Author name code: woodard ADS astronomy entries on 2022-09-14 author:"Woodard, Martin" ------------------------------------------------------------------------ Title: Towards realistic estimates of solar global oscillation mode-coupling measurement noise Authors: Woodard, M. F. Bibcode: 2021MNRAS.505.3433W Altcode: Time series of the medium-l spherical-harmonic decomposition of SOHO/MDI Doppler images were used to investigate the noise of solar-oscillation mode-coupling measurements and to reveal the coupling signatures of global-scale Rossby-waves and magnetic activity. A theoretical model of mode-coupling noise was developed starting from the assumption that the Doppler oscillation signal obeys Gaussian statistics. The measured coupling noise was found to agree with the model noise at the level of 20 per cent. The noise of mode-coupling measurements obtained from MDI data turns out to be considerably larger than that of hypothetical, 'ideal' measurements, which could only be obtained from observations with more coverage of the Sun's surface than current instruments provide. The noise analysis was carried out for a simple suboptimal mode-coupling estimation procedure which is described in some detail. A more rigorous, maximum-likelihood, approach to mode-coupling measurement, which generalizes a method currently used to extract global oscillation mode information, is also described and its relationship to the simple mode-coupling analysis is discussed. Title: Inferring Solar Differential Rotation through Normal-mode Coupling Using Bayesian Statistics Authors: Kashyap, Samarth G.; Das, Srijan Bharati; Hanasoge, Shravan M.; Woodard, Martin F.; Tromp, Jeroen Bibcode: 2021ApJS..253...47K Altcode: 2021arXiv210108933K Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global-scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via "a-coefficients." However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analyzing eigenfunction corrections, which respect cross-coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields a-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer a-coefficients due to differential rotation from cross-spectra for both f-modes and p-modes. We demonstrate that this technique works reasonably well for modes with angular degrees ℓ = 50-291. The inferred a3-coefficients are found to be within 1 nHz of the frequency-splitting values for ℓ > 200. We also show that the technique fails at ℓ < 50 owing to the insensitivity of the measurement to the perturbation. These results serve to further establish mode-coupling as an important helioseismic technique with which to infer internal structure and dynamics, both axisymmetric (e.g., meridional circulation) and non-axisymmetric perturbations. Title: Sensitivity of helioseismic measurements of normal-mode coupling to flows and sound-speed perturbations Authors: Hanasoge, Shravan M.; Woodard, Martin; Antia, H. M.; Gizon, Laurent; Sreenivasan, Katepalli R. Bibcode: 2017MNRAS.470.1404H Altcode: 2017arXiv170508204H In this article, we derive and compute the sensitivity of measurements of coupling between normal modes of oscillation in the Sun to underlying flows. The theory is based on first-born perturbation theory, and the analysis is carried out using the formalism described by Lavely & Ritzwoller (1992). Albeit tedious, we detail the derivation and compute the sensitivity of specific pairs of coupled normal modes to anomalies in the interior. Indeed, these kernels are critical for the accurate inference of convective flow amplitudes and large-scale circulations in the solar interior. We resolve some inconsistencies in the derivation of Lavely & Ritzwoller (1992) and reformulate the fluid-continuity condition. We also derive and compute sound-speed kernels, paving the way for inverting for thermal anomalies alongside flows. Title: Helioseismic Constraints on the Depth Dependence of Large-Scale Solar Convection Authors: Woodard, Martin F. Bibcode: 2017SPD....4811304W Altcode: A recent helioseismic statistical waveform analysis of subsurface flow based on a 720-day time series of SOHO/MDI Medium-l spherical-harmonic coefficients has been extended to cover a greater range of subphotospheric depths. The latest analysis provides estimates of flow-dependent oscillation-mode coupling-strength coefficients b(s,t;n,l) over the range l = 30 to 150 of mode degree (angular wavenumber) for solar p-modes in the approximate frequency range 2 to 4 mHz. The range of penetration depths of this mode set covers most of the solar convection zone. The most recent analysis measures spherical harmonic (s,t) components of the flow velocity for odd s in the angular wavenumber range 1 to 19 for t not much smaller than s at a given s. The odd-s b(s,t;n,l) coefficients are interpreted as averages over depth of the depth-dependent amplitude of one spherical-harmonic (s,t) component of the toroidal part of the flow velocity field. The depth-dependent weighting function defining the average velocity is the fractional kinetic energy density in radius of modes of the (n,l) multiplet. The b coefficients have been converted to estimates of root velocity power as a function of l0 = nu0*l/nu(n,l), which is a measure of mode penetration depth. (nu(n,l) is mode frequency and nu0 is a reference frequency equal to 3 mHz.) A comparison of the observational results with simple convection models will be presented. Title: Possible signature of solar oblateness in the Sun's oscillation frequency splittings Authors: Woodard, M. F. Bibcode: 2016AdSpR..58.1464W Altcode: Departures from spherical symmetry split the frequencies of the Sun's normal oscillation modes. In addition to the well-studied, dominant splitting of the mode frequencies, due to the first-order advection of internal wave motion, a number of second-order effects of rotation on the frequency splittings, predominantly the solar oblateness, are expected. Whereas the largest rotational frequency splittings have an odd dependence on the azimuthal order, m, of the modes, the second-order effects should have an even dependence. The biggest, and thus far the only well-studied, even-m effect on splittings, is due to the solar-cycle variations in magnetic activity near the Sun's surface, which need to be modeled with some care to bring out the signature of solar oblateness. A crude analysis of the even mode-frequency splittings, obtained from approximately 15 years of SOHO/MDI spherical-harmonic time series, was undertaken. To extract the small even-m splittings of interest from the dominant, solar-cycle effects, which have a strong mode-frequency dependence, the former were assumed to depend only weakly on mode frequency and to have no time dependence. Perhaps the most important finding of the study is that the MDI data are capable of yielding statistically significant estimates of solar oblateness. Indeed the oblateness estimates obtained from the analysis presented here appear to be roughly consistent with both theoretical expectations and with direct measurements of the oblateness. There is also a hint of a pole-equator temperature difference in the seismic measurements, at the level recently suggested by Miesch and Hindman. Title: Evidence for large-scale subsurface convection in the Sun Authors: Woodard, M. F. Bibcode: 2016MNRAS.460.3292W Altcode: 2016arXiv160506192W A helioseismic statistical waveform analysis of subsurface flow was performed on two 720-day time series of SOHO/Michelson Doppler Imager medium-l spherical-harmonic coefficients. The time series coincide with epochs of high and low solar activity. Time-dependent coupling-strength coefficients b^t_s(n,l) of modes of the same radial order n and degree l, but different azimuthal order m, were inferred from the waveform analysis. These coefficients are sensitive to flows and general aspherical structure. For odd values of s ≪ l, the coefficient b^t_s(n,l) measures an average over depth of the amplitude of one spherical-harmonic (s, t) component of the toroidal flow velocity field. The depth-dependent weighting function defining the average velocity is the fractional kinetic energy density in radius of modes of the (n, l) multiplet. A mean-square (n, l)-dependent flow velocity was inferred from the b-coefficients for s in the range 5 through 35 for each n and l in the respective ranges 1 through 5 and 120 through 149 for the epochs of high and low activity. A further averaging, over l, yielded a root-mean-square flow velocity as a function of n for each epoch, which increases from about 20 m s-1 at n = 1 to 35 m s-1 at n = 5. The inferred velocities are consistent with (though perhaps do not demand) a cellular pattern of flow extending over the vertical range of mode sensitivity, estimated to be about 4 per cent of the solar radius below the photosphere. Title: Helioseismic Evidence for Large-Scale Solar Subsurface Turbulence Authors: Woodard, Martin F. Bibcode: 2016SPD....47.0722W Altcode: A statistical waveform analysis of subsurface flow was performed on two 720-day time series of SOHO/MDI Medium-l spherical-harmonic coefficients. The time series coincide with epochs of high and low solar activity.Time-dependent coupling-strength coefficients b(s,t;n,l) of modes of the same radial order n and degree l, but different azimuthal order m, were inferred from the waveform analysis. These coefficients are sensitive to flows and general aspherical structure. For odd values of s ≪ l, the coefficient b(s,t;n,l) measures an average over depth of the amplitude of one spherical-harmonic (s,t) component of the toroidal flow velocity field. The depth-dependent weighting function defining the average velocity is the fractional kinetic energy density in radius of modes of the (n,l) multiplet. A mean-square (n,l)-dependent flow velocity was inferred from the b-coefficients for s in the range 5 through 35 for each n and l in the respective ranges 1 through 5 and 120 through 149 for the epochs of high and low activity. A further averaging, over l, yielded a root mean square flow velocity as a function of n for each epoch, which average increases from about 20 m/s at n=1 to 35 m/s at n=5. The inferred velocities are consistent with (though perhaps do not demand) a cellular pattern of flow extending over the vertical range of mode sensitivity, estimated to be a few percent of the solar radius below the photosphere. Title: Detectability of Large-Scale Solar Subsurface Flows Authors: Woodard, M. Bibcode: 2014SoPh..289.1085W Altcode: The accuracy of helioseismic measurement is limited by the stochastic nature of solar oscillations. In this article I use a Gaussian statistical model of the global seismic wave field of the Sun to investigate the noise limitations of direct-modeling analysis of convection-zone-scale flows. The theoretical analysis of noise is based on hypothetical data that cover the entire photosphere, including the portions invisible from the Earth. Noise estimates are derived for measurements of the flow-dependent couplings of global-oscillation modes and for combinations of coupling measurements that isolate vector-spherical-harmonic components of the flow velocity. For current helioseismic observations, which sample only a fraction of the photosphere, the inferred detection limits are best regarded as optimistic limits. The flow-velocity fields considered in this work are assumed to be decomposable into vector-spherical-harmonic functions of degree less than five. The problem of measuring the general velocity field is shown to be similar enough to the well-studied problem of measuring differential rotation to permit rough estimates of flow-detection thresholds to be gleaned from past helioseismic analysis. I estimate that, with existing and anticipated helioseismic datasets, large-scale flow-velocity amplitudes of a few tens of should be detectable near the base of the convection zone. Title: Global-Oscillation Eigenfunction Measurements of Solar Meridional Flow Authors: Woodard, M.; Schou, J.; Birch, A. C.; Larson, T. P. Bibcode: 2013SoPh..287..129W Altcode: 2012SoPh..tmp..179W We describe and apply a new helioseismic method for measuring solar subsurface axisymmetric meridional and zonal flow. The method is based on a theoretical model of the response of global-oscillation eigenfunctions to the flow velocity and uses cross spectra of the time-varying coefficients in the spherical-harmonic expansion of the photospheric Doppler-velocity field. Eigenfunction changes modify the leakage matrix, which describes the sensitivity of the spherical-harmonic coefficients to the global-oscillation modes. The form of the leakage matrix in turn affects the theoretically expected spherical-harmonic cross spectra. Estimates of internal meridional and zonal flow were obtained by fitting the theoretical flow-dependent cross spectra to spherical-harmonic cross spectra computed from approximately 500 days of full-disk Dopplergrams from the Helioseismic and Magnetic Imager (HMI) on the SDO spacecraft. The zonal-flow measurements, parameterized in the form of "a" coefficients, substantially agree with measurements obtained from conventional global-mode-frequency analysis. The meridional-flow estimates, in the form of depth-weighted averages of the flow velocity, are similar to estimates obtained from earlier analyses, for oscillation modes that penetrate the outermost one-third of the convection zone. For more deeply penetrating modes, the inferred flow velocity increases significantly with penetration depth, indicating the need for either a modification of the simple conveyor-belt picture of meridional flow or improvement in the cross-spectral model. Title: Using Distortion of Global Mode Eigenfunctions to Estimate Large Scale Flows Authors: Schou, J.; Woodard, M. F.; Baldner, C. S.; Larson, T. P. Bibcode: 2013enss.confE.103S Altcode: The accurate measurement of large scale flows, such as differential rotation and meridional flow, throughout the solar interior is important for understanding the solar interior and the processes relevant for the solar dynamo. Ordinarily normal modes would be expected to give more accurate measurement of large scale flows than local helioseismic techniques. Unfortunately, mode frequencies are not sensitive to the meridional flow and so traditional methods do not work. Here we describe our progress on using the distortion of the eigenfunctions to measure flows. In particular on identifying the source of the large systematic errors previously reported and determining the effect of light travel time, center to limb phase variations and other physical effects. Title: Meridional Circulation From Normal Mode Analysis Authors: Schou, Jesper; Woodard, M. F.; Larson, T. P. Bibcode: 2012AAS...22020505S Altcode: We have recently been able to make significant progress in the determining the solar meridional flow from the perturbations it causes to the eigenfunctions of normal modes. In this poster we will describe some of our recent progress and show that it will likely be possible to determine the flow over a substantial part of the solar interior with a precision sufficient

to address important questions. However, we also see what appears to be significant systematic errors and will describe our efforts at understanding those. Title: Measuring Meridional Flow Using Global Modes Authors: Schou, J.; Woodard, M. F.; Birch, A. C.; Larson, T. P. Bibcode: 2011sdmi.confE..71S Altcode: In the past the meridional flow has been determined near the photosphere by direct observations and below the solar surface using local helioseismic methods. To first order normal mode frequencies are not sensitive to the meridional flow, and so they are not useful for this purpose. However, the eigenfunctions are sensitive to the meridional flow to first order. Here we describe our progress on a project to measure the eigenfunction perturbations and infer the meridional flow with depth. Title: Helioseismic Measurement of Large-Scale Solar Flows Authors: Woodard, M. F. Bibcode: 2009ASPC..416...15W Altcode: Large-scale flows in the deep convection zone, differential rotation and meridional circulation in particular, are relatively accessible to seismic probing and provide observational constraints on convection-zone dynamics and structure. Flows and other aspherical perturbations dynamically couple the Sun's normal oscillation modes. Mode coupling produces signatures in power spectra of time series of coefficients in the spherical harmonic decomposition of the photospheric velocity field, the outstanding signature being the consequence of azimuthal-order (m) dependence of the mode frequencies due to differential rotation. Mode coupling also produces a signature in cross spectra of distinct spherical-harmonic time series. In this paper I discuss the problem of using helioseismic cross-spectra to map large-scale flow in the Sun and prospects for measuring deep meridional flow. Some preliminary estimates of meridional and zonal flow from MDI and GONG spherical-harmonic time series are presented. Title: Seismic Detection of Solar Mesogranular-Scale Flow Authors: Woodard, M. F. Bibcode: 2009ApJ...706L..62W Altcode: Helioseismic correlation data computed from the Solar and Heliospheric Observatory/Michelson Doppler Imager high-resolution Doppler images were inverted for solar flows of horizontal scale between 5 and 45 Mm. The photospheric Doppler velocity inferred from the inversions was compared with the original Doppler images at each scale. For horizontal scales greater than about 15 Mm, the seismically inferred and directly observed flow maps look similar, as has been seen in studies of supergranular flow. At smaller scales, the similarity disappears, but regression analysis reveals significant correlation between the maps, demonstrating that solar p- and f-mode oscillations contain useful information about these flows. The slope of the computed regression plots is close to unity, indicating that mesogranular-scale flows extend at least a megameter below the photosphere. Title: Large Scale Flows From Eigenfunction Fitting Authors: Schou, Jesper; Woodard, M. F.; Birch, A. C. Bibcode: 2009SPD....40.0705S Altcode: It is well known that the eigenfunctions, as well as the eigenfrequencies, of normal modes are perturbed by flows and other asphericities. In the case of meridional flows, it is known that while the perturbations to the frequencies are second order, the eigenfunctions are perturbed at first order in the strength of the meridional flow. This leads to the question of whether one might be able to measure meridional flows using the observed eigenfunction perturbations. Here we discuss how well we expect to be able to do this and show some preliminary analysis of MDI data. Title: Helioseismic Detection of Mesogranular Flow Authors: Woodard, Martin F. Bibcode: 2009SPD....40.0701W Altcode: Helioseismic correlation data computed from a sequence of SOHO/MDI high resolution Doppler images were inverted for subphotospheric flow on horizontal scales ranging from 45 Mm down to 5 Mm. The measurements are sensitive to flows over a depth range of roughly one Mm. The resulting seismically-reconstructed and directly-observed photospheric Doppler velocity maps are similar for horizontal scales less than about 15 Mm, as was seen in previous studies of supergranulation-scale flow.

For smaller scales it becomes difficult to see directly-observed Doppler features in the seismic maps. But regression analysis of the maps does reveal a component of subsurface flow which correlates with the photospheric flow. The ratio of the seismically-inferred to directly-observed Doppler velocity is typically 0.7 to 0.8, indicating a slight decrease in the flow velocity with depth in the first megameter below the photosphere. This finding is consistent with the results of the recent convection simulations of Benson, Stein, and Nordlund.

This work is supported by NASA contract NNH07CD06C. Title: Probing Supergranular Flow in the Solar Interior Authors: Woodard, M. F. Bibcode: 2007ApJ...668.1189W Altcode: Helioseismic correlation data computed from SOHO MDI high-resolution Doppler images were inverted for solar subsurface supergranular flow, using an improved forward model. The statistical error of the inversion is set by the stochastic nature of solar wave excitation, which limits the detection of supergranular flow with these data to the first 4 or 5 Mm below the photosphere. The photospheric supergranulation pattern, averaged over the ~34 hour duration of the data set, is found to persist over the detectable depth range. Contrary to some recent findings, there is no indication of a flow reversal at a depth between 5 and 6 Mm. However, since the depth of the putative reversal is at the edge of the detectable range for these data, the discrepancy is probably not significant. Title: Inferring subsurface supergranular flow directly from helioseismic correlation data Authors: Woodard, M. F. Bibcode: 2006ESASP.624E..51W Altcode: 2006soho...18E..51W No abstract at ADS Title: The Seismic Correlation Signature of Moderate-Scale Flow in the Sun Authors: Woodard, M. F. Bibcode: 2006ApJ...649.1140W Altcode: Flows in the solar convection zone introduce correlation between distinct horizontal-wavevector and frequency components of wave motion in the Sun. We derive expressions for the linear sensitivity of helioseismic correlation data to subphotospheric supergranular-scale flow, for use in helioseismic inversions. The sensitivity calculations are based on a model of randomly excited p- and f-mode oscillations in a plane-parallel solar envelope with flows. Woodard described a simple inversion of helioseismic correlation data for depth- and time-independent supergranular flow. The sensitivity calculations presented in this paper generalize previous calculations to the case of a flow velocity that depends on depth and varies slowly with time. We describe a method of estimating the data sensitivity kernel from helioseismic observations. The observed and theoretical kernels agree to within ~20%. Title: Meridional flow measurements with statistical waveform analysis Authors: Mitra-Kraev, U.; Thompson, M. J.; Woodard, M. F. Bibcode: 2006ESASP.624E..57M Altcode: 2006soho...18E..57M No abstract at ADS Title: Cross-Spectral Signatures in Global Helioseismology Data of Large-Scale Flow in the Sun Authors: Woodard, M. F. Bibcode: 2005AGUSMSP11B..12W Altcode: Large-scale flows in the Sun's interior have been studied using a variety of helioseismic techniques, including spectral analysis of spherical harmonic time series of photospheric velocity oscillations. Detailed maps of differential rotation have been obtained from measurements of the frequencies of resonance peaks in the power spectra. Flows can also affect power spectra in subtler ways, e.g., by their influence on the widths of resonance peaks. In addition to their spectral signature, flows and other aspherical perturbations also produce cross-spectral signatures, via the mode-coupling effect of a flow. Cross power spectra of time series of coefficients in the spherical-harmonic decomposition of SOHO/MDI medium-ℓ velocity images have been computed and are being compared with theoretical predictions. The results of a preliminary comparison of observed and theoretically predicted cross spectra for differential rotation and meridional circulation will be presented. A program to systematically map large-scale solar internal flow using cross-spectral data will be described. The author acknowledges useful discussions with colleagues, especially Doug Braun, Yuhong Fan, Aaron Birch, and Jesper Schou. He is also grateful to Jesper Schou for help in acquiring MDI data products and to NASA for support under contract NAS5-3114. The Solar Oscillations Investigation- Michelson Doppler Imager experiment on SOHO is supported by NASA contract NAG5-3077 at Stanford University. SOHO is a project of international cooperation between ESA and NASA. Title: Depth Dependence of Solar Supergranular Flow Authors: Woodard, M. F. Bibcode: 2005AGUSMSP11C..04W Altcode: Helioseismic inversions for subsurface supergranular motion have been performed using spectral-domain covariance estimates of the surface wave field as input data. The surface wave measurements used in this study were obtained from a 34-hr sequence of SOHO/MDI high-resolution Doppler images covering an approximately 210 Mm x 210 Mm corotating patch of the photosphere near disk center. The utilized signal, derived by filtering the Doppler sequence, is dominated by p- and f-modes in the frequency range 2.5 - 5.5 mHz and the range 600 - 1800 of angular degree. The helioseismic forward model used for the inversions is based on a wave equation which includes a stochastic driving term and a damping term. The effect of subsurface flows on wave propagation is treated in the single-scattering Born approximation. Several approaches to inverting helioseismic covariance data were used, including the method of optimally-localized averages. The inversions provide evidence for supergranular-scale motion to a detection depth of 6 Mm beneath the photosphere. The measured correlation coefficient between the surface motion and the motion at depth is observed to decrease with depth, but, contrary to some previous findings, does not change sign over the detectable depth range. The author is indebted to Yuhong Fan and Aaron Birch for extensive discussions about the theoretical aspects of wave propagation in flows and to the former for vital solar model outputs. The study was supported by NSF grant ATM-0223127. Title: Origin of irradiance variations from disk photometer data Authors: Woodard, Martin F.; Libbrecht, Ken G. Bibcode: 2003ESASP.517..117W Altcode: 2003soho...12..117W No abstract at ADS Title: Spatial and temporal variations in the solar brightness Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 2003SoPh..212...51W Altcode: We have investigated long-term variations of solar brightness as a function of both time and solar latitude using eight years of ground-based photometric data in conjunction with space-based irradiance data. In particular, we have examined whether the combination of sunspot brightness deficits and facular brightness excesses is sufficient to explain the solar cycle irradiance variations. After correcting for the contribution from sunspots, we find that the irradiance data can be adequately explained by a model in which the remaining brightness variations are due entirely to facular contributions confined to the magnetically active latitudes. Thus we find no support for the hypothesis that there are convectively driven hot bands in the active latitudes, and our data show brightness variations that are well described by a facular contrast function. Title: Inferring Inhomogeneous Structure in the Sun Directly from Correlations in the Seismic Signal Authors: Woodard, M. F. Bibcode: 2002AAS...200.8902W Altcode: 2002BAAS...34..791W A program for probing inhomogeneous subsurface structure in the Sun based on the direct modeling of correlations in the seismic signal will be outlined and some preliminary results presented. The method has been applied , for instance, to the detection of subsurface supergranular-scale flow using helioseismology datasets from the Michelson Doppler Interferometer on the SOHO satellite. Strategies for studying flow fields on both larger and smaller size and time scales will be discussed. This effort is supported by NASA under grant NAG5-8485. Title: Seismic Investigation of Solar Convection Authors: Woodard, Martin Bibcode: 2002smra.progE..26W Altcode: No abstract at ADS Title: Solar Subsurface Flow Inferred Directly from Frequency-Wavenumber Correlations in the Seismic Velocity Field Authors: Woodard, M. F. Bibcode: 2002ApJ...565..634W Altcode: We describe a general strategy for inverting helioseismic data for laterally inhomogeneous (aspherical) structure, in which the physical state of the solar interior is compared directly with inhomogeneity-induced correlations in the observable seismic wave field. The method was used to invert for subsurface supergranular-scale flow using Solar and Heliospheric Observatory Michelson Doppler Imager high spatial resolution Doppler images. Although many approximations were made in carrying out the inversion, general agreement (a correlation coefficient of 0.68) was found between the Doppler component of the seismically inferred flow in the photosphere and the directly observed surface Doppler signal. Title: Evolution of the sun's near-surface asphericities over the activity cycle Authors: Goode, P. R.; Didkovsky, L. V.; Libbrecht, K. G.; Woodard, M. F. Bibcode: 2002AdSpR..29.1889G Altcode: Solar oscillations provide the most accurate measures of cycle dependent changes in the sun, and the Solar and Heliospheric Observatory/Michelson Doppler Imager (MDI) data are the most precise of all. They give us the opportunity to address the real challenge — connecting the MDI seismic measures to observed characteristics of the dynamic sun. From inversions of the evolving MDI data, one expects to determine the nature of the evolution, through the solar cycle, of the layers just beneath the sun's surface. Such inversions require one to guess the form of the causal perturbation — usually beginning with asking whether it is thermal or magnetic. Matters here are complicated because the inversion kernels for these two are quite similar, which means that we don't have much chance of disentangling them by inversion. However, since the perturbation lies very close to the solar surface, one can use synoptic data as an outer boundary condition to fix the choice. It turns out that magnetic and thermal synoptic signals are also quite similar. Thus, the most precise measure of the surface is required. We argue that the most precise synoptic data come from the Big Bear Solar Observatory (BBSO) Solar Disk Photometer (SDP). A preliminary analysis of these data implies a magnetic origin of the cycle-dependent sub-surface perturbation. However, we still need to do a more careful removal of the facular signal to determine the true thermal signal. Title: Energy Distribution of Solar Oscillation Modes Inferred from Space-Based Measurements Authors: Woodard, M. F.; Korzennik, S. G.; Rabello-Soares, C.; Kumar, P.; Tarbell, T. D.; Acton, S. Bibcode: 2001AGUSM..SP21C05W Altcode: We have measured the energy distribution of solar p- and f-mode oscillations of angular degree and temporal frequency in the range 100<l<800 and 2<ν (mHz)<4 using helioseismology data from the SOI/MDI instrument on the SOHO satellite. At temporal frequency ν ≈ 3 mHz, the surface velocity power per oscillation mode increases slightly with angular degree between l = 100 and l=200, but decreases rather steeply with l above l=200, in approximate agreement with earlier findings from ground-based measurements. From this we infer that the time-averaged energy per mode, which is theoretically related to the modal surface velocity power, decreases steeply with l at fixed frequency, over the entire observed l range. Specifically, at ν =3.1 mHz the energy per mode drops by a factor of ≈ 10 between l=150 and l=650, a circumstance not quantitatively understood at present. This research was supported by NASA and Stanford University. Title: Energy Distribution of Solar Oscillation Modes Inferred from Space-based Measurements Authors: Woodard, M. F.; Korzennik, S. G.; Rabello-Soares, M. C.; Kumar, P.; Tarbell, T. D.; Acton, S. Bibcode: 2001ApJ...548L.103W Altcode: We have measured the energy distribution of solar p- and f-mode oscillations of angular degree and temporal frequency in the range 100<l<800 and 2<ν(mHz)<4 using helioseismology data from the Solar Oscillations Investigation-Michelson Doppler Imager instrument on the Solar and Heliospheric Observatory satellite. At temporal frequency ν~3 mHz, the surface velocity power per oscillation mode increases slightly with angular degree between l=100 and l=200 but decreases rather steeply with l above l=200, in approximate agreement with earlier findings from ground-based measurements. From this we infer that the time-averaged energy per mode, which is theoretically related to the modal surface velocity power, decreases steeply with l, at fixed frequency, over the entire observed l-range. Specifically, at ν=3.1 mHz, the energy per mode drops by a factor of ~10 between l=150 and l=650, a circumstance not quantitatively understood at present. Title: Theoretical Signature of Solar Meridional Flow in Global Seismic Data Authors: Woodard, Martin F. Bibcode: 2000SoPh..197...11W Altcode: Approximate expressions are derived for the perturbations in solar p- and f-mode oscillation eigenfunctions, due to large-scale, meridional flows which are symmetric about the equator. The essential signature of the perturbed eigenfunctions in global helioseismic data is derived and the prospects for detecting meridional flow using global seismic techniques are discussed. Title: Photometric measurements of the solar disk at BBSO. Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 2000BAAS...32..803W Altcode: No abstract at ADS Title: Photometric Measurements of the Solar Disk at BBSO Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 2000SPD....31.0112W Altcode: Precise photometric measurements of the Sun have been obtained with the BBSO Solar Disk Photometer (SDP) since 1993. We describe our program to characterize sunspot, facular, and non-facular contributions to the spatial and temporal variations in solar irradiance using these data. This work is supported by NSF and NASA. Title: Solar Differential Rotation Derived from H-alpha Full Disk Images by Means of Local Correlation Tracking Authors: Woodard, M. F.; Denker, C.; Strous, L. H.; BBSO Collaboration; LMSAL Collaboration Bibcode: 1999AAS...19410007W Altcode: 1999BAAS...31..997W We present the application of Local Correlation Tracking (LCT) techniques to time series of contrast-enhanced H-alpha full disk images taken as part of the synoptic observing program at Big Bear Solar Observatory (BBS0) during the summer of 1998. A typical set of daily H-alpha full disk images consists of 600 to 800 individual frames, taken 30 to 60 s apart, with a 2k x 2k pixel Kodak 4.2 MegaPlus CCD camera at BBSO's Singer telescope. For each pair of successive images, we compute displacement vectors over a 64 x 64 element cartesian grid covering the solar disk. The resulting daily-averaged flow maps show predominantly solar differential rotation and proper motions in active regions. We remap the flow maps to heliographic coordinates and determine a Legendre polynomial expansion of the daily differential rotation profile. We present preliminary findings regarding differential rotation based on different types of features seen in H-alpha, such as quiet-sun fibrils, plages, and dark filaments. We discuss the relation of our differential rotation profiles to profiles derived by other methods and address the question of time variability. The work at BBSO is supported by ONR under grant N00014-97-1-1037, by NSF under grant ATM 97-14796, and by NASA under grant NAG 5-4919. Louis Strous is supported by NASA NAG5-3077 to Stanford University. Title: Evidence for non-potential magnetic fields in the quiet Sun Authors: Woodard, M. F.; Chae, Jongchul Bibcode: 1999SoPh..184..239W Altcode: A comparison of BBSO Hα centerline filtergrams and videomagnetograms was made to investigate the existence of non- potential magnetic fields in the quiet Sun near magnetic network. We use the fibril structure in the Hα images as a proxy for the horizontal chromospheric magnetic field which we compare with the horizontal field obtained by potential extrapolation of the observed, line-of-sight photospheric field. The quiet-Sun field was found to be consistently and significantly non-potential in each of the three fields of view studied. A transient extreme ultraviolet (EUV) brightening, known as a blinker, occurred during the observations of a region where the field is highly non-potential, suggesting a connection between magnetic reconnection and non-potentiality. Title: Effect of Subsurface Inhomogeneities on the Statistics of Solar Oscillation Power Spectra Authors: Woodard, M. F. Bibcode: 1998SoPh..180...19W Altcode: We characterize the statistical properties of the frequency-wave number periodogram of randomly-driven waves in a hypothetical, one-dimensional, spatially inhomogeneous, stationary medium. The derived properties are used to formulate a conceptually simple test of solar oscillation power spectra to estimate the separate contributions of true dissipation and inhomogeneous structure to the linewidths of high-degree p- and f-mode oscillations. Title: Spatial Dependence of Solar-Cycle Changes in the Sun's Luminosity Authors: Taylor, S. F.; Varsik, J. R.; Woodard, M. F.; Libbrecht, K. G. Bibcode: 1998SoPh..178....1T Altcode: We report observations of the large-scale spatial dependence of the Sun's luminosity variations over the period 1993-1995. The measurements were made using a new scanning disk solar photometer at Big Bear Solar Observatory, specially designed to measure large-scale brightness variations at the 10−4 level. Since the level of solar activity was very low for the entire observation period, the data show little solar cycle variation. However, the residual brightness signal ΔI/I (after subtracting the mean, first, and second harmonics) does show a strong dependence on heliocentric angle, peaking near the limb. This is as one would expect if the residual brightness signal (including the excess brightness coming from the active latitudes) were primarily facular in origin. Additional data over the next few years, covering the period from solar minimum to maximum, should unambiguously reveal the large-scale spatial structure of the solar cycle luminosity variations. Title: Implications of Localized, Acoustic Absorption for Heliotomographic Analysis of Sunspots Authors: Woodard, M. F. Bibcode: 1997ApJ...485..890W Altcode: Acoustic tomography is emerging as an important tool for mapping the three-dimensional structure of the Sun. Widely used in seismic studies of the Earth, tomography is being applied to probe the structure of magnetically active regions (e.g., sunspots), large-scale convective motions, and the structure of the solar atmosphere. By interpreting solar tomographic data by analogy with terrestrial tomographic data, namely in terms of acoustic travel times, Duvall et al. (1996) argue that strong downflows of matter are present beneath sunspots, a conclusion that lends support to the idea that convective downdrafts play a role in forming and maintaining sunspots.

Helioseismic waves are thought to be generated continuously, in a turbulent layer beneath the solar photosphere. This circumstance and the fact that sunspots absorb acoustic energy make heliotomographic data more difficult to interpret than their terrestrial counterpart. Therefore, the seismic evidence for strong downflows below sunspots should be viewed with skepticism.

We perform a theoretical analysis of waves in a simple absorbing medium, which explicitly deals with both the generation and propagation of waves. The results of the analysis suggest that localized acoustic absorption can qualitatively mimic the effect of a travel time perturbation. However, absorption (by itself, or in combination with flows) appears to be incapable of quantitatively reproducing the sunspot observations, indicating the operation of some as yet unidentified mechanism. Title: Distributiong Flight Dynamics Products via the World-Wide Web Authors: Woodard, M.; Matusow, D. Bibcode: 1996ESASP.394.1382W Altcode: 1996smog.conf.1382W No abstract at ADS Title: Doppler Acoustic Diagnostics of Subsurface Solar Magnetic Structure Authors: Lindsey, C.; Braun, D. C.; Jefferies, S. M.; Woodard, M. F.; Fan, Y.; Gu, Y.; Redfield, S. Bibcode: 1996ApJ...470..636L Altcode: We used the Bartol-NSO-NASA South Pole helioseismic observations of 1991 January to probe the subsurface structure of active regions to depths of ∼15,000 km. The helioseismic signature we particularly examine is intended to register acoustic Doppler effects caused by horizontal flows associated with the active region. We propose to show that the Doppler acoustic signature of horizontal flows is particularly well suited for deep subsurface diagnostics in terms of vertical discrimination of the structure. This study is based primarily on observations of NOAA Active Regions 6431, 6432, 6440, and 6442 between 1991 January 1 and January 8. We interpret the acoustic signatures we find in terms of a general outflow of the solar medium surrounding the active region. The acoustic signatures are strongly dependent on wavenumber, which suggests an outflow that is quite weak near the surface, the upper 4000 km of the subphotosphere, but which increases strongly with depth to velocities of several hundred meters per second at 15,000 km. This depth profile evolves rapidly as the active region matures. Young active regions show a strong outflow signature for waves that explore depths between 4000 and 8000 km. As the active region matures, the outflow vacates these intermediate layers and submerges to depths mostly below 8000 km.

We examine the location of AR 6442 for a possible preemergence signature. We also show evidence for extended, relatively superficial flows in the quiet Sun between the active region bands directed roughly into the active region bands. Title: Advances in Helioseismology Authors: Libbrecht, K. G.; Woodard, M. F. Bibcode: 1994snft.book..428L Altcode: No abstract at ADS Title: Observations of Time Variation in the Sun's Rotation Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 1993Sci...260.1778W Altcode: Observations of solar p-mode frequency splittings obtained at Big Bear Solar Observatory in 1986 and during 1988-90 reveal small (~1 percent) changes in the sun's subsurface angular velocity with solar cycle. An asymptotic inversion of the splitting data yields the latitude dependence of the rotation rate and shows that the largest changes in the angular velocity, ≈4 nanohertz, occurred between 1986 and the later years, at high (≈60^circ) solar latitudes. Earlier helioseismic observations suggest that solar cycle changes in the ratio of magnetic to turbulent pressure in the solar convection zone are large enough to account for the magnitude of the observed angular velocity variations but a detailed model of the phenomenon does not exist. Title: Seismic evidence of modulation of the structure and angular velocity of the Sun associated with the solar cycle Authors: Gough, D. O.; Kosovichev, A. G.; Sekii, T.; Libbrecht, K. G.; Woodard, M. F. Bibcode: 1993ASPC...40...93G Altcode: 1993IAUCo.137...93G; 1993ist..proc...93G No abstract at ADS Title: The Form of the Angular Velocity in the Solar Convection Zone Authors: Gough, D. O.; Kosovichev, A. G.; Sekii, T.; Libbrecht, K. G.; Woodard, M. F. Bibcode: 1993ASPC...42..213G Altcode: 1993gong.conf..213G No abstract at ADS Title: Solar Activity and Oscillation Frequency Splittings Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 1993ApJ...402L..77W Altcode: Solar p-mode frequency splittings, parameterized by the coefficients through order N = 12 of a Legendre polynomial expansion of the mode frequencies as a function of m/L, were obtained from an analysis of helioseismology data taken at Big Bear Solar Observatory during the 4 years 1986 and 1988-1990 (approximately solar minimum to maximum). Inversion of the even-index splitting coefficients confirms that there is a significant contribution to the frequency splittings originating near the solar poles. The strength of the polar contribution is anti correlated with the overall level or solar activity in the active latitudes, suggesting a relation to polar faculae. From an analysis of the odd-index splitting coefficients we infer an upper limit to changes in the solar equatorial near-surface rotatinal velocity of less than 1.9 m/s (3 sigma limit) between solar minimum and maximum. Title: Advances in Helioseismology Authors: Libbrecht, K. G.; Woodard, M. F. Bibcode: 1991Sci...253..152L Altcode: Globally coherent oscillation modes were discovered in the sun about a decade ago, providing a unique seismological probe of the solar interior. Current observations detect modes that are phase-coherent for up to 1 year, with surface velocity amplitudes as low as 2 millimeters per second, and thousands of mode frequencies have been measured to accuracies as-high as 1 part in 10^5. This article discusses the properties of these oscillation modes and the ways in which they are adding to our understanding of the structure and dynamics of the sun. Title: Is There an Acoustic Resonance in the Solar Chromosphere? Authors: Woodard, M. F.; Libbrecht, K. G. Bibcode: 1991ApJ...374L..61W Altcode: By comparing helioseismology data from 1986, 1988, and 1989, it was found that the frequecy dependence of the frequency perturbation of solar p-modes caused by solar activity drops abruptly for modes of frequency above approximately 3.9 mHz. The drop in the frequency dependence of the frequency shift may result from solar cycle changes in the chromosphere, provided that the chromosphere acts as a cavity in which p-modes are trapped (Goldreich et al.). No evidence is found in the temporal power spectrum of a time series of narrow-band Ca II K-line filtergrams of a resonance which would reveal the existence of a chromospheric cavity. This circumstance constrains the possible physical explanations of the frequency shifts. Title: Short-Term Changes in Solar Oscillation Frequencies and Solar Activity Authors: Woodard, M. F.; Kuhn, J. R.; Murray, N.; Libbrecht, K. G. Bibcode: 1991ApJ...373L..81W Altcode: It is shown that the frequencies of solar rho-mode oscillations change significantly over periods as short as one month. These changes correlate significantly with variations in the strength of surface solar activity as measured by the average, over the sun's visible surface, of the magnitude of the line-of-sight magnetic field component from magnetograms. The frequency and mean magnetic variations are found to obey a linear relationship. It is seen that the mean frequency shift at any time depends on the history of solar activity over an interval of, at most, several months prior to the measurement and conclude that the dominant mechanism of the frequency shift is correlated with surface magnetic activity. Title: Frequencies of Solar Oscillations Authors: Libbrecht, K. G.; Woodard, M. F.; Kaufman, J. M. Bibcode: 1990ApJS...74.1129L Altcode: Solar oscillations have been observed at three different spatial scales at Big Bear Solar Observatory during 1986-1987 and, using three data sets, a new and more accurate table of solar oscillation frequencies has been compiled. The oscillations, which are presented as functions of radial order n and spherical harmonic degree l, are averages over azimuthal order and therefore approximate the normal mode frequencies of a nonrotating, spherically symmetric sun, near solar minimum. The table contains frequencies for most of the solar p and f modes with l between 0 and 1860, n between 0 and 26, and oscillation mode frequencies between 1.0 and 5.3. Title: Solar-cycle effects on solar oscillation frequencies Authors: Libbrecht, K. G.; Woodard, M. F. Bibcode: 1990Natur.345..779L Altcode: Measurements of solar oscillations taken in 1986 and 1988 show systematic changes in the Sun's acoustic-mode frequencies of the order of 1 part in 10,000. These data reveal that the frequency shifts are the result of latitude-dependent changes in the structure of the Sun which are correlated with the Sun's magnetic-activity cycle. Title: Depth and Latitude Dependence of the Solar Internal Angular Velocity Authors: Rhodes, Edward J., Jr.; Cacciani, Alessandro; Korzennik, Sylvain; Tomczyk, Steven; Ulrich, Roger K.; Woodard, Martin F. Bibcode: 1990ApJ...351..687R Altcode: One of the design goals for the dedicated helioseismology observing state located at Mount Wilson Observatory was the measurement of the internal solar rotation using solar p-mode oscillations. In this paper, the first p-mode splittings obtained from Mount Wilson are reported and compared with those from several previously published studies. It is demonstrated that the present splittings agree quite well with composite frequency splittings obtained from the comparisons. The splittings suggest that the angular velocity in the solar equatorial plane is a function of depth below the photosphere. The latitudinal differential rotation pattern visible at the surface appears to persist at least throughout the solar convection zone. Title: Observations of Solar Cycle Variations in Solar p_Mode Frequencies and Splittings Authors: Libbrecht, K. G.; Woodard, M. F. Bibcode: 1990LNP...367..145L Altcode: 1990psss.conf..145L We discuss here two sets of helioseismology data acquired at Big Bear Solar Observatory during the summers of 1986 and 1988. Each data set consists of roughly 60,000 fulldisk Doppler images of the sun, accumulated over a four-month time span. These data clearly show that solar p-mode frequencies change with time, and that the measured frequency shifts v = v 88 - v 86 depend strongly on frequency and only weakly on for 5 60. The frequency dependence is well described by v ∞ M-1(v), where M(v) is the mode mass for low-ℓ modes. Such a frequency dependence is expected if the effective sound speed perturbation is located predominantly near the solar surface. It should be possible to invert the frequency shift measurements to determine some aspects of the structure of solar activity as a function of depth. The data also show that the even-index splitting coefficients depend strongly on frequency, again being well described by α2j (v) ∞ M-1(v). This functional form is expected if the sound speed perturbation responsible for Δv is localized in solar latitude. Latitude inversions of the time-dependent splitting and Δv measurements show that the perturbation is strongest in the active latitudes, but includes a weak polar component. Title: Distortion of High-Degree Solar p-Mode Eigenfunctions by Latitudinal Differential Rotation Authors: Woodard, Martin F. Bibcode: 1989ApJ...347.1176W Altcode: The eigenfunctions of nonradial oscillation of the sun depart from their customarily assumed spherical harmonic form as a result of solar rotation. This paper treats the effect on oscillation eigenfunctions of slow, axisymmetric differential rotation which is also mirror symmetric across the solar equatorial plane. A given oscillation eigenfunction, perturbed by rotation, can be expressed as a superposition of eigenfunctions of a (hypothetical) nonrotating state of the sun. The assumption of axialsymmetry implies that each of the superposed functions has the same azimuthal order, m, while mirror symmetry dictates that the values of l, the spherical harmonic degree, involved in the superposition be either all odd or all even. The treatment here is further specialized to the case of high-degree p- and f-mode oscillations, on which the effect of the Coriolis acceleration is small. It is shown that, by ignoring the Coriolis effect and making other sensible approximations, an analytic expression can be obtained for the rotationally distorted mode eigenfunctions, which is expected to be a good approximation for typical solar oscillations of degree l greater than 100. The effect of the calculated mode distortion on the measurement of solar rotation is explored. Title: On the measurement of solar rotation using high-degree p-mode oscillations. Authors: Woodard, Martin F.; Libbrecht, Ken G. Bibcode: 1988ESASP.286...67W Altcode: 1988ssls.rept...67W The authors describe the progress made and some of the difficulties encountered in measuring the solar rotation rate with p-modes of degree 100 ⪉ l ⪉ 400, using a set of high-resolution solar images taken at Big Bear Solar Observatory. The main conclusion drawn from an analysis of one day of data is that the equatorial angular velocity is essentially equal to the observed surface rate over the radius range 0.87 ⪉ r/R ⪉ 0.99 to within a few percent. Because of likely systematic errors at the 1% level these data do not allow us to distinguish between a surrface rotation rate equal to that measured using magnetic tracers and that based on the Doppler shift of photospheric spectral lines. Title: Radial and latitudinal gradients in the solar internal angular velocity. Authors: Rhodes, Edward J., Jr.; Cacciani, Alessandro; Korzennik, Sylvain G.; Tomczyk, Steven; Ulrich, Roger K.; Woodard, Martin F. Bibcode: 1988ESASP.286...73R Altcode: 1988ssls.rept...73R The authors recently presented the results of an analysis of the frequency splittings of intermediate-degree (3 < degree ≤ 170) p-mode oscillations which were obtained from a 16-day subset of our 1984 Mt. Wilson 60-foot tower observations. These results showed evidence for both radial and latitudinal gradients in the solar internal angular velocity. In particular, the results indicated that, from 0.6 Rsun to 0.95 Rsun, the solar internal angular velocity increases systematically from 440 to 463 nHz, corresponding to a positive radial gradient of ≍66 nHz/Rsun for that portion of the solar interior. A previous analysis indicated that the latitudinal differential rotation gradient which is seen at the solar surface persists throughout the convection zone, although there was some indication that the differential rotation might disappear entirely below the base of the convection zone. Here the authors extend their previous analysis to include comparisons with additional observational studies and they also present comparisons between our earlier results and the results of additional inversions of several of the observational datasets. All of these comparisons reinforce the previous conclusions regarding the existence of radial and latitudinal gradients in the internal angular velocity. Title: On the Constancy of Intermediate-Degree p-Mode Frequencies during the Declining Phase of Solar Cycle 21 Authors: Rhodes, Edward J., Jr.; Woodard, Martin F.; Cacciani, Alessandro; Tomczyk, Steven; Korzennik, Sylvain G.; Ulrich, Roger K. Bibcode: 1988ApJ...326..479R Altcode: A comparison of two sets of frequencies of intermediate-degree (6 ≤ l ≤ 89) solar p-mode oscillations obtained in late 1981 and in mid-1984 shows agreement at the level of 0.02 μHz, or better than one part in 105. In particular, the frequencies of 573 modes obtained at the South Pole during 1981 December 24 - 25, (reported by Duvall, Harvey, and Pomerantz in 1987) were compared with the frequencies of the same modes as observed at the Mount Wilson Observatory 60 foot Solar Tower from 1984 July 29 through August 13. It is concluded that the data are consistent with no change in intermediate-degree p-mode frequencies between late 1981 and mid-1985. Title: Frequencies of low-degree solar acoustic oscillations and the phase of the solar cycle Authors: Woodard, Martin F. Bibcode: 1988SoPh..114...21W Altcode: A study of the solar total irradiance data of the Active Cavity Radiometer Irradiance Monitor (ACRIM) on the Solar Maximum Mission (SMM) satellite shows a small but formally significant shift in the frequencies of solar acoustic (p-mode) oscillations between the epochs of maximum and minimum solar activity. Specifically, the mean frequency of the strongest p-mode resonances of low spherical-harmonic degree (l = 0-2) is approximately 1.3 parts in 104 higher in 1980, near the time of sunspot maximum, than in ∼1985, near sunspot minimum. The observed frequency shift may be an 11-yr effect but the precise mechanism is not clear. Title: Time Variations of the Frequencies of Low-Degree Solar P-Modes Authors: Woodard, M. F.; Noyes, R. W. Bibcode: 1988IAUS..123..197W Altcode: A comparison of three separate years (1980, 1984, 1985) of SMM/ACRIM solar total irradiance data reveals small but significant changes in the frequencies of low-degree solar p-modes. Title: Measurements of Solar Internal Rotation Obtained with the Mt-Wilson 60-FOOT Solar Tower Authors: Rhodes, E. J., Jr.; Cacciani, A.; Woodard, M.; Tomczyk, S.; Korzennik, S.; Ulrich, R. K. Bibcode: 1988IAUS..123...41R Altcode: The authors have obtained estimates of the solar internal rotational velocity from measurements of the frequency splittings of p-mode oscillations. Specifically, they have analyzed a 10-day time series of full-disk Dopplergrams obtained during July and August 1984 at the 60-Foot Tower Telescope of the Mt. Wilson Observatory. Title: Constancy of Intermediate-degree p-Mode Frequencies During the Declining Phase of Solar Cycle 21 Authors: Rhodes, E. J., Jr.; Woodard, M. F.; Cacciani, A.; Tomczyk, S.; Korzennik, S.; Ulrich, R. K. Bibcode: 1987BAAS...19Q.933R Altcode: No abstract at ADS Title: Angular Velocity of the Solar Interior Obtained by an Asymptotic Inversion of P-Mode Frequency Shifts Authors: Woodard, M. F.; Rhodes, E. J., Jr.; Tomczyk, S.; Korzennik, S.; Cacciani, A.; Ulrich, R. K. Bibcode: 1987BAAS...19..934W Altcode: No abstract at ADS Title: Estimates of the solar internal angular velocity obtained with the Mt. Wilson 60-foot solar tower Authors: Rhodes, Edward J., Jr.; Cacciani, Alessandro; Woodard, Martin; Tomczyk, Steven; Korzennik, Sylvain; Ulrich, R. K. Bibcode: 1987ASSL..137...75R Altcode: 1987isav.symp...75R Estimates are obtained of the solar internal angular velocity from measurements of the frequency splittings of p-mode oscillations. A 16-day time series of full-disk Dopplergrams obtained during July and August 1984 at the 60-foot tower telescope of the Mt. Wilson Observatory is analyzed. Power spectra were computed for all of the zonal, tesseral, and sectoral p-modes from l = 0 to 89 and for all of the sectoral p-modes from l = 90 to 200. A mean power spectrum was calculated for each degree up to 89. The frequency differences of all of the different nonzonal modes were calculated for these mean power spectra. Title: Frequencies of low-degree solar acoustic oscillations and the phase of the solar cycle Authors: Woodard, Martin F. Bibcode: 1987SoPh..114...21W Altcode: A study of the solar total irradiance data of the Active Cavity Radiometer Irradiance Monitor (ACRIM) on the Solar Maximum Mission (SMM) satellite shows a small but formally significant shift in the frequencies of solar acoustic (p-mode) oscillations between the epochs of maximum and minimum solar activity. Specifically, the mean frequency of the strongest p-mode resonances of low spherical-harmonic degree (l = 0-2) is approximately 1.3 parts in 104 higher in 1980, near the time of sunspot maximum, than in ≡1985, near sunspot minimum. The observed frequency shift may be an 11-yr effect but the precise mechanism is not clear. Title: Evidence for Radial Gradients in the Solar Internal Rotational Velocity Authors: Rhodes, E. J., Jr.; Tomczyk, S.; Woodard, M. F.; Cacciani, A.; Korzennik, S.; Ulrich, R. K. Bibcode: 1986BAAS...18Q1010R Altcode: No abstract at ADS Title: Change of solar oscillation eigenfrequencies with the solar cycle Authors: Woodard, M. F.; Noyes, R. W. Bibcode: 1986ASIC..169..303W Altcode: 1986ssds.proc..303W No abstract at ADS Title: Extraterrestrial Evidence Regarding the Frequency of Collisional Events in the Solar System Authors: Lavielle, B.; Regnier, S.; Simonoff, G. N.; Marti, K.; Woodard, M. Bibcode: 1985Metic..20Q.692L Altcode: No abstract at ADS Title: Change of solar oscillation eigenfrequencies with the solar cycle Authors: Woodard, Martin F.; Noyes, Robert W. Bibcode: 1985Natur.318..449W Altcode: Solar acoustic eigenfrequencies depend on the internal structure of the Sun, which may change during the 11-yr cycle of magnetic activity as a result of various effects associated with the solar dynamo. Observations of low-degree acoustic frequencies were made, using the ACRIM instrument on the Solar Maximum Mission (SMM) satellite, in 1980 (near solar maximum) and 1984 (near solar minimum). The analysis of these data, presented here, indicates that the frequencies of l = 0 and l = 1 acoustic modes in the 5-min band have decreased from 1980 to 1984, by ~0.42 µHz or 1.3 parts in 104. This finding may have important implications for our understanding of the mechanism of the solar activity cycle. Title: Clusters and Cycles in the Cosmic Ray Age Distributions of Meteorites Authors: Woodard, M. F.; Marti, K. Bibcode: 1985ICRC....5..402W Altcode: 1985ICRC...19e.402W Statistically significant clusters in the cosmic ray exposure age distributions of some groups of iron and stone meteorites are observed, suggesting epochs of enhanced collisions and breakups. Fourier analyses of the age distribution of chondrites reveal no significant periods, nor does the same analysis when applied to iron meteorite clusters. Title: Observations of low-degree modes from the Solar Maximum Mission. Authors: Woodard, M. Bibcode: 1984sses.nasa..195W Altcode: 1984sss..conf..195W Mean frequencies, amplitudes, and linewidths for the solar 5 min p mode oscillations of degree 0, 1, and 2 have been obtained from approx. 280 days of SMM-ACRIM total irradiance data. The frequencies are in good agreement with measurements obtained from velocity data. The amplitudes of the modes lie along a well defined envelope of power vs. frequency, which peaks at 3.1 mHz and has a width of 0.7 mHz (FWHM). The r.m.s. amplitude of the highest peak in the spectrum (n=21, l=1) is approx. 3 ppm of the total flux. The linewidths of the narrowest l=O modes are approx. 1 micro Hz (FWHM). A broad continuum of power caused both by solar surface granulation and by instrumental noise interferes with the analysis of 5 min modes. The continuum spectral power in a 1 micro Hz band near 3 mHz corresponds to an apparent r.m.s. variation of approx. 0.5 parts per million of the mean solar flux. Title: Total irradiance observations of low degree p-modes Authors: Woodard, M.; Hudson, H. Bibcode: 1984MmSAI..55...67W Altcode: A spectral analysis of 10 months of brightness data from the Active Cavity Radiometer Irradiance Monitor on the SMM satellite has revealed new information about the low-degree (l = 0,1, and 2) p-mode oscillations of the Sun. The mean frequencies, irradiance amplitudes, and line widths of the largest amplitude modes have been determined with previously unattained accuracy. Title: The Inconstant Solar Constant Authors: Willson, R. C.; Hudson, H.; Woodard, M. Bibcode: 1984S&T....67..501W Altcode: The Active Cavity Radiometer Irradiance Monitor (ACRIM) of the Solar Maximum Mission satellite measures the radiant power emitted by the sun in the direction of the earth and has worked flawlessly since 1980. The main motivation for ACRIM's use to measure the solar constant is the determination of the extent to which this quantity's variations affect earth weather and climate. Data from the solar minimum of 1986-1987 is eagerly anticipated, with a view to the possible presence of a solar cycle variation in addition to that caused directly by sunspots. Title: Upper limit on solar interior rotation Authors: Woodard, M. Bibcode: 1984Natur.309..530W Altcode: The power spectrum of solar total irradiance (flux) variations front the Active Cavity Radiometer Irradiance Monitor (ACRIM) on the Solar Maximum Mission (SMM) spacecraft shows individual 5-min p- mode oscillations 1,2of spherical harmonic degree l= 0-2 and radial order n= 16-26. An m-state splitting analysis based on the widths of the (n,l) multiplets in the spectrum of ACRIM data yields a mean (sidereal) interior rotation rate between 0 and 2.2 times the observed 0.456- µHz equatorial surface rate, consistent with the rapid rotation rate originally claimed by Claverie et al.3based on a (controversial) splitting interpretation of the these 5-min modes seen in line-of-sight velocity. Rotationally split p- and g- and f-modes have been identified in the temporal power spectrum of the limb-darkening data of Bos and Hill4, and from these splittings two internal rotation curves5,6 have been deduced which imply a solar gravitational quadrupole moment J2large enough to spoil the precise agreement between general relativity and observations of planetary motion. The splitting of the low- l 5-min p- modes implied by these curves is inconsistent with the upper limit derived here, and the reported conflict with einsteinian theory, is therefore, premature. Title: Short-Period Oscillations in the Total Solar Irradiance. Authors: Woodard, Martin Frederic Bibcode: 1984PhDT........34W Altcode: 1985DiAbI..45.2582W This dissertation deals with normal-mode vibrations of the Sun, as measured by their effect on the solar total irradiance. I have studied "5-minute" p-mode oscillations of radial order n (TURNEQ) 16 - 26 and degree l = 0 - 2 in the 1980 data from the Active Cavity Radiometer Irradiance Monitor (ACRIM) on the Solar Maximum Mission satellite (SMM). The frequencies of separate (n,l) peaks in the spectrum of ACRIM data agree with those of previous observers to within ('(TURN))3 parts in 10('4). For the oscillatory peaks of highest signal quality the frequency error is estimated to be less than one part in 10('4). Discrepancies between observed and theoretical frequencies are already much greater than discrepancies among observed frequencies. Quasi -degenerate m states, of which the (n,l) peaks are in principle comprised, cannot be distinguished in the ACRIM power spectrum, contrary to previous claims. The mean amplitudes of individual (n,l) peaks have been established with 10 - 20% accuracy; r.m.s. fractional irradiance variations of up to 3 parts per million occur. The observed amplitude ratio of 5-min modes in velocity and intensity agrees roughly with preliminary theoretical estimates, but more theory and observation are required. The 5-min oscillation data from the ACRIM have been interpreted in terms of a theoretically motivated picture in which the modes behave like independently and randomly excited harmonic oscillators. The relative frequency width of the most prominent l = 0 modes is about one part in 2000, implying an oscillator time constant (e-folding time for energy decay) of 1 to 2 days. This lifetime is approximately confirmed by direct observations of amplitude variations in these modes. The lifetime of these modes was previously a matter of controversy. Solar rotation will split the m states belonging to a given (n,l) multiplet, making the l > 1 peaks broader than they would otherwise be. From the widths of the l = 1 peaks I deduce an upper limit on the separation of adjacent m states of 1.0 (mu)Hz with ('(TURN))99% confidence. This is 2.2 times the splitting that solid-body rotation at the observed ('(TURN))1/25-day sidereal equatorial surface rate would imply. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI. Title: Frequencies, amplitudes and linewidths of solar oscillations from total irradiance observations Authors: Woodard, M.; Hudson, H. S. Bibcode: 1983Natur.305..589W Altcode: Ten months of solar total irradiance data from the Solar Maximum Mission satellite have generated accurate frequencies, amplitudes and linewidths for individual ~5-min solar p-mode ocillations of low degree. The modes can be described as independent and chaotically excited oscillators, and provide no evidence for the fine structure taken to imply rapid internal rotation of the Sun. Title: Upper Limits on the Internal Rotation Rate of the Sun Authors: Woodard, M. F.; Hudson, H. S. Bibcode: 1983BAAS...15..951W Altcode: No abstract at ADS Title: Solar Surface Granulation and Variations of Total Irradiance Authors: Hudson, H. S.; Woodard, M. F. Bibcode: 1983BAAS...15R.715H Altcode: No abstract at ADS Title: Solar Oscillations Observed in the Total Irradiance Authors: Woodard, M.; Hudson, H. Bibcode: 1983SoPh...82...67W Altcode: 1983IAUCo..66...67W The total solar irradiance measurements obtained by the active-cavity radiometer on board the Solar Maximum Mission have been analyzed for evidence of global oscillations. We find that the most energetic low-degree p-mode oscillations in the five-minute band have amplitudes of a few parts per million of the total irradiance, and we positively detect modes with l = 0, 1, and 2. The distribution in l differs from that of the velocity spectrum, with relatively more power at lower l values. The individual modes have narrow line widths, corresponding to values of Q greater than a few thousand, or lifetimes of at least a week. We do not detect the 160-min oscillation in the power spectrum, and place an upper limit of 5 parts per million (99.9% confidence) on its amplitude. Title: What are Solar Irradiance Observations of Global Oscillations Telling Us? Authors: Woodard, M. F.; Hudson, H. S. Bibcode: 1982BAAS...14..864W Altcode: No abstract at ADS Title: The Effects of Sunspots on Solar Irradiance Authors: Hudson, H. S.; Silva, S.; Woodard, M.; Willson, R. C. Bibcode: 1982SoPh...76..211H Altcode: Sunspots have an obvious direct effect upon the visible radiant energy falling upon the Earth. We show how to estimate this effect and compare it quantitatively with recent observations of the solar total irradiance (Willson et al., 1981). The sunspots explain about half of the total observed variance of one-day averages. Since the sunspot effect on irradiance produces an asymmetry of the solar radiation, rather than (necessarily) a variation of the total luminosity, we have also estimated the sunspot population on the invisible hemisphere. This extrapolation allows us to estimate the true luminosity deficit produced by sunspots, in a manner that tends toward the correct long-term average value. We find no evidence for instantaneous global re-emission to compensate for the sunspot flux deficit. Title: Observations of p-mode Oscillations in the Total Solar Irradiance Authors: Woodard, M.; Hudson, H.; Willson, R. C. Bibcode: 1981BAAS...13..858W Altcode: No abstract at ADS Title: Variability of Total Solar Irradiance: the Flux Deficit of Sunspots Authors: Hudson, H. S.; Woodard, M.; Willson, R. C. Bibcode: 1980BAAS...12Q.898H Altcode: No abstract at ADS