Preprocessing of magnetograms for magnetohydrostatic extrapolations

<p>Magnetohydrostatic (MHS) extrapolations are developed to model 3D magnetic fields and plasma structures in the solar low atmosphere by using measured vector magnetic fields on the photosphere. However, the photospheric magnetogram may be inconsistent with the MHS assumption. By applying Gauss‘ theorem to an isolated active region, we obtain a set of surface integrals of the magnetogram as criteria for a MHS system. The integrals are a subset of Aly’s criteria for a force-free field (FFF). Based on the new criteria, we preprocess the magnetogram to make it more consistent with the MHS assumption and, at the same time, close to the original data. As a byproduct, we also find the boundary integral that is used to compute the energy of a FFF usually underestimates the magnetic energy of an active region.</p>

An Example of the Generalized Fractional Laplacian on Rn

Using the normalization of the fractional Laplacian (-△)su(x) over the space Ck(Rn ) for all s > 0 and s ≠ 1, 2, ···, we evaluate (-△)se-||x||2 based on the multidimensional surface integrals over the unit sphere and special functions.

On the Generalized Riesz Derivative

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

Overview on the spectral combination of integral transformations

&lt;p&gt;Geodetic boundary-value problems (BVPs) and their solutions are important tools for describing and modelling the Earth&amp;#8217;s gravitational field. Many geodetic BVPs have been formulated based on gravitational observables measured by different sensors on the ground or moving platforms (i.e. aeroplanes, satellites). Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface integrals with Green&amp;#8217;s kernel functions. When solving this problem for higher-order derivatives of the gravitational potential as boundary conditions, more than one solution is obtained. Solutions to gravimetric, gradiometric and gravitational curvature BVPs (Martinec 2003; &amp;#352;prl&amp;#225;k and Nov&amp;#225;k 2016), respectively, lead to two, three and four formulas. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.&lt;/p&gt;&lt;p&gt;In this contribution we present combinations of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green&amp;#8217;s kernel functions by a spectral combination method. We investigate an optimal combination of different orders and directional derivatives of potential. The spectral combination method is used to combine terrestrial data with global geopotential models in order to calculate geoid/quasigeoid surface. We consider that the first-, second- and third-order directional derivatives are measured at the satellite altitude and we continue them downward to the Earth&amp;#8217;s surface and convert them to the disturbing gravitational potential, gravity disturbances and gravity anomalies. The spectral combination method thus serves in our numerical procedures as the downward continuation technique. This requires to derive the corresponding spectral weights for the n-component estimator (n = 1, 2, &amp;#8230; 9) and to provide a generalized formula for evaluation of spectral weights for an arbitrary N-component estimator. Properties of the corresponding combinations are investigated in both, spatial and spectral domains.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;