Abstract
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.
Highlights
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
The purpose of time-distance helioseismology (Duvall et al 1993; Gizon & Birch 2005, and references therein) is to infer the subsurface structure and dynamics of the Sun using spatial-temporal correlations of the random wave field observed at the solar surface
As one term is of order T −2 and the other one of order T −3, it follows that Z2 will dominate for long observation times
Summary
The purpose of time-distance helioseismology (Duvall et al 1993; Gizon & Birch 2005, and references therein) is to infer the subsurface structure and dynamics of the Sun using spatial-temporal correlations of the random wave field observed at the solar surface. Wave travel times between pairs of points (denoted τ) are measured from the cross-covariance function. 4 and the Appendices, we derive analytical formulae for the noise covariance matrices of travel times and products of travel times Observables: cross-covariance function, travel times, and products of travel times. The cross-covariance function between two points at the surface of the Sun is a multiplication in the Fourier domain (Duvall et al 1993): C(x1, x2, ω). We are interested in the noise covariance matrix for travel times τα (x1, x2) and products of travel times τα (x1, x2)τα (x3, x4), where τ is defined by Eq (3). Let τ1 := τα (x1, x2), τ2 := τα (x3, x4), and more generally τi := ταi (x2i−1, x2i)
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