Algorithm 1: KF-MVDR-MF algorithm for underwater localisation estimation.

Input: True source position and system parameters, raw acoustic data Output: Estimated source location 1 Set the true position of the sound source and parameters; 2 Acquire and read simulation data; 3 Truncate the signal and perform segmented processing; 4 Calculate the number of Fourier-transform points and segmented-processing parameters; 5 Compute the narrowband covariance matrix for the truncated signal; 6 Calculate the energy weighting: $$w_X\left(f\right)={\displaystyle \frac{{\left|X\left(f\right)\right|}^2}{{\sum \left|X\left(f\right)\right|}^2}}$$ 7 Estimate the covariance matrix: $$R_x={\displaystyle \frac{1}{N_{\mathrm{tr}}}}\sum _{n=1}^{N_{\mathrm{tr}}}{R}_{x,n}$$ 8 Compute the sound-field replica vector and the KF-MVDR-MF weight vector: $$\begin{array}{cc}\hfill p\left(r\right)& ={\displaystyle \frac{1}{\rho \sqrt{8\pi r}}}{e}^{-j\pi /4}\varphi \left(z\right){\varphi }^H\left(z_r\right){\displaystyle \frac{e^{-jkr}}{\sqrt{kr}}},\hfill \\ \hfill w_C& ={\displaystyle \frac{p\left(r\right)}{\parallel p\left(r\right)\parallel }},\hfill \\ \hfill w_M& ={\displaystyle \frac{R_x^{-1}{w}_C}{w_C^H{R}_x^{-1}{w}_C}}.\hfill \end{array}$$ 9 Convert results to decibel units: $${\mathrm{KF-}\mathrm{MVDR-}\mathrm{MF}}_{\mathrm{dB}}=10{log}_{10}\left(\mathrm{KF-}\mathrm{MVDR-}\mathrm{MF}\right)$$ 10 Plot KF-MVDR-MF results and output the estimated sound-source location;