Generalization of the noise model for time-distance helioseismology

In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here we consider travel times and also products of travel times as observables. They contain information about e.g. the statistical properties of convection in the Sun. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process which is stationary and Gaussian. We generalize the existing noise model (Gizon and Birch 2004) by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to $1/T$, $1/T^2$, and $1/T^3$, where $T$ is the duration of the observations. For typical observation times of a few hours, the term proportional to $1/T^2$ dominates and $Cov[\tau_1 \tau_2, \tau_3 \tau_4] \approx Cov[\tau_1, \tau_3] Cov[\tau_2, \tau_4] + Cov[\tau_1, \tau_4] Cov[\tau_2, \tau_3]$, where the $\tau_i$ are arbitrary travel times. This result is confirmed for $p_1$ travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis

Linear Sensitivity of Helioseismic Travel Times to Local Flows

Time-distance helioseismology is a technique for measuring the time for waves to travel from one point on the solar surface to another. These wave travel times are affected by advection by subsurface flows. Inferences of plasma flows based on observed travel times depend critically on the ability to accurately model the effects of subsurface flows on time-distance measurements. We present a Born approximation based computation of the sensitivity of time distance travel times to weak, steady, inhomogeneous subsurface flows. Three sensitivity functions are obtained, one for each component of the 3D vector flow. We show that the depth sensitivity of travel times to horizontally uniform flows is given approximately by the kinetic energy density of the oscillation modes which contribute to the travel times. For flows with strong depth dependence, the Born approximation can give substantially different results than the ray approximation.Comment: 6 pages, 6 figure

Systematic Center-to-Limb Variation in Measured Helioseismic Travel Times and Its Effect on Inferences of Solar Interior Meridional Flows

We report on a systematic center-to-limb variation in measured helioseismic travel times, which must be taken into account for an accurate determination of solar interior meridional flows. The systematic variation, found in time-distance helioseismology analysis using SDO/HMI and SDO/AIA observations, is different in both travel-time magnitude and variation trend for different observables. It is not clear what causes this systematic effect. Subtracting the longitude-dependent east-west travel times, obtained along the equatorial area, from the latitude-dependent north-south travel times, obtained along the central meridian area, gives remarkably similar results for different observables. We suggest this as an effective procedure for removing the systematic center-to-limb variation. The subsurface meridional flows obtained from inversion of the corrected travel times are approximately 10 m/s slower than those obtained without removing the systematic effect. The detected center-to-limb variation may have important implications in the derivation of meridional flows in the deep interior, and needs a better understanding.Comment: accepted for publication by ApJ Letter

Local helioseismic and spectroscopic analyses of interactions between acoustic waves and a sunspot

Using a high cadence imaging spectropolarimetric observation of a sunspot and its surroundings in magnetically sensitive (FeI 6173 A) and insensitive (FeI 7090 A) upper photospheric absorption lines, we map the instantaneous wave phases and helioseismic travel times as a function of observation height and inclination of magnetic field to the vertical. We confirm the magnetic inclination angle dependent transmission of incident acoustic waves into upward propagating waves, and derive (1) proof that helioseismic travel times receive direction dependent contributions from such waves and hence cause errors in conventional flow inferences, (2) evidences for acoustic wave sources beneath the umbral photosphere, and (3) significant differences in travel times measured from the chosen magnetically sensitive and insensitive spectral lines.Comment: 12 pages, 5 figures, To appear in the Astrophysical Journal Letter

Impact of Locally Suppressed Wave sources on helioseismic travel times

Wave travel-time shifts in the vicinity of sunspots are typically interpreted as arising predominantly from magnetic fields, flows, and local changes in sound speed. We show here that the suppression of granulation related wave sources in a sunspot can also contribute significantly to these travel-time shifts, and in some cases, an asymmetry between in and outgoing wave travel times. The tight connection between the physical interpretation of travel times and source-distribution homogeneity is confirmed. Statistically significant travel-time shifts are recovered upon numerically simulating wave propagation in the presence of a localized decrease in source strength. We also demonstrate that these time shifts are relatively sensitive to the modal damping rates; thus we are only able to place bounds on the magnitude of this effect. We see a systematic reduction of 10-15 seconds in $p$-mode mean travel times at short distances ($\sim 6.2$ Mm) that could be misinterpreted as arising from a shallow (thickness of 1.5 Mm) increase ($\sim$ 4%) in the sound speed. At larger travel distances ($\sim 24$ Mm) a 6-13 s difference between the ingoing and outgoing wave travel times is observed; this could mistakenly be interpreted as being caused by flows.Comment: Revised version. Submitted to Ap

Stability or regularity of the daily travel time in Lyon? Application of a duration model

Escaping unidimensional analysis limits and linear regression irrelevancy, the duration model incorporates impacts of covariates on the duration variable and permits to test the dependence of daily travel times on elapsed time. In the perspective of a discussion of Zahavi's hypothesis, the duration model approach is applied to the daily travel times of Lyon (France). The relationships between daily travel times and socio-economic attributes and activity duration only support the “weak version of TTB stability hypothesis”. Furthermore the non-monotonic estimated hazard questions the minimisation of daily travel times.Duration model; Non-parametric, semi-parametric and parametric approaches; Travel time budget; Zahavi's hypothesis

User equilibrium traffic network assignment with stochastic travel times and late arrival penalty

The classical Wardrop user equilibrium (UE) assignment model assumes traveller choices are based on fixed, known travel times, yet these times are known to be rather variable between trips, both within and between days; typically, then, only mean travel times are represented. Classical stochastic user equilibrium (SUE) methods allow the mean travel times to be differentially perceived across the population, yet in a conventional application neither the UE or SUE approach recognises the travel times to be inherently variable. That is to say, there is no recognition that drivers risk arriving late at their destinations, and that this risk may vary across different paths of the network and according to the arrival time flexibility of the traveller. Recent work on incorporating risky elements into the choice process is seen either to neglect the link to the arrival constraints of the traveller, or to apply only to restricted problems with parallel alternatives and inflexible travel time distributions. In the paper, an alternative approach is described based on the ‘schedule delay’ paradigm, penalising late arrival under fixed departure times. The approach allows flexible travel time densities, which can be fitted to actual surveillance data, to be incorporated. A generalised formulation of UE is proposed, termed a Late Arrival Penalised UE (LAPUE). Conditions for the existence and uniqueness of LAPUE solutions are considered, as well as methods for their computation. Two specific travel time models are then considered, one based on multivariate Normal arc travel times, and an extended model to represent arc incidents, based on mixture distributions of multivariate Normals. Several illustrative examples are used to examine the sensitivity of LAPUE solutions to various input parameters, and in particular its comparison with UE predictions. Finally, paths for further research are discussed, including the extension of the model to include elements such as distributed arrival time constraints and penalties