Modeling the Subsurface Structure of Sunspots

While sunspots are easily observed at the solar surface, determining their subsurface structure is not trivial. There are two main hypotheses for the subsurface structure of sunspots: the monolithic model and the cluster model. Local helioseismology is the only means by which we can investigate subphotospheric structure. However, as current linear inversion techniques do not yet allow helioseismology to probe the internal structure with sufficient confidence to distinguish between the monolith and cluster models, the development of physically realistic sunspot models are a priority for helioseismologists. This is because they are not only important indicators of the variety of physical effects that may influence helioseismic inferences in active regions, but they also enable detailed assessments of the validity of helioseismic interpretations through numerical forward modeling. In this article, we provide a critical review of the existing sunspot models and an overview of numerical methods employed to model wave propagation through model sunspots. We then carry out a helioseismic analysis of the sunspot in Active Region 9787 and address the serious inconsistencies uncovered by Gizon et al. (2009a, 2009b). We find that this sunspot is most probably associated with a shallow, positive wave-speed perturbation (unlike the traditional two-layer model) and that travel-time measurements are consistent with a horizontal outflow in the surrounding moat.     

Large-Scale Zonal Flows Near the Solar Surface

Migrating bands of weak, zonal flow, associated with the activity bands in the solar cycle, have been observed at the solar surface for some time. More recently, these flows have been probed deep within the convection zone using global helioseismology and examined in more detail close to the surface with the techniques of local helioseismology. We compare the near-surface results from global and local helioseismology using data from the Michelson Doppler Imager and the Global Oscillation Network Group with surface Doppler velocity measurements from the Mount Wilson 150-foot tower and find that the results are in reasonable agreement, with some explicable differences in detail. All of the data sets show zones of faster rotation approaching the equator from mid-latitudes during the solar cycle, with a variation at any given location that can be approximately, but not completely, described by a single sinusoid and an amplitude that does not drop off steeply below the surface.     

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.     

Local Fractional Frequency Shifts Used as Tracers of Magnetic Activity

Using data from SOI-MDI (Haber et al., 2000), we compute the local frequencies of high-degree p modes and f modes. The frequencies are obtained through ring-diagram mode fitting. The Dense-Pack data set consists of a mosaic of 189 overlapping tiles, each tracked separately at the surface rotation rate over 1664-min time intervals during the Dynamics Programs. Each tile is 16° square and the tile centers are separated by 7.5° in latitude and longitude. For each sampling day and for each tile, we have computed the frequency shift measured relative to the temporal and spatial average of the entire set of frequencies. The motion of active regions as they rotate across the solar disk is vividly traced by these measurements. Active regions appear as locations of large positive frequency shifts. If the shifts are averaged over the solar disk and are scaled down to the appropriate wave number regime, the magnitude and frequency dependence of the shifts are consistent with the measured changes in global oscillation frequencies that occur over the solar cycle. As with the frequency shifts of low-degree global oscillations, the frequency dependence of the shifts indicates that the physical phenomena inducing the shifts is confined to the surface layers of the Sun.     

A GCV based method for nonlinear ill-posed problems

This paper discusses the inversion of nonlinear ill-posed problems. Such problems are solved through regularization and iteration and a major computational problem arises because the regularization parameter is not known a priori. In this paper we show that the regularization should be made up of two parts. A global regularization parameter is required to deal with the measurement noise, and a local regularization is needed to deal with the nonlinearity. We suggest the generalized cross validation (GCV) as a method to estimate the global regularization parameter and the damped Gauss-Newton to impose local regularization. Our algorithm is tested on the magnetotelluric problem. In the second part of this paper we develop a methodology to implement our algorithm on large-scale problems. We show that hybrid regularization methods can successfully estimate the global regularization parameter. Our algorithm is tested on a large gravimetric problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Eco-environmental surface modelling requires integration of both extrinsic and intrinsic informations

Extrinsic information,from satellite observations and the outputs of spatial models,and intrinsic information,from ground observations and spatial sampling,provide two different but complementary streams of information about the Earth’s surface.These extrinsic and intrinsic informations are indispensable for the modelling of the Earth’s surface systems along with an appropriate method for integrating these two kinds of information.Building on this idea,the fundamental theorem for Earth’s surface system modeling(FTESM)was proposed,from which several corollaries have been deduced,corresponding to spatial interpolation,spatial upscaling,spatial downscaling,data fusion and model-data assimilation,respectively(Yue et al.,2016).     

Model Fusion and Joint Inversion

Inverse problems are inherently non-unique, and regularization is needed to obtain stable and reasonable solutions. The regularization adds information to the problem and determines which solution, out of the infinitely many, is obtained. In this paper, we review and discuss the case when a priori information exists in the form of either known structure or in the form of another inverse problem for a different property. The challenge is to include such information in the inversion process. To use existing known structure, we review the concept of model fusion, where we build a regularization functional that fuses the inverted model to a known one. The fusion is achieved by four different techniques. Joint inversion of two data sets is achieved by using iterative data fusion. The paper discusses four different methods for joint inversion. We discuss the use of correspondence maps or the petrophysics of the rocks, as well as structure. In particular, we suggest to further stabilize the well-known gradient cross product and suggest a new technique, Joint Total Variation, to solve the problem. The Joint Total Variation is a convex functional for joint inversion and, as such, has favorable optimization properties. We experiment with the techniques on the DC resistivity problem and the borehole tomography and show how model fusion and joint inversion can significantly improve over existing techniques.     

Subsurface Meridional Flow from HMI Using the Ring-Diagram Pipeline

We have determined the meridional flows in subsurface layers for 18 Carrington rotations (CR 2097 to 2114) analyzing high-resolution Dopplergrams obtained with the Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO). We are especially interested in flows at high latitudes up to 75^° in order to address the question whether the meridional flow remains poleward or reverses direction (so-called counter cells). The flows have been determined in depth from near-surface layers to about 16 Mm using the HMI ring-diagram pipeline. The measured meridional flows show systematic effects, such as a variation with the B ₀-angle and a variation with central meridian distance (CMD). These variations have been taken into account to lead to more reliable flow estimates at high latitudes. The corrected average meridional flow is poleward at most depths and latitudes with a maximum amplitude of about $20~\mathrm{m\,s}^{-1}$ near 37.5^° latitude. The flows are more poleward on the equatorward side of the mean latitude of magnetic activity at 22^° and less poleward on the poleward side, which can be interpreted as convergent flows near the mean latitude of activity. The corrected meridional flow is poleward at all depths within ±?67.5^° latitude. The corrected flow is equatorward only at 75^° latitude in the southern hemisphere at depths between about 4 and 8 Mm and at 75^° latitude in the northern hemisphere only when the B ₀ angle is barely large enough to measure flows at this latitude. These counter cells are most likely the remains of an insufficiently corrected B ₀-angle variation and not of solar origin. Flow measurements and B ₀-angle corrections are difficult at the highest latitude because these flows are only determined during limited periods when the B ₀ angle is sufficiently large.     

Scandium-bearing carbonates of the Tarr albitite complex,southeast Sinai

Unusually high Sc-concentrations (30 to 150 ppm) were found in the carbonate minerals (dolomite, breunnerite, calcite) of the Tarr albitite complex. Scandium is probably part of the carbonate crystal structure substituting for (Mg, Fe²⁺). Scandium is considered to have originated from a source of mantle-affinity, thus supporting an earlier suggestion that the Tarr albitite complex is related to an ophiolite at depth.