Extraction of Reflective Modes Using a Short Vertical Line Array in Shallow Water with a Thermocline

The waveguide invariant called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> is used in many passive and active sonar applications where the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> is required. In shallow water, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> is classically considered as a single value but several studies show that when a thermocline exists, it is better modeled by a distribution because of its dependence on mode pairs. However, it is not a simple issue to estimate the precise distribution of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> in the real ocean, which limits application scenarios of the waveguide invariant. To solve this problem, based on the difference between reflective modes and refractive modes in grazing angles, we design a beamformer using a short vertical line array to filter out the refractive modes while the reflective modes with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta=1$</tex> are preserved. Simulations demonstrate the effectiveness of the proposed method in shallow waveguide with a thermocline.