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Time-delay estimation has essential applications in the field of radar, sonar and robotics. For a very distant source, time-delay vector estimation across an M-sensor array is realised using the generalised cross-correlation (GCC) function and the estimates combined with their covariance matrix, expressed in terms of a priori known signal and noise spectra, to yield a linear minimum variance unbiased estimator, known as the Gauss–Markov Estimate. In the absence of a priori information, spectral estimation has to be done at one of the sensors. For close range sources, the use of amplitude attenuation information across the array will improve this estimate further. This study presents a new amplitude information related Gauss–Markov estimate, which calculates the power spectral density (PSD) at all the sensors of the array and gives more accurate time delay estimates in terms of the mean-square error when compared to the earlier constant amplitude-based technique using the PSD at the closest sensor. The performance has been evaluated against signal-to-noise ratio for varying distance of a source from the receiving array. The results have been verified by simulations and experiments for a two-dimensional source localisation problem.

Recent advances in mantle convection modeling led to the release of a new generation of convection codes, able to self-consistently generate plate-like tectonics at their surface. Those models physically link mantle dynamics to surface tectonics. Combined with plate tectonic reconstructions, they have the potential to produce a new generation of mantle circulation models that use data assimilation methods and where uncertainties in plate tectonic reconstructions are taken into account. We provided a proof of this concept by applying a suboptimal Kalman filter to the reconstruction of mantle circulation (Bocher et al., 2016). Here, we propose to go one step further and apply the ensemble Kalman filter (EnKF) to this problem. The EnKF is a sequential Monte Carlo method particularly adapted to solve high-dimensional data assimilation problems with nonlinear dynamics. We tested the EnKF using synthetic observations consisting of surface velocity and heat flow measurements on a 2-D-spherical annulus model and compared it with the method developed previously. The EnKF performs on average better and is more stable than the former method. Less than 300 ensemble members are sufficient to reconstruct an evolution. We use covariance adaptive inflation and localization to correct for sampling errors. We show that the EnKF results are robust over a wide range of covariance localization parameters. The reconstruction is associated with an estimation of the error, and provides valuable information on where the reconstruction is to be trusted or not.

We formulate and numerically solve a two-dimensional boundary value problem of Stefan type with nonlinear heat sources of a special kind and a variable heat exchange coefficient. The model under study arises in cryosurgery in the process of freezing some living biological tissue by a cryoinstrument of cylindrical shape placed on the surface of the tissue. The model takes into account the actually observed effect of spatial localization of heat. Some results of the computer simulation are presented.