Multi-scale deep learning for estimating horizontal velocity fields on the solar surface

Abstract

Context. The dynamics in the photosphere is governed by the multi-scale turbulent convection termed as granulation and supergranulation. It is important to derive three-dimensional velocity vectors to understand the nature of the turbulent convection and to evaluate the vertical Poynting flux toward the upper atmosphere. The line-of-sight component of the velocity can be obtained by observing the Doppler shifts. However, it is difficult to obtain the velocity component perpendicular to the line of sight, which corresponds to the horizontal velocity in disk center observations. Aims. We present a new method based on a deep neural network that can estimate the horizontal velocity from the spatial and temporal variations of the intensity and vertical velocity. We suggest a new measure for examining the performance of the method. Methods. We developed a convolutional neural network model with a multi-scale deep learning architecture. The method consists of multiple convolutional kernels with various sizes of receptive fields, and performs convolution for spatial and temporal axes. The network is trained with data from three different numerical simulations of turbulent convection. Furthermore, we introduced a novel coherence spectrum to assess the horizontal velocity fields that were derived for each spatial scale. Results. The multi-scale deep learning method successfully predicts the horizontal velocities for each convection simulation in terms of the global correlation coefficient, which is often used to evaluate the prediction accuracy of the methods. The coherence spectrum reveals the strong dependence of the correlation coefficients on the spatial scales. Although the coherence spectra are higher than 0.9 for large-scale structures, they drastically decrease to less than 0.3 for small-scale structures, wherein the global correlation coefficient indicates a high value of approximately 0.95. By comparing the results of the three convection simulations, we determined that this decrease in the coherence spectrum occurs around the energy injection scales, which are characterized by the peak of the power spectra of the vertical velocities. Conclusions. The accuracy for the small-scale structures is not guaranteed solely by the global correlation coefficient. To improve the accuracy on small scales, it is important to improve the loss function for enhancing the small-scale structures and to utilize other physical quantities related to the nonlinear cascade of convective eddies as input data.