)e1
V
whose substitution in Eq. (4) leads to the following generalized Capon algorithm [8]:
θˆ = arg max P (θ), P (θ) = eT1 C −1(θ)e1.
(5)
θ
V − V0
2 F
≤ ε. (7)
To exclude the trivial solution V = 0 to (7), it is necessary to impose additional condition on the 2 admissible norm of expected matrix V0: V0 F > ε. In the subsequent analysis, without loss of generality, we 2 assume that V0 F = K , and, hence, ε < K .
Estimate (5) implies a priori knowledge of the dependence of signal matrix V(θ) on parameter θ. However, under most practical scenarios, this matrix is replaced (because of incomplete data on the propagation channel) by a certain presumed matrix V0(θ), calculated for the nominal acoustic characteristics of the waveguide. The presence of a mismatch between V(θ) and V0(θ) considerably impairs the performance of the above localization method. Below, we construct a robust algorithm version that improves the stability of estimation procedure and partially compensates for the effect of deterministic discrepancy.
To determine the optimum robust signal matrix, we construct Lagrange function
3. CONSTRUCTION OF THE ADAPTIVE SIGNAL MATRIX
[(Γˆ −x 1 + ν I)V − ν V0 ]E = 0,
In constructing the algorithm that minimizes mismatch effects, we assume that true matrix V(θ) differs from expected matrix V0(θ) by a some unknown covariance matrix error whose norm is bounded by a given 2 constant: V(θ) − V0 (θ) F ≤ ε, where ε is a non-negative regularization parameter (whose choice will be discussed later on) and ⋅ F denotes the Frobenius norm. Adaptation to a priori unknown reception conditions consists in construction of a robust signal matrix V(θ, ε), ensuring maximum output power (5) and satisfying the above constraint:
where I is the identity matrix. Since E ≠ 0, the above equation will obviously be satisfied if:
max{eT1 (V +Γˆ −x 1V)−1e1} subject to V
V − V0
2 F
≤ ε. (6)
(Here and below, argument θ is omitted for brevity.) We construct an approximate solution to optimization problem (6); for this purpose, in expression (6), we replace the objective function by its lower boundary. We use the Cauchy–Bunyakowsky inequality:
1 = e1
4
(
)(
)
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2 F
− ε),
where ν is a real valued positive multiplier. Differentiation of L with respect to V (for fixed ν ) and equating the result to zero, gives
Γˆ −x 1VE + ν V − ν V0 = 0, E = e1e1T ≡ (e1, 0, , 0). (8) Multiplying Eq. (8) by matrix E on the right and tak2 ing into account that (in view of e1 = 1) E 2 = E, we have
V(θ, ε) = ν(Γˆ −x 1 + ν I)−1 V0(θ) −1 ≡ V0(θ) − (ν Γˆ x + I) V0 (θ).
(9)
In obtaining (9), we used relation
(Γˆ −x 1 + ν I)−1 = ν −1[I − (ν Γˆ x + I)−1], which is a consequence of the general formula for inversion of a sum of matrices:
( A + B ) −1 = A −1 − A −1 (B −1 + A −1 ) A −1, −1
where A = ν I, B = Γˆ −x 1. Lagrange multiplier ν can be found from the condition
V − V0
2 F
−1
= (ν Γˆ x + I) V0
2 F
= ε,
(10)
which is a consequence of the constraint imposed on the norm of the mismatch matrix.
≤ e1T Ce1 e1T C −1e1 ,
ACOUSTICAL PHYSICS
L(V, ν) = e1T V +Γˆ −x 1Ve1 + ν( V − V0
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taking into account that (Γˆ −x 1 + ν I)−1 = ˆ ˆ +, we reduce the ˆ (Λˆ −1 + ν I)−1 Ψ ˆ+=Ψ ˆ (I + ν Λˆ )−1 ΛΨ Ψ expression for C(θ, ε) to the form convenient for calculations:
15 km 1481 m/s 70 m
44.5 m
с(z) 1 2 160 m
f = 250 Hz
3 N = 24
1467 m/s
cb = 1750 ± 50 m/s, ρb = 1.5 ± 0.3 g/сm3, β = 0.25 ± 0.15 dB/λ Fig. 1. Geometry of the numerical experiment.
To determine the root of Eq. (10), we perform an eigenvalue decomposition of the sample covariance matrix: N
ˆˆˆ+ = Γˆ x = ΨΛΨ
∑ λˆ ψˆ ψˆ , j
j
+ j
j =1
(11)
λˆ 1 ≥ λˆ 2 ≥ ≥ λˆ N > 0,
ˆ = (ψ ˆ 1, , ψ ˆ N ) is the N × N matrix containwhere Ψ ing the eigenvectors of Γˆ x , and Λˆ = diag(λˆ 1, , λˆ N ) is the diagonal matrix of the corresponding eigenvalues. Then, Eq. (10) can be reduced to the form N
g(ν) =
mj
2
∑ (1 + νλˆ ) j =1
2
ˆ +j V0. = ε, m j = ψ
(12)
j
For ν > 0 , g(ν) is a monotonically decreasing function, since N
∂ g(ν) = −2 ∂ν j =1
∑
2 λˆ j m j < 0, (1 + νλˆ j )3
in addition, g(0) = ∑ j =1 m j ≡ V0 F > ε, and, for ν → ∞, g(ν) → 0 < ε. Hence, for ν > 0 , equation (12) has a unique solution. The localization region of the corresponding root is determined by the inequalities N
2
2
V0 F − ε V − ε
