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Seismic imaging of the Sun's far hemisphere and its applications in space weather forecasting

Abstract The interior of the Sun is filled acoustic waves with periods of about 5 min. These waves, called “p modes,” are understood to be excited by convection in a thin layer beneath the Sun's surface. The p modes cause seismic ripples, which we call “the solar oscillations.” Helioseismic observatories use Doppler observations to map these oscillations, both spatially and temporally. The p modes propagate freely throughout the solar interior, reverberating between the near and far hemispheres. They also interact strongly with active regions at the surfaces of both hemispheres, carrying the signatures of said interactions with them. Computational analysis of the solar oscillations mapped in the Sun's near hemisphere, applying basic principles of wave optics to model the implied p modes propagating through the solar interior, gives us seismic maps of large active regions in the Sun's far hemisphere. These seismic maps are useful for space weather forecasting. For the past decade, NASA's twin STEREO spacecraft have given us full coverage of the Sun's far hemisphere in electromagnetic (EUV) radiation from the far side of Earth's orbit about the Sun. We are now approaching a decade during which the STEREO spacecraft will lose their farside vantage. There will occur significant periods from thence during which electromagnetic coverage of the Sun's far hemisphere will be incomplete or nil. Solar seismology will make it possible to continue our monitor of large active regions in the Sun's far hemisphere for the needs of space weather forecasters during these otherwise blind periods.


Introduction
Magnetic regions in the Sun's outer atmosphere exert major impacts on space weather in the near-Earth environment. Near-term impacts on Earth are predominantly due to magnetic regions in the Sun's near hemisphere. However, because the Sun rotates, magnetic regions in the far hemisphere cross into the near hemisphere with little warning-except for our ability to monitor the Sun's far hemisphere much as we do the near hemisphere. For the past several years NASA's twin STEREO spacecraft have been in positions to view the entirety of the far hemisphere from the far side of Earth's orbit about the Sun. The STEREO spacecraft are gradually drifting back to Earth's side of our orbit about the Sun and in a few years will lose their farside vantage. There will occur significant periods from thence during which electromagnetic coverage of the Sun's far hemisphere will be incomplete or nil.
Our helioseismic observatories offer an attractive resource for the continued monitor of the Sun's far hemisphere in periods during which electromagnetic observations of the Sun's far hemisphere will be unavailable or incomplete. We have developed algorithms capable of analyzing seismic observations of the Sun's near hemisphere to map large active regions in the Sun's far hemisphere. This offers a promising resource for the continued monitoring of the Sun's far hemisphere for the indefinite future.
This chapter will give a brief review of the basic principles of solar seismology, the history of their conceptual and observational development, and their adaptation as an opportune spin-off for synoptic mapping of active regions in the Sun's far hemisphere that can serve the practical needs of space weather forecasting. We will proceed then to illustrate how seismic monitoring of the Sun's far hemisphere works in practice. We will end the chapter with a summary of some of the major practical applications of far-hemispheric monitoring in space weather forecasting. Unlike electromagnetic radiation in a vacuum, sound waves in the solar interior are continuously refracted, following curve paths that return them to the surface in distance usually much shorter than that of the straight-line trajectory tangent of the path at the source, S. Waves with periods around 5 min are specularly reflected at the surface. They continue for multiple skips beneath the surface before they eventually lose their coherence or decay.

Basic Principles of Farside Solar Seismology
The ability to monitor the Sun's far hemisphere by analyzing observations of seismic oscillations over its near hemisphere is a classical spin-off from a broad field of solar research called helioseismology. The acoustic waves of which the solar oscillations are a surface signature are understood to be excited almost entirely by convective motion in a narrow layer comprising just the outer few hundreds of kilometers of the Sun's convection zone. (The convection zone is understood to extend from about what we identify as the Sun's surface at its outer extreme, to a depth of about 0.3 solar radius, i.e., ∼210,000 km, beneath said surface.) The particular p modes of use for monitoring the Sun's far hemisphere emanate downward from the outer convection zone, penetrating deep into the Sun's interior. But, unlike electromagnetic waves in a vacuum, acoustic waves in the solar interior are subject to strong refraction. They generally travel along upwardly curving "ray paths," (see Figure 1) most of them returning to the surface in less than a radian along the solar surface. Upon reaching the surface, they undergo a specular reflection back into the Sun's interior, retaining most of their coherence. From thence, they adhere closely to a continued ray path identical to the previous one, beginning anew from the point of reflection, proceeding the same direction as before beneath the great circle along the Sun's surface connecting the origin with the reflection point. By multiple such specular reflections, a vast swarm of p-mode waves excited by convection travel beneath a myriad of solar circumferences, reverberating between the Sun's near and far hemispheres multiple times before they eventually lose their coherence and decay.
The p modes obey basic principles very similar to those obeyed by electromagnetic waves, such as those we recognize as visible light. We are no better familiar with these principles than by virtue of the great variety optical techniques we have applied to visible light in our development of remote sensing. Our optical devices, typically based upon one or more lenses, sample electromagnetic disturbances in a given region, , the "pupil" of the device, over some surface, , containing , to image the source distribution of the seismic radiation (i.e., the p modes) understood to have brought these disturbances to  in  from a considerable distance. We can similarly use the solar oscillations observed at the Sun's surface to image the source distributions of said seismic radiation, again over domains a considerable distance from that over which the oscillations are observed.
With this background, we can state formally the basic undergoings that make seismic monitoring of the Sun's far hemisphere possible.
1. Convection in the extreme upper convection zone produces acoustic waves (p modes) with periods of several minutes that penetrate deep into the Sun's interior. 2. The p modes obey principles highly analogous to those from which electromagnetic optics benefit. 3. The p modes communicate between opposite hemispheres of the Sun, retaining their coherence from one to the other. 4. The p modes interact strongly with active regions when they encounter them at the Sun's surface and carry the signature of that interaction with them as they reflect back into the solar interior, retaining it from one hemisphere to the other.
10.1002/2016SW001547 5. The p modes make a clearly observable seismic signature where they encounter the Sun's surface, identified as "the solar oscillations." The component of this oscillatory signature in the Sun's near hemisphere is mapped in spatial and temporal detail by our helioseismic observatories.
The major difference between the familiar electromagnetic optics we use in the terrestrial environment and those applied to the signatures of acoustic waves observed at the Sun's surface is that the latter are essentially computational. Acoustic lenses are routinely used to image the sources of acoustic waves in the ocean, for example, just as electromagnetic lenses are used to image the sources of visible light in air or in a vacuum. It is not yet practical to submerge acoustic lenses into the Sun's interior. Nor can we submerge acoustically sensitive surfaces to map the acoustic images projected by any such optics. What we can rather do is to apply the disturbances we infer from the solar oscillations on the Sun's near surface to the surface of an appropriate acoustic model of the solar interior and computationally simulate the waves in the model interior to be associated therewith. And we can sample acoustic disturbances over any surface at any depth in a computer model of the solar interior to an extent we have not the technology to do in the Sun itself.
A great deal of our ability to monitor the Sun's far hemisphere by the foregoing faculties is owed to the development of helioseismology, an area of research that began in the 1960s whose primary theme has been the use of observations of solar oscillations at the Sun's surface, and our understanding of these as a manifestation of acoustic waves that inhabit the solar interior, to infer the thermal structure of the solar interior and its flows from surface to core. We will proceed with a brief historical review of this field and describe basic properties of the acoustic waves of which it is diagnostic.

Helioseismology
What we now call the solar oscillations were discovered independently by Leighton et al. [1962] and Evans et al. [1963], by observing temporal variations in Doppler shifts of photospheric lines over the Sun's surface. These oscillations, found to have periods ranging from about 2 to 8 min, were soon recognized as the surface signature of acoustic waves in the Sun's photosphere [Noyes and Leighton, 1963]. The waves of which the solar oscillations are a signature were eventually understood to be excited by convection a few hundreds of kilometers beneath the Sun's surface [Stein et al., 2004], some of which penetrate hundreds of thousands of kilometers into the Sun's interior [Ulrich, 1970].
In applying the term "acoustic" to such waves, we must admit that the spectrum of the oscillations in question is far lower in frequency, by a factor of about 10 5 , than the familiar acoustic spectrum to which our ears are sensitive. The applicability of acoustic in spite of this rather capitalizes on the waves of which the disturbances are a signature being "compressional" in the same sense as that which we call "sound," here on Earth. That is, the predominant restoring force acting on the disturbances of which the waves are comprised is that of gas pressure, and this acts against an inertia that is entailed in the mass density of the medium.
Ulrich [1970] introduced the idea of using the solar oscillations as a diagnostic of the Sun's interior structure. Deubner [1975] and Rhodes et al. [1977] developed this at length, as a diagnostic of both the thermal structure of the solar interior and of the dependence of differential rotation with depth. In the late 1980s and early 1990s, solar seismology began to develop along two significantly separate theoretical and observational lines, distinguishing between "global" and "local" helioseismology. Global helioseismology views the Sun as a collection of harmonic oscillators, capitalizing on highly accurate measurements of the frequencies of its myriad of normal modes to develop a model of its overall thermal structure. The normal mode frequencies also give us a credible model of how the solar interior rotates. Global diagnostics give us sharp discrimination in depth and some in latitude but none in longitude. It has given us maps of the solar interior rotation rate [Rhodes et al., 1979], affirming that the Sun is a differentially rotating fluid. We understand that the equatorial interior rotates significantly faster than the poles, as it does at the surface, down to the base of the convection zone, that the region of fastest rotation is about 10,000 km beneath the equatorial photosphere, and beneath that the rotational angular velocity decreases. Global helioseismology also tells us the thermal structure of the solar interior [Christensen-Dalsgaard et al., 1993], identifying the base of the convection zone at a radius of ∼0.7R ⊙ from Sun center. It also reveals a reversal of the sound speed gradient in the Sun's core, implying a high concentration of helium therein.
What is recognized as "local helioseismology" looks at the foregoing acoustic waves more from the perspective of optics, to focus on relatively compact anomalies in an otherwise nominal medium LINDSEY AND BRAUN SEISMIC IMAGING OF THE SUN'S FARSIDE 763 Figure 2. Wavefronts, represented here as expanding turquoise rings, emanate from seismic sources embedded in a model of the solar interior. Where these wavefronts break to the surface, they manifest outwardly propagating surface ripples beginning directly above respective sources and propagating outward along the surface therefrom. Curved trajectories terminated by arrow heads represent segments of the ray paths that characterize refraction in the solar medium (see Figure 1). Reproduced from Lindsey et al. [2011], courtesy of InTech (http://www.intechopen.com). [Braun et al., 1988;Braun and Birch, 2008]. It is this local perspective of solar seismology under which the ability to seismically image active regions in the far hemisphere has been developed [Lindsey and Braun, 1990;Braun et al., 1992]. Active regions represent relatively localized anomalies in a solar environment that is otherwise highly uniform over any given layer in radial distance, i.e., very close to spherically symmetric.

Helioseismic Holography
The idea of adapting basic principles of wave optics to helioseismic observations as described in section 2 is called computational helioseismic holography. The basic concept was introduced by Roddier [1975] as a suggested means of imaging acoustic sources submerged beneath the solar photosphere. Lindsey and Braun [1990] proposed this as a means of imaging sunspots in the far photosphere, i.e., viewing them acoustically through the solar interior from the surface of the Sun's near hemisphere. Further developments are described by Braun et al. [1992], Lindsey et al. [1996], and Lindsey and Braun [1997] (relating seismic holography to the more recently introduced "time-distance helioseismology" of Duvall et al. [1993] and Duvall et al. [1996]) and further by Chang et al. [1997], Braun et al. [1998], Chou [2000], Lindsey [2000a, 2000b], Braun and Lindsey [2001], Lindsey and Braun [2000b], Skartlien [2001Skartlien [ , 2002, and others.
Helioseismic holography is intended to take advantage of the fine local discrimination given to us by diffraction-limited optics. The local discrimination our eyesight gives us in the electromagnetic spectrum is a benefit of the preservation of phase coherence [Zernike, 1938;van Cittert, 1939] as light travels from its source to our eyes. A similar benefit is found in acoustic waves traveling through the solar interior and reflecting from the Sun's surface.
In its general form, helioseismic holography can be thought of as a development emerging from the following conceptual exercise in coherent wave optics [see Lindsey and Braun, 2000b]: We imagine an idealized acoustic model of the solar surface and its underlying interior with a few monopolar acoustic sources embedded therein. This is illustrated in Figure 2. Acoustic radiation emanating from these two sources is expressed in terms of an acoustic field, , disturbances in which are represented as ring-like (in a planar slice, bubble like in three dimensions) wavefronts that expand outward, hence both upward and downward as time proceeds. Segmental samples of the upwardly curving ray paths (which are perpendicular to the wavefronts) are included in Figure 2, terminated by arrow heads to indicate the upwardly curving direction into which the wavefronts are propagating. The only observable manifestation of (r, t) is the disturbances that appear at the surface,  0 . These disturbances first appear at points on  0 directly above the sources from which they emanated and propagate from thence outward along  0 . These surface ripples traveling across  0 are the essence captured by helioseismic observations in such a scenario.
We now consider a detailed record, 0 , of the surface value of taken from the foregoing helioseismic observations over some domain, , in  0 for a duration of a few oscillatory periods. Formal computational acoustic holography, as prescribed by Lindsey et al. [2011], consists of the following composite operation: 1. Apply the surface disturbance, 0 , in some domain,  ∈  0 , in time reverse to the surface,  0 , of an acoustic model of the solar medium that itself is devoid of sources, absorbers, or other significant anomalies. 2. Allow the acoustic model to propagate the resulting disturbances backward into its interior. 3. Then, sample the regressed acoustic field in some domain within the model. . Coherent computational regression of the surface acoustic field into a computationally accessible model of the solar interior. A record, 0 , of the surface acoustic field is applied computationally to the model surface in time reverse to drive a disturbance that propagates backward into the model interior. This coherent acoustic egression, H + (r, t) can be sampled anywhere in the model interior. We call the acoustic power, |H + (r, t)| 2 , of this regressed acoustic field the "egression power" of 0 . This is averaged over time along a "sampling surface" (red) whose depth is the same as that of the leftward seismic source in Figure 3 and plotted radially beneath it (bottom of frame). Reproduced from Lindsey et al. [2011], courtesy of InTech (http://www.intechopen.com).
We call the time-reversed acoustic field, H + (r, t), in the model the "coherent acoustic egression" of 0 . H + (r, t) is a coherent reconstruction of a component of that arrived at the solar surface,  0 , with every apparent intention of egressing through it-hence the term "egression,"-from the interior of the medium, supposing that the medium provided somewhere on the other side of  0 for it to continue into-or, perhaps, the mechanism to simply absorb it. (The idealized acoustic model is supposed to register the signature we have identified as 0 , whereafter the wave that caused it then goes away-and does not come back some time later to complicate the interpretation of 0 as to its supposed source.) Figure 3 illustrates steps 2 and 3, above, for a sampling domain on a surface,  z , of constant depth, z, radially beneath  0 . In fact, Figure 3 attempts to convey diagramatically significant respects in which the regressed acoustic field differs from the actuality diagrammed in Figure 2. Approaching the source locations from above, the regressed acoustic field does not fully condense into the point sources of its counterparts in Figure 3. This is an essential consequence of the acoustic model in Figure 3 not including point absorbers to play the time reverse role of the point sources in Figure 2. Rather than disappearing back into their sources, the waves in the regressed acoustic field continue downward through their respective source layers and into the underlying model interior. This behavior is entirely similar to that which applies in familiar lense optics.
H + (r, t) nevertheless condenses into conspicuously compact kernels at the locations of the respective sources. The kernel representing the leftward source in Figure 3 is clear in a plot of |H + (r, t)| 2 along a sampling surface at the depth of said source (see left side of the plot at the bottom of Figure 3). For the rightward source, which lies deeper than the sampling surface (right side of egression-power plot), the egression-power signature remains extant but is out of focus, hence diffuse-as it would similarly be for a source above the sampling surface.
The actual degree to which H + in fact succeeds in emulating the time reverse of in practice is contingent upon several factors: These begin with (1) the limited completeness of the foregoing surface application, in terms of the limited pupil  over which 0 is applied (in time reverse) to  0 , and in the ability of the observations to fully characterize all aspects of the surface disturbance and (2) any loss in phase coherence retained by the wave as it propagates through the medium from source to pupil.
Incompleteness of the surface application has an analogy in electromagnetic optics in diffraction due to an entrance pupil that is invariably limited. This is customarily expressed in terms of smearing of the image [Born and Wolf , 1975b]. Loss of phase coherence is analogous to the quality lost by electromagnetic radiation to some degree when it encounters optical anomalies, such as in atmospheric turbulence or clouds [Born and Wolf , 1975a]. This appears to be negligible for acoustic waves at depths exceeding a few hundreds of kilometers into the Sun's interior. In the acoustic context, the vast bulk of the solar interior appears to be of a remarkably high optical quality.

Seismic Holography Applied to the Sun's Far Hemisphere
Geometrically speaking, seismic holography of the Sun's far hemisphere can be considered a straightforward extension of the simple examples illustrated by Figures 2 and 3 if the sampling surface in Figure 3 is distended first downward, past the center of the Sun, and from thence all the way to the surface of the far hemisphere.
However, when this is done, the upward curvature of the ray paths, which is somewhat subtle for the short ray path segments in Figures 2 and 3, becomes important in ways we have yet to confront. For mapping the Sun's far hemisphere, curvature of the ray paths becomes critical, as does the curvature inherent in any spherically symmetric medium.

Acoustic Refraction
Curvature of ray paths due to refraction (see Figure 1) is fairly familiar on Earth in mirages seen above a flat expanse such as the Mojave Desert in June, due to excess heating of air a few feet above the sand. Said heating reduces the density of the medium and its refractive index, making the speed of light through that air slightly greater than at a greater height. This causes a ray path nominally parallel to flat ground to curve upward, as prescribed by Snell's law: In the familiar instance of a vertically stratified medium in a plane-parallel geometry, Snell's law can be expressed by where signifies the angle of the tangent to the ray path, Γ, at a given point, P, from directly vertical; c signifies the speed of light in the medium at the height of P; and K is a constant that identifies the ray path under consideration. (For example, if the value of c at P is c 0 , and the ray path desired passes through P at an angle 0 from vertical, then K = sin 0 ∕c 0 all along that ray path.) When c decreases with height, sin does likewise, as does itself; hence, the ray path becomes more vertical with increasing height and therefore curves upward.
In the terrestrial atmosphere, the variation in c is not very large, and it is only because of the high acuity of our eyes that we can resolve mirages. For the species of acoustic waves in the solar interior that most readily communicate between opposing hemispheres in the solar interior the effects of refraction are anything but subtle: These waves must travel a long distance from one surface encounter to the next, penetrating deep beneath the surface between encounters. The practical model to represent this behavior replaces the flat expanse in the Mojave Desert with a spherically symmetric medium that approximates the sound speed, c, as a function of r, the radial distance of the reference point, P, from the center of the Sun.
In such a spherically symmetric medium, Snell's law transforms to where is now the angle between Γ and the local radial direction at P and r is the radial distance of P from the center of spherical symmetry, i.e., the center of the solar interior. As in the plane-parallel case, a constant sound speed prescribes a straight ray path, while a sound speed that decreases with increasing r prescribes one that curves upward. Any ray path, Γ, that is aimed downward from the Sun's surface but not directly toward the Sun's center, i.e., sin 0 > 0, curves progressively farther away from the radially downward direction with increasing depth, eventually reaching a minimum radial distance, i.e., where sin has become unity. From this point it continues its upward curvature, eventually returning to the surface. In this geometry, K all along the optical path, Γ, is fixed at its surface value, with 0 the nonzero angle of the ray path from the radial at the Sun's surface and c 0 is the speed of sound thereat.
The solar interior has a strong negative temperature gradient with increasing radial distance, r, extending from its core to its surface. The speed of sound in the Sun's core is understood to drop from about 500 km/s in its core to about 8 km/s at its surface. Figure 4 shows a sample of ray paths, and associated wavefronts, for the family of waves that propagate from a focus in the far hemisphere to points in the near hemisphere in two skips, with a specular reflection halfway in between. Consecutive wavefronts are separated by 5 min. The total one-way travel time from the source in the far hemisphere to the surface of the near hemisphere is in the range (3.5 ± 0.25) h.

The Specular Reflections and Multiple-Skip Seismology
The diagram in Figure 4 illustrates both a major difficulty imposed by refraction of waves due to the strong sound speed gradient in the solar interior and how this difficulty can be alleviated by taking advantage of the specular reflections that occur when these waves encounter the Sun's surface. The curvatures of the ray paths due to refraction are seen to range from approximately (1.0-1.5)∕R ⊙ from those with the longest skip distance to those with the shortest. This radically shortens the distance traveled by waves emanating from the surface along ray paths that begin with angles, 0 , more than a degree from vertically downward, in only a single skip. The tangents to the ray paths at their surfaces range from 0.64 ∘ for the ones that penetrate deepest (red) to 2.9 ∘ for the shallowest (turquoise). If the sound speed, c, in the solar interior were constant, all of the ray paths emanating from the focus in Figure 4 would proceed straight along said tangents from their origin at said focus to arrive in the near hemisphere within ∼6 ∘ of the antipode of the focus in the far hemisphere. Because of refraction, none of these ray paths reach the near hemisphere in a single skip. However, by continuing a single additional skip from their first surface encounters, after a specular reflection, they all arrive into the opposing hemisphere, within 75 ∘ of the antipode of the focus, which is within the practical observing range of helioseismic observations from directly over said antipode.
In point of fact, there actually is a family of waves that travels directly from the far hemisphere to the near hemisphere in a single skip. These are confined to surface tangent angles, 0 , within 0.33 ∘ of vertically downward. This narrow cone comprises a much smaller fraction of the acoustic spectrum than that which encompasses angles up to 2.9 ∘ but requires the additional skip. The additional skip, then, opens approximately (2.9∕0.33) 2 = 75 times the spatial acoustic spectrum inhabited by waves that propagate from the focus to the near hemisphere in only a single skip. This results in seismic images with proportionately greater spatial resolution, ΔA, (in area) and "statistical weight" (W) for a given integration time of the seismic signature: We will return to this consideration in sections 3.1 and 3.2.

Acoustic Scattering and Phase Correlation Seismology
Up to now, we have treated the objects of seismic monitoring as simple acoustic sources, which can be mapped simply by integrating the acoustic power, of the regressed acoustic field integrated over some period of time This is what is plotted in red at the bottom of Figure 3, and this form of the diagnostic is called "acoustic power holography." Acoustic power holography is also sensitive to acoustic absorbers, showing them in silhouette. Active regions are known to be strong absorbers of obliquely incident acoustic waves [Braun et al., 1988;Braun and Birch, 2008] as they reflect them back into the solar interior, i.e., waves with skip distances of a few tens of Mm. Inconveniently, this strong absorption appears to abate for waves approaching normal (locally vertical) incidence , i.e., those with skip distances sufficient to take them to the opposing hemisphere within two skips. At this writing, the details of this behavior are only well enough understood for us to say that two-skip egression-power mapping of the Sun's far hemisphere is inadequate to realistically reveal active regions that are clearly visible by other means. Their signatures in egression power are like that of a white cat in a white room. (We are indepted to former summer student Mark Fagan for this apt analogy.) They are invisible.
Active regions are, nevertheless, strong scatterers of acoustic waves reflecting back into the solar interior from their photospheres, whether the incidence is normal or oblique. Their primary effect of benefit to seismic monitoring of the Sun's far hemisphere is to locally shift the phase of the reflection by a fraction of a radian, with respect to that of the quiet photosphere. This shift expedites the arrival of the echo back into the near hemisphere by a few seconds. A convenient physical model for this effect supposes that the active region photosphere is physically depressed some tens of kilometers by magnetic forces so that an upcoming wave encounters the reflecting layer a few seconds before its counterpart impinging upward into the quiet Sun. This "acoustic Wilson depression" appears to be an extension of the classical Wilson depression observed in sunspot umbrae [Lindsey et al., 2010]. The standard diagnostic of the phase shift attached to such a perturbation involves a comparison between acoustic radiation impinging into the focus and its outcoming echo. The echo is the egression, H + , introduced above (see yellow arrows in Figure 4). The amplitude of acoustic radiation coherently converging into the focus-we call this the "coherent acoustic ingression" and denote it by "H − "-is the time-forward analogy of H + . It is the surface disturbance, 0 , in the near hemisphere propagated into the acoustic model forward in time (see green arrows in Figure 4), as opposed to the time reverse application of 0 prescribed at the surface to derive the acoustic egression. The basic phase relationship between the acoustic radiation, H − , impinging into the focus and its outcoming echo is to be found in the statistical cross correlation, between H − and H + . The phase shift between H − and H + can be expressed by integrating the Fourier transform,Ĉ, of C, over the positive half of its frequency spectrum: ) .

Farside Solar Seismology in Practice
The first seismic maps of the Sun's far hemisphere were published by Lindsey and Braun [2000a] [see also Braun and Lindsey, 2001] at the turn of the 21st century. These were computed from helioseismic observations by the Michelson-Doppler Interferometer (MDI) [Scherrer et al., 1995] [González Hernández et al., 2007].
Over the 15 years since this synoptic mapping has begun, as the quality of the helioseismic observations has improved, the seismic maps have also improved. Other complementary diagnostics for seismic mapping of active regions in the far hemisphere have also been successfully opened: Hartlep et al. [2008] and Ilonidis et al. [2009] have developed a successful seismic monitor based on helioseismic tomography to map active regions . NOAA AR 11498 is seen at (190 ∘ W, 12 ∘ S), crossing the farside meridian (top), approaching the east limb (middle), and rotating into direct view (bottom) in the near hemisphere. In these renderings, the phase shift prescribed by equation ( (7)) is expressed as a "travel time perturbation," , of the approximately 7 h round trip travel time required for acoustic waves to complete the circuit diagrammed in Figure 4 ≡ where T 0 is taken to be 5 min and it is understood that 0 = 2 ∕T 0 . The phase perturbation imposed by NOAA AR 11498 is equivalent to a reduction of up to ∼12 s enjoyed by the echo returning from AR 11498 in the ∼7 h round trip from the near hemisphere to the far hemisphere and back.

The Spatial Resolution of Farside Seismic Holography
The spatial resolution for the wave configuration in Figure 4 is most directly expressed by its electromagnetic analogy, i.e., the Abbe diffraction limit where 0 is the temporal frequency of the seismic radiation; T 0 ≡ 1∕ 0 is its temporal period, which we take to be 5 min, as it is in Figure 4; 0 is the wavelength, c 0 T 0 , approaching the surface; and we call 0 the "opening angle" of the optical configuration so represented. With 0 = 2.9 ∘ , then, Δs works out to 10 ∘ of arc over the Sun's surface, to represent the family of waves that suffer a single specular reflection to reach the near hemisphere in two skips. For the family of waves that reach the near hemisphere in only a single skip, i.e., = 0.33 ∘ , the diffraction limit explodes to 87 ∘ . This only marginally resolves the entire far hemisphere into more than a single region and is far from resolving the largest active regions. In summary, then, the facility of acoustically viewing the Sun's far hemisphere through at least one specular reflection is crucial to practical seismic monitoring of the Sun's far hemisphere out to anything approaching the antipode of disk center.

The Sensitivity of Farside Seismic Holography
Notwithstanding the adaptability of the most basic principles of optics in the electromagnetic spectrum, seismic monitoring of the Sun's far hemisphere is encumbered by fundamental limitations that make it far less sensitive than electromagnetic monitoring. This is mostly because of the vastly lower spectral bandwidth to which helioseismic observations are confined. The fundamental limitation in sensitivity is due to the limited statistical weight, W, introduced at the end of section 2.3.2, attached to a record of random noise over a limited spectral bandwidth, Δ , for a limited time, Δt. Table 1 shows a comparison of key temporal and spatial parameters contributing to W in a 1 ms electromagnetic snapshot by SDO/HMI light and a 1 day seismic integration.
For the seismic signature representing a single element of spatial resolution, we define W by Helioseismologists often call the uncertainty in a seismic signature imposed by this limit "realization noise." In the case of the phase, , of a cross correlation such as that expressed by equation ( (7)), the realization uncertainty is in radians, which translates to an uncertainty in the travel time perturbation, , of The seismic signature of NOAA AR 11498 in Figure 5 appears to inhabit an area of ∼(10 ∘ ) 2 , approximately the diffraction limit quantified in section 3.1, above. By its appearance in Figure 5, this signature has a signal-to-noise ratio of roughly 10. To get a preliminary idea of the overwhelming advantage that electromagnetic imaging has over seismic, it is instructive, though far from a complete analysis, to start with a comparison of W for visible electromagnetic radiation imaged by HMI with the same for seismic signatures. Table 1 lists appropriate electromagnetic values to accommodate this W(electromagnetic) W(seismic) = (5 × 10 7 ∕370 2 ) (1728∕120, 000 2 ) ≈ 1.7 × 10 10 ,

LINDSEY AND BRAUN SEISMIC IMAGING OF THE SUN'S FARSIDE
in which we are now including an account for the much finer spatial resolution instruments such as SDO/HMI and SDO/AIA give us in the visible electromagnetic spectrum.
In fact, realistic statistical weights for visible and UV electromagnetic intensities emanating from the Sun in the visible and UV spectra, while far greater than for seismic images, are nothing like those for the phase shifts that characterize seismic realization noise. This is, firstly, because photons are quantized. The quantum-electromagnetic analogy to equation ( (11)) is where ΔI represents the photon-statistical noise in the intensity, I , and now represents simply the expectation number of photons emanating from a black body of temperature T during the interval dt into the solid angle spanned by the imager's angular resolution, as prescribed by its aperture. Already in the visible spectrum, the factor 2∕(e h ∕kT − 1) is only 0.03.
In the EUV spectrum, which is where solar activity is by far the most conspicuous-and is, accordingly, what really matters to space weather-it is most convenient to express signal-to-noise ratio in terms of simply the number of photons, N , detected as follows: In fact, electromagnetic observations are encumbered by noise sources that well exceed even those prescribed by photon statistics, as a result of fine structure variations in the magnetic flux distribution in the quiet Sun. We nevertheless find that the mean EUV intensities of strong active regions averaged are typically a few hundred times their mean RMS variation in the quiet Sun.

Comparative Applications in the Sun's Near Hemisphere
A great deal of our understanding of the farside seismic maps comes from comparative seismic imaging of active regions in the Sun's near hemisphere. These mappings benefit from shorter skip distances, hence greater opening angles, 0 , giving us much finer spatial resolution, and can therefore discriminate the seismic signatures of different parts of an active region. Figure 6 shows seismic, visible light (continuum intensity and line-of-sight magnetic field) and EUV images of NOAA AR 11158 in the near hemisphere. One of the remarkable findings shown by nearside images, long before the first farside images were accomplished, was that, notwithstanding the far stronger magnetic fields of sunspots, the predominant component of the seismic signature of an active region is manifested invariably by the plage surrounding the sunspots, not the sunspots themselves. Our rough understanding of this is that the seismic signature can be roughly thought of as a measure of the photospheric depression of the active region due to Lorentz forces. While a strong magnetic field produces a deeper photospheric depression, the depth of the depression rapidly saturates with increasing magnetic pressure (B 2 ∕(8 )), because of an approximately exponential increase in opposing gas pressure with depth. So at least in terms of acoustic indications, the sunspot umbra is depressed not so very much further than the magnetically weaker plage. The latter, then, predominates the overall acoustic signature simply because it inhabits many times the area of the sunspots. Indeed, it was recognition of this by Braun and Lindsey [2000b] that made the strong case to Lindsey and Braun [2000a] that active regions in the Sun's far hemisphere ought to be clearly visible in seismic maps of the far hemisphere while prospects for detecting just sunspots were marginal.
Another feature that is evident from Figure 6 is that seismic signatures of active regions tend to reflect the magnetic morphology of the active region much more closely than the source distribution of excess UV radiation emanating from the active region chromosphere and transition region. This is roughly consistent with the model, briefly mentioned above, that even plages are magnetically depressed, as least in terms of the depth at which upcoming acoustic waves are effectively reflected back into the solar interior. This is certain to be a radical oversimplification of a reality that involves the subtleties of coupling between fast and slow modes [Cally, 2000;Cally and Bogdan, 1997;Schunker et al., 2008;Spruit and Bogdan, 1992;Lindsey et al., 2010]. This could someday be clarified by stereo observations of plages in visible light from spaceborne platforms with differing vantages in heliocentric orbit. In the mean time, the simple model of a magnetically depressed plage photosphere appears to be useful for at least some purposes.
Greater seismic sensitivity can be attained, if at the expense of temporal resolution, by averaging the helioseismic signature over several days. For a considerable domain of purposes, the sensitivity gained by integrating the seismic signature for 5 days more than makes up for the loss in temporal resolution. Five day cumulations of 24 h integrations increase the number of active regions that can be clearly recognized transiting the far hemisphere from a few dozen in a solar cycle to several hundreds. This radical increase is because of the demographics of active region strengths rather than because of some kind of statistical magic. It is simply because the number of active regions exceeding a given magnetic flux, Φ, for example, increases so rapidly as Φ decreases.
But there is another feature of solar activity that further justifies the sacrifice of temporal resolution: If magnetic flux that has emerged into the photosphere could disappear anything like as suddenly as it can emerge, then the 5 day cumulations would be open to the liability of showing a signature when none was extant. In fact, this happens rarely or never. Once a large intrusion of magnetic flux has emerged into the photosphere, however, suddenly, it does not appear to be in the general character of solar magneto-convection then to retract it, eject it, nor otherwise to disappear it except through predominantly diffusive processes that operate over a period of days or weeks [Wang et al., 1991[Wang et al., , 1994. Moreover, it appears that, to the moderately trained eye, the 5 day cumulations are not very much less sensitive to rapidly emerging flux than the individual 1 day integrations. This is consistent with the consideration that the 1 day "snapshots" for any day can in fact be algebraically recovered from the series of 5 day cumulations.
In summary, then, for space weather forecasting purposes, the 5 day cumulations are very useful, notwithstanding a formal sacrifice in temporal resolution. These, then, will be the focus of the next section.

Applications of Farside Solar Seismology in Space Weather Research and Forecasting
Applications of monitoring of the Sun's far hemisphere to space weather research and forecasting are already many and appear to be rapidly growing. Before we elaborate on these we note that monitoring of the Sun's far hemisphere, while useful for the task of forecasting the appearance of active regions in the Sun's near hemisphere, is not, of itself, a forecasting facility. It remotely senses active regions in the far hemisphere at the time when the waves whose signatures it subsequently observes encounter them in the far hemisphere. So like other familiar sensing instruments (thermometers, barometers, wind vanes, etc.), it renders characterizations of its subjects in the recent past, not the future. Forecasting is something that remains to be done when it has delivered its product, and the success of this is contingent upon the ability of the subject to change unpredictably in the interim. With this in mind, we briefly describe two promising aspects of forecasting applications of seismic monitoring of the Sun's far hemisphere presently under development.

The Relationship Between Seismic Signatures and UV Irradiance
Generally, the earliest known space weather manifestation to be anticipated by a strong helioseismic signature in the Sun's far hemisphere is the effect that the associated active region will have on UV irradiance at Earth. Realistic forecasting of UV and EUV irradiance on Earth is crucial to projecting conditions in the Earth's ionosphere, thermosphere, and exosphere. Excess UV and EUV emission from the Sun inflates the Earth's exosphere, increasing drag on spacecraft-and space debris-in low Earth orbit by up to about an order of magnitude and causing their orbital elements to drift accordingly. Forecasting variations in the solar UV and EUV irradiance is therefore useful for projections of the orbital elements of spacecraft and tens of thousands of individual items of space debris through periods during which observations of these are forestalled by poor weather. We understand, for example, that NASA moves the International Space Station approximately once per month on the average to avoid a prospective collision with one or more pieces of space junk. In fact, Figure 8 somewhat understates the prospective urgency to be aware of active regions such as those shown in Figure 7: Because excess EUV emission emanates largely from heated gas that is relatively optically thin, it tends to appear bright from Earth vantage very soon after it crosses the eastern limb, i.e., long before it becomes likely that a coronal mass ejection (CME) from the region or ejecta from a flare is likely to hit Earth. Without the farside monitor, regions like FS-101 and 103 can gestate unseen for up to 2 weeks before appearing unexpectedly at the eastern limb just hours before giving rise to a sudden, unwelcome boost to the EUV irradiance at Earth. In summary, the farside monitor can extend our warning of large, fully developed active region approaching the Sun's eastern limb from about a day to possibly nearly 2 weeks, depending on when and where in the far hemisphere the newly emerged flux breaks the surface. A study by Fontenla et al. [2009]  confirms a prompt improvement in a solar EUV irradiance forecast by including consideration of the seismic signature of newly emerged magnetic flux in the Sun's far hemisphere. This has yet to be implemented on a synoptic basis but is encouraging.
Work by Liewer et al. [2012Liewer et al. [ , 2014 to calibrate seismic signatures with their EUV intensities has benefited from STEREO observations of the far hemisphere concurrent with the seismic maps. In a study of some 22 seismic signatures recognized by Stanford's Large Active Region Discriminator in early 2011 and early 2012, they found a high spatial correlation of these with regions of excess EUV (He II 304 Å) intensity, with no false alarms. However, they also identified a similar number of regions with conspicuous, if on-the-average somewhat weaker, EUV excesses that elicited no seismic signature at all or one that was insignificant. This has to be expected

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at some point for a seismic monitor that is not nearly as sensitive as the observations in electromagnetic radiation against which they are compared (see section 3.2).
To complicate matters, there are instances of seismic signatures whose strengths are similar but whose EUV excess varied by a factor of up to about 3. In fact, very similar disparities characterizes X-ray intensities of active regions in the near hemisphere with similar magnetic fluxes [Fisher et al., 1998;Falconer et al., 1997] and are therefore rather expected of seismic signatures. These disparities appear to have a strong relationship to details of the magnetic configuration, such as magnetic shear across neutral lines [Falconer et al., 1997], which begs not only for the ability to discriminate magnetic polarity to identify neutral lines but also for discrimination of the horizontal component of the magnetic field and its directivity in order to assess its shear. (Note, for example, the strong enhancement of He II 304 Å emission along the major neutral line of the bipolar middle lobe of NOAA AR 11158, comparing Figures 6 (bottom right) and 6 (top right). Extrapolating the heating cited by Falconer et al. [1997] from the corona to the transition region would suggest that the heating that gives rise to the strong He II 304 Å emission along this neutral line is contingent upon the degree of magnetic shearing across it.) These are qualities of active regions which seismic signatures by themselves cannot directly discriminate and are the subject of further discussion in the next section.

Relationship Between Seismic Signatures and Magnetic Flux Distributions
We think we understand the relationship seismic signatures have with magnetic flux distributions better than that which they have with the distributions of excess UV and EUV flux. The former have been modeled with some degree of success [e.g., Lindsey et al., 2010] in terms of the active region photosphere being depressed by magnetic pressure. As discussed in the previous section, the latter depends upon heating of magnetic chromospheres, transition regions and coronae, the physics of which is poorly understood, and whose morphology appears to be more complex, as discussed in the previous section. Like the seismic signatures, the magnetic forces thought to cause Wilson depressions in sunspot umbrae are independent of the sign of the polarity. We think this explains the relative similarity in the morphology of the seismic signature in Figure 6 (top left) to that of the HMI line-of-sight magnetogram in Figure 6 (top right).
Initial efforts to calibrate seismic signatures of active regions with their magnetic fluxes [González Hernández et al., 2007] showed a positive but poor correlation between the two. Control comparisons between magnetic configurations of active regions before and after farside transit [González Hernández et al., 2007] suggest that this was a result of considerable evolution in magnetic structure in the week or so separating the seismic observations in the far hemisphere and the magnetic observations in the near hemisphere. This has motivated a study that takes advantage of seismic signatures of active regions in the Sun's near hemisphere such as those shown in Figure 6 to clarify this relationship, wherein the magnetic maps are concurrent with the seismic maps. This study is in its beginning stages.
Notwithstanding the quantitative variation in magnetic flux that can take place between an active region's tenure in the far hemisphere and its appearance in the near hemisphere, the recognition of a strong seismic signatures in the far hemisphere is found to be a highly reliable indication of simply the existence of a significant concentration of magnetic flux at its location when it crosses into direct view from Earth [Liewer et al., 2012[Liewer et al., , 2014, usually one that is subsequently recognized as an active region by NOAA and designated accordingly. Based upon this, the Stanford Farside Seismic Monitor includes a "Large Active Region Discriminator" that recognizes a class of seismic signatures, the approximately 400 strongest signatures during solar activity cycle 24. These signatures are found to anticipate the appearance and location of significant magnetic flux concentrations in the near hemisphere with (97.5 ± 0.5)% reliability. Maps and tables of these are posted and archived at http://jsoc.stanford.-edu/data/farside/AR_Maps and http://jsoc.stanford.edu/data/ farside-/AR_Lists, respectively.
One aspect of the connection between seismic signatures and magnetic flux distributions is the relationship between strong magnetic regions and the possibility of a flare or CME. From the moment at which an active region appears on the Sun's eastern limb, we typically have several days before the densest component of a CME emanating from the region is likely to hit the Earth. So knowledge of conditions in the far hemisphere is not nearly as urgent from the standpoint of flares and CMEs as it is for forecasting UV and EUV irradiance.
Monitoring of the far hemisphere, however, can apparently be prospectively useful for forecasting the solar wind, including high-speed streams, which emanate from coronal holes. Nick Arge and Carl Henney, at AFRL in Albuquerque, with Gordon McDonald, at New Mexico State University in Las Cruces, are developing analytical techniques for modeling the coronal magnetic field globally using both nearside and farside maps [see Arge et al., 2013]. The nearside maps show magnetic flux distributions over the near photosphere, including polarity, and their models remember and evolve this distribution appropriately as active regions in the near hemisphere cross into the far hemisphere. The role of the farside signatures is to apprise the modeling algorithm of magnetic flux that is newly emerged in the far hemisphere. The problem for realistic modeling of coronal fields is the need to know the sign of the polarity of the flux. Seismic signatures are insensitive to this (see Figure 6 (top left) and Figure 9 (bottom left)). However, the Hale polarity law offers a promising resource by which this may be guessed with sufficient dependability to give us more realistic models of the coronal magnetic field than would follow from the supposition that there was no significant new magnetic flux where a new strong seismic signature has emerged.

Direct Space Weather Effects of Activity in the Far Hemisphere
In section 1, we pointed out that near-term impacts of space weather on Earth are predominantly due to magnetic regions in the Sun's near hemisphere, meaning that while the impact of active regions in the far hemisphere could eventually be major, this was not to be expected until they crossed into the near hemisphere-which a large one invariably would, hence its relevance to forecasting. There are, nevertheless, instances of active regions impacting space weather on Earth while still in the far hemisphere to a degree that is significant in some human contexts. CMEs launched from active regions in the Sun's far hemisphere have been known to accelerate protons to high energies and inject them into magnetic streamlines that bring them to Earth [Dresing et al., 2012;Cohen et al., 2014;Wiedenbeck et al., 2013;Richardson et al., 2012]. An especially conspicuous such event was a massive halo CME on 15-16 August 2001, which showered Earth with protons having energies of hundreds of MeV. Figure 10 shows some of the observations we have of this event from SOHO and NOAA's Geostationary Operational Environmental Satellites (GOES; see http://www.polarlicht-vorher-sage.de-/goes_archive). Figure 10c shows a snapshot of the CME by the Large Angle Spectrometric Coronagraph (LASCO) aboard SOHO. Figure 10a plots fluxes of high-energy protons appearing minutes after the top of the CME first cleared the LASCO occulting disk. Figure 10b plots (a) Fluxes of high-energy protons, which began to arrive minutes after first appearance of the CME to SOHO. (b) Excess X-ray flux from the SOHO vantage is essentially nil, because the CME emanated (we think) from the Sun's far hemisphere; hence, any X-rays from the active region were radiated into the far side of the solar system. concurrent fluxes of soft X-rays from the Sun. The lack of significant X-ray emission from this event (Figure 10b) is understood to indicate that the CME emanated from the Sun's far hemisphere. The active region from which the CME errupted must in fact have released intense X-rays. However, X-rays travel in straight lines from their sources, and so we understand that these must have been radiated into the far side of the solar system, hence their invisibility from Earth vantage. Because protons are charged particles, their trajectories, in the magnetic environment of the heliosphere, can (indeed, generally do) deviate radically from straight lines emanating from the near neighborhood of the active region from which the CME errupted [Laitinen et al., 2016]. Figure 10d shows the seismic map of the far hemisphere for the 24 h period centered on 15 August 2001 computed from SOHO MDI observations. It shows the signature of a newly emerged active region appearing south (below) and somewhat east (left) of farside disk center, on its way to the Sun's eastern (left) limb from behind. This appears to be the likely source region of the 15 August 2001 CME. NOAA designated this AR 09591 soon after it subsequently crossed the eastern limb and became directly visible from Earth.
In fact, the proton storm from the 15 August 2001 CME happened to be of significant concern to the crew of the International Space Station (ISS), members of which were undertaking an extravehicular activity (EVA) 10.1002/2016SW001547 when it began. Space weather impacts from the far hemisphere such as these, though rare, get to be more than annoying when they show up uninvited during an EVA in Earth orbit. This suggests that planning of EVAs could benefit from a knowledge of exceptionally large active regions even in the Sun's far hemisphere.

The Future of Farside Solar Seismology
It should be understood that seismic monitoring of the Sun's far hemisphere is encumbered by stringent technical limitations, familiarly expressed in terms such as "diffraction" and "realization noise," that render it far inferior to electromagnetic imaging in the visible and UV electromagnetic spectra. The signal-to-noise ratio in an image taken by a spaceborne EUV camera in a 1 s exposure exceeds that of an acoustic image integrated for a full day by 2 to 3 orders of magnitude. Nevertheless, the seismic monitors presently mapping the Sun's far hemisphere will reliably detect and identify something approaching 400 active regions transiting the Sun's far hemisphere even in a relatively weak solar cycle. These active regions account for most of the excess UV and EUV emission from the Sun in the years of solar maximum and are usually the ones of predominant concern for space weather forecasting.
For the past several years NASA's twin STEREO spacecraft have been in positions to view the entirety of the far hemisphere from the far side of Earth's orbit about the Sun, making helioseismic mapping of the Sun's far hemisphere unnecessary. After about 2019, STEREO coverage of the Sun's far hemisphere will begin to diminish as these spacecraft drift back to Earth's side of our orbit about the Sun, eventually to become practically nil. And the continuation of STEREO observations in the following decade, when these spacecraft recover their farside vantage, is far from guaranteed.
We are now approaching significant periods during which seismic monitoring will be our only means of detecting and accurately locating newly emerged magnetic flux in the Sun's far hemisphere more than a few hours before it appears on the Sun's eastern limb. The Solar Wind Anisotropies (SWAN) Experiment aboard SOHO can detect the presence of active regions in the Sun's far hemisphere by backscattering of Ly-radiation from the far side of the solar system. It has not nearly the spatial discrimination of helioseismic diagnostics, but that gives us a good measure of the total EUV brightness emanating from the far hemisphere [Bertaux et al., 2000]. Solar Orbiter, scheduled at this writing to be launched in October of 2018, will orbit the Sun with a period at first of about a year and eventually of about 150 days. It will carry a "Polarimetric and Helioseismic Imager" and have extensive coverage of the Sun's far hemisphere during about half of its orbit for about the succeeding decade. There will be significant periods during which Solar Orbiter and STEREO combined will continue to fully cover the Sun's far hemisphere. There will nevertheless be significant periods in which direct electromagnetic coverage of the Sun's far hemisphere will be next to nil.
On the other hand, Solar Orbiter will be useful for calibrating seismic signatures of active regions in the Sun's far hemisphere made from Earth vantage with concurrently observed magnetic qualities of the same, very much like STEREO that is presently allowing us to do with EUV intensities.
One naturally hopes that human society will eventually develop in such a direction that plain electromagnetic monitoring of the Sun's entire surface-of both near and far hemispheres-will be constantly maintained by spaceborne instruments spread over multiple vantages. This will always tell us much more about solar activity than will ever be possible by just seismic monitoring from near-Earth vantage. The benefits of doing this in a world of billions would already far exceed its expense today. It nevertheless remains attractive to think that seismic monitoring of the Sun's far hemisphere from the near-Earth neighborhood, either from ground-based or spaceborne seismic observatories, will continue for centuries. Seismic monitoring, taking advantage of helioseismic observatories of the future, can be readily available as a backup when other resources fail, and solar observations from Earth or Earth orbit will continue to have a high economy relative to observations from heliocentric orbit. And, necessary or not, seismic monitoring of the Sun's far hemisphere is, plainly speaking, just a terrific load of fun, an ardently stimulating and educational exercise for any student who has reached an age to begin to understand solar activity, solar rotation, how these come together to make space weather what it is for Earth or any other planet, and the role these phenomena have come to play in modern technological civilization during the twentieth century and since. We think that seismic monitoring of the Sun's far hemisphere will enjoy a long future in our relationship with the heliosphere and its crucial role in space weather, and that centuries hence its history in space weather forecasting and research will be looked back upon with distinct pleasure.