# Comparison of Travel-Time and Amplitude Measurements for Deep-Focusing Time–Distance Helioseismology

## Abstract

The purpose of deep-focusing time–distance helioseismology is to construct seismic measurements that have a high sensitivity to the physical conditions at a desired target point in the solar interior. With this technique, pairs of points on the solar surface are chosen such that acoustic ray paths intersect at this target (focus) point. Considering acoustic waves in a homogeneous medium, we compare travel-time and amplitude measurements extracted from the deep-focusing cross-covariance functions. Using a single-scattering approximation, we find that the spatial sensitivity of deep-focusing travel times to sound-speed perturbations is zero at the target location and maximum in a surrounding shell. This is unlike the deep-focusing amplitude measurements, which have maximum sensitivity at the target point. We compare the signal-to-noise ratio for travel-time and amplitude measurements for different types of sound-speed perturbations, under the assumption that noise is solely due to the random excitation of the waves. We find that, for highly localized perturbations in sound speed, the signal-to-noise ratio is higher for amplitude measurements than for travel-time measurements. We conclude that amplitude measurements are a useful complement to travel-time measurements in time–distance helioseismology.

## Keywords

Helioseismology Oscillations, solar## 1 Introduction

Time–distance helioseismology (Duvall *et al.*, 1993) is a branch of local helioseismology (*e.g.* Gizon and Birch, 2005) that aims at probing the complex subsurface structures of the solar interior. The time–distance method measures the travel times of acoustic waves between any pair of points on the solar surface from the cross-covariance function of the observed oscillation signals. Seismic travel times contain information about the local physical properties of the medium and have thus been broadly used in helioseismology (*e.g.* Gizon and Birch, 2002; Birch, Kosovichev, and Duvall, 2004; Gizon, Birch, and Spruit, 2010).

A consistent issue with local helioseismology is the signal-to-noise ratio. When examining near-surface structures such as supergranular flows (Duvall *et al.*, 1996; Langfellner, Gizon, and Birch, 2015), averaging is typically performed around an annulus, where the cross-covariance is calculated between the center point and the average signal in the annulus. This technique is highly sensitive to near-surface perturbations. To probe greater depths, one would seek a different averaging technique that has peak sensitivity at any chosen target depth. Such a technique is known as deep-focusing and was first described by Duvall (1995), who outlined a procedure in which points on the surface are chosen such that a large number of connecting ray paths intersect at the target (focus) point, with the expectation that sensitivity is large near the target depth. The deep-focusing time–distance technique has been employed to study the meridional flow in the solar interior (*e.g.* Hartlep *et al.*, 2013; Zhao *et al.*, 2013) and sunspot structure (*e.g.* Moradi and Hanasoge, 2010). Jensen (2001) investigated the application of the deep-focusing method to improve inversions for large sunspots. Using the Rytov approximation, he found sensitivity in a shell around the target point but zero sensitivity at the target point, consistent with wavefront healing seen in the Born approximation in geophysics and helioseismology (Liang *et al.*, 2013). To resolve this drawback, Hughes, Pijpers, and Thompson (2007) suggested an optimized technique for deep focusing that allocates weightings for each measurement. They obtained improvements in the results by considering travel-time measurements of synthetic experiments.

In addition to the travel times, the cross-covariance function contains additional information that may be of use to helioseismology. For instance, in terrestrial seismology cross-covariance amplitudes have been used to characterize seismic waves (*e.g.* Nolet, Dahlen, and Montelli, 2005). The importance of the amplitudes was examined by Dalton, Hjörleifsdóttir, and Ekström (2014), who concluded that assumptions and simplifications in the measurement of surface-wave amplitudes affect the attenuation structure found through inversions. Moreover, Dahlen and Baig (2002) investigated the Fréchet sensitivity kernels using the geometrical ray approximation for travel-time and amplitude measurements. They found a maximum sensitivity along the point-to-point ray path when examining the amplitude of seismic-wave cross-correlation. In contrast to travel times, few studies have considered the amplitude measurements of the cross-covariance function in helioseismology. Liang *et al.* (2013) measured the spatial maps of wave travel times and amplitudes from the cross-covariance function of the wave field around a sunspot in the NOAO Active Region 9787. Using 2D ray theory, they observed an amplitude reduction that was attributed to the defocusing of wave energy by the fast-wave-speed perturbation in the sunspot. Recent work by Nagashima *et al.* (2017) described a linear procedure to measure the amplitude of the cross-covariance function of solar oscillations. This linear relation between the cross-covariance function and the amplitude allows the derivation of Born sensitivity kernels using the procedure of Gizon and Birch (2002), which provides a straightforward interpretation for the amplitude measurements.

The deep-focusing time–distance technique using amplitude measurements is lacking in time–distance helioseismology. Furthermore, the deep-focusing analysis has been considered only using the ray theory, which is a high-frequency approximation and does not take into account finite-wavelength effects. As a result, the ray approximation may be inaccurate for amplitude calculations (*e.g.* Tong *et al.*, 1998). In this study, we use the deep-focusing time–distance technique to compare signal and noise for travel-time and amplitude measurements under the Born approximation. Section 2 describes the definition of travel-time and amplitude measurements and explains the deep-focusing technique and the noise model. The setup and derivation of sensitivity kernels are explained in Section 3 and the results are presented in Section 4. Conclusions are given in Section 5.

## 2 Travel-Time and Amplitude Measurements

### 2.1 Definitions

*et al.*(2017):

### 2.2 Deep-Focusing Averages

### 2.3 Noise Model

Here we describe the noise in the averaged measurements for travel time and amplitude. Random noise in helioseismology is due to the stochastic excitation of acoustic waves by turbulent convection. The noise model developed by Gizon and Birch (2004) is based on the reasonable assumption that the reference wave field \([\phi_{0}]\) is described by a stationary Gaussian random process.

*et al.*(2014) showed that the covariance is explicitly given by

## 3 Travel-Time and Amplitude Sensitivity Kernels for Sound-Speed Perturbations to a Uniform Background Medium

### 3.1 Wave Equation and Reference Green’s Function

*e.g.*Gizon

*et al.*, 2017)

### 3.2 Perturbation to the Cross-Covariance Function

### 3.3 Travel-Time and Amplitude Sensitivity Kernels

## 4 Example Calculations

### 4.1 Choice of Numerical Values and Parameters

*i.e.*the

*Solar Dynamics Observatory*(SDO)/

*Helioseismic and Magnetic Imager*(HMI) cadence.

### 4.2 Point-to-Point Sensitivity Kernels

### 4.3 Deep-Focusing Sensitivity Kernels

### 4.4 Kernel Widths as Functions of Target Depth

The vertical widths of the deep-focusing sensitivity kernels for travel time [\(L_{\tau}\)] and amplitude [\(L_{a}\)] are defined in Figure 4c. This width indicates the extent of the central regions of a kernel, within which it keeps the same (negative) sign. The smaller \(L_{\tau}\) (or \(L_{a}\)), the higher the spatial resolution of the travel-time (or amplitude) deep-focusing technique.

*i.e.*when \(\Delta\phi= n \pi\), \(n\in\mathbb{Z}\). In particular the sensitivity is zero along the direct ray path (\(n=0\)). The width of the travel-time kernel coincides with the boundary of the first (\(n=1\)) Fresnel zone:

*i.e.*\(\Delta\phi= \pi/2 + m \pi\), with \(m\in\mathbb{Z}\). Setting \(m=0\) gives the width of the amplitude kernel:

The dependence of the widths on target radius \(z_{t}\) is understood through \(D = 4 \sqrt{R^{2}- z_{t}^{2}}\), so that \(L\propto(R^{2} - z_{t}^{2})^{1/4}\). As seen in Figure 6, the model values for \(L_{\tau}\) and \(L_{a}\) from Equations 34 – 35 are in reasonable agreement with the numerical values, including the relationship \(L_{a} = L_{\tau}/\sqrt {2}\approx 0.7 L_{\tau}\).

### 4.5 Noise Covariance

*a posteriori*why we chose points in the pupil that are separated by \(\lambda_{0}/4\approx8.3\) Mm to avoid under sampling.

### 4.6 Localized Sound-Speed Anomaly at \(z_{0}=0.7~\mathrm{R}_{\odot}\)

In order to quantify the bias and variance of the travel-time and amplitude measurements in the present deep-focusing setup, we compute the travel-time and amplitude perturbations generated by sound-speed perturbations of our choosing (forward modeling).

Figure 9a shows the deep-focusing travel-time measurements \([\overline{\delta\tau}]\) and the corresponding noise levels (standard deviations) for different target locations \(\boldsymbol{r} _{t} = (0,0,z_{t})\), where \(0.4< z_{t}/\mathrm{R_{\odot}}<1\). Due to the hollow nature of the deep-focusing travel-time kernel, the signal is weaker at the depth where the perturbation is located than in the surroundings. The bulk of the perturbation is within \(| z_{0}-z_{t} | < L_{\tau}(z_{0}) /2 \approx 0.13~\mathrm{R}_{\odot}\). On the other hand, a maximum signal for the amplitude measurements is obtained at the radius where the perturbation is placed (Figure 9b) due to the concentrated sensitivity of the deep-focusing kernel for amplitude measurements (Figure 4b).

To compare the two types of measurements, travel-time *versus* amplitude measurements, the signal-to-noise ratios are plotted in Figure 9c. We find that the signal-to-noise ratio is higher and better localized for the amplitude measurements than for the travel-time measurements, given the highly localized perturbation in sound speed that we chose in this section.

### 4.7 Sound-Speed Anomaly in a Shell at Radius \(r_{0}=0.7~\mathrm{R}_{\odot}\)

The corresponding travel-time and amplitude perturbations, as well as the noise levels for \(T=\) four years, are shown in Figures 10a and 10b. We see that the travel-time and amplitude signals peak below \(z_{t}=0.7~\mathrm{R}_{\odot}\): the deep-focusing averaging scheme is not unbiased. For a shell-like perturbation, the signal-to-noise ratio is twice as large for the travel-time measurements as for the amplitude measurements (Figure 10(c)).

## 5 Conclusion

In this article we considered toy models in a uniform background medium to study the localization and noise properties of the deep-focusing time–distance technique. We considered two measurement quantities extracted from the cross-covariance function: travel times and amplitudes. The sensitivity kernels for sound speed were derived under the first Born approximation.

We computed the spatial sensitivity of travel-time and amplitude to perturbations in sound speed with respect to a uniform background medium. We find that the travel-time sensitivity to sound-speed perturbations is zero at the target location and negative in a surrounding region with diameter \(L_{\tau} \approx(\lambda_{0} D)^{1/2}\), where \(\lambda_{0}\) is the wavelength and \(D\) is the travel distance between the points used in the deep-focusing averaging. On the other hand, the amplitude sensitivity peaks at the target location and is negative in a region with diameter \(L_{a} \approx(\lambda_{0} D/2)^{1/2}\), resulting in a higher signal-to-noise ratio for small-scale perturbations. We conclude that amplitude measurements are a useful complement to travel-time measurements in local helioseismology.

In future studies, we intend to extend this work to a standard solar model using accurate computations of Green’s functions in the frequency domain. We also intend to study the capability of the deep-focusing technique to recover flows in the solar interior. Deep-focusing travel times have already been used to recover meridional circulation (*e.g.* Rajaguru and Antia, 2015). No significant improvement is expected from using deep-focusing amplitude measurements to recover such slowly varying flows. However, amplitude measurements should help resolve flows that vary on scales smaller than the wavelength, *e.g.* convective flows.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. We thank Thomas L. Duvall, Aaron C. Birch, Zhi-Chao Liang, Chris S. Hanson, and Kaori Nagashima for useful discussions and comments. MP is a member of the International Max Planck Research School (IMPRS) for Solar System Science at the University of Göttingen. The computer infrastructure was provided by the German Data Center for SDO funded by the German Aerospace Center (DLR) and by the Ministry of Science of the State of Lower Saxony, Germany.

## Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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