Doppler Spectrum Reconstruction for Precipitation with Gaussian Process Models

Nowadays, fast-scanning phased array radars at airports are used for point-like target (e.g., birds and drones) detection and tracking. The phased array is typically one-dimensional, and the beam is formed in the elevation axis. The radar scans the azimuth mechanically. The fast scanning feature in the azimuth (typically up to 60 [rpm]) allows radars of this kind to have a faster update of the atmosphere around them, which is needed for point-like target detection and tracking. Due to the fast scanning nature in azimuth, the radar has a limited dwell time per range-azimuth cell. Therefore, it limits the capability of the radar to measure the Doppler spectrum during precipitation events accurately. With conventional techniques, the radar needs to scan the azimuth slowly to measure the Doppler spectrum of the precipitation-like events accurately. A new approach based on Gaussian processes has been investigated for reconstructing the Doppler spectrum of precipitation-like events for a very fast scanning radar. This approach shows that the Doppler spectrum can be reconstructed accurately by using a few samples of the radar echoes in time by exploiting some prior knowledge of the spectrum. The radar echoes in time are modelled with a Gaussian process having a pre-defined covariance structure.

f(t) = GP(0, K(Θ))

Here f(t) is the time domain radar echoes of a typical precipitation event with time t, GP stands for Gaussian process, and K is the kernel covariance function with parameters (Θ). The kernel function is derived using the prior knowledge of typical Doppler spectrum shape for precipitation events. The parameters (Θ) of this covariance matrix (known as the hyper-parameters for the Gaussian process model) are estimated first from the available observations. The following marginal log-likelihood is maximized to obtain the optimized hyper-parameters of the kernel function.

log(p(z|Θ)) = -1/2(z ^H K ^-1(Θ) z + |K| + N log(2π))

In the above equation, z is the observation samples, N is the number of samples, the superscript ^H is the Hermitian operator.

The next step is to reconstruct the spectrum using appropriate point estimates of the hyper-parameters. The reconstruction of the total spectrum is carried out by using theoretical models of the covariance function in the frequency domain. The frequency domain posterior of a Gaussian process prior model (in the time domain) is also Gaussian in nature. The posterior mean and covariance of the spectrum are derived and are available in closed form. As the posterior of the spectrum is known in closed form and uses a limited number of observations, the computational load is not very high. As Gaussian process models use a Bayesian framework for inference, it inherently deals with irregular/ missing observations and accounts for its uncertainty in the posterior estimate. The estimated spectrum is compared with the classical Discrete Fourier Transform (DFT) technique, and it is found that the proposed method reproduces the spectrum shape accurately with only a few samples. In contrast, the DFT approach requires a considerable number of samples.

A sample result based on simulated weather observations is shown in the attached figure. The "x" axis is normalized Doppler frequency, and the "y" axis is the power spectrum in linear scale normalized. The ground truth spectrum (plotted with pink dotted lines) is constructed using 1024 coherent time samples with the DFT approach. As the ground truth result here is constructed with a stochastic signal having a finite time interval, the spectrum contains multiple peaks inside the Gaussian envelope. The blue dotted line is the spectrum constructed with only eight samples (zero-padded till 1024) using the DFT approach. The black dotted line is the posterior mean spectrum constructed with only eight consecutive samples using the proposed approach. The green lines are several different realizations of the posterior spectrum using the proposed approach. It can be observed that the estimated spectrum using the proposed approach reconstructs the shape of the original spectrum accurately by only using a few samples in time. The presentation and the final version will explain more results and analysis of the proposed approach.