Travel-time Shifts Research Articles

Adaptive traveltime inversion

We have developed a method to obtain a misfit function for robust waveform inversion. In this method, called adaptive traveltime inversion (ATI), a matching filter that matches predicted data to measured data is computed. If the velocity model is relatively accurate, the resulting matching filter is close to a Dirac delta function. Its traveltime shift, which characterizes the defocusing of the matching filter, is computed by minimization of the crosscorrelation between a penalty function such as [Formula: see text] and the matching filter. ATI is constructed by minimization of the least-squares errors of the calculated traveltime shift. Further analysis indicates that the resulting traveltime shift corresponds to a first-order moment, the mean value of the resulting matching filter distribution. We extend ATI to a more general misfit function formula by computing different order moment of the resulting matching filter distribution. Choosing the penalty function in adaptive waveform inversion (AWI) as [Formula: see text], the misfit function of AWI is the second-order moment, the variance of the resulting matching filter distribution with zero mean. Because our ATI method is based on a global comparison using deconvolution, such as AWI, it can resolve the “cycle skipping” issue. We evaluate our ATI misfit function and compare it with state-of-the-art options such as least-squares inversion (L2 norm), wave-equation traveltime inversion, and AWI using schematic examples before moving to more complex examples, such as the Marmousi model. For the Marmousi model, starting with a 1D [Formula: see text] model, with data without low frequencies (no energy below 3 Hz), a meaningful estimation of the P-wave velocity model is recovered. Our ATI misfit function (first-order moment) indicates comparable performance with the AWI misfit function (the second-order moment). We also include a real data example from the Gulf of Mexico to demonstrate the effectiveness of our method.

Frequency Dependence of Helioseismic Measurements of the Center-to-Limb Effect and Flow-induced Travel-time Shifts

Abstract Time–distance helioseismology measures acoustic travel times to infer the structure and flow field of the solar interior; however, both the mean travel times and the travel-time shifts suffer systematic center-to-limb variations, which complicate the interpretation and inversions of the time–distance measurements. In particular, the center-to-limb variation in travel-time shifts (CtoL effect) has a significant impact on the inference of the Sun’s meridional circulation, and needs to be removed from the helioseismic measurements, although the observational properties and the physical cause of the CtoL effect have yet to be investigated. In this study, we measure the CtoL effect in the frequency domain using Doppler-velocity data from the Solar Dynamics Observatory/Helioseismic and Magnetic Imager, and study its properties as a function of disk-centric distance, travel distance, and frequency of acoustic waves. It is found that the CtoL effect has a significant frequency dependence—it reverses sign at a frequency around 5.4 mHz and reaches maximum at around 4.0 mHz before the sign reversal. The tendency of frequency dependence varies with disk-centric distance in a way that both the sign-reversal frequency and the maximum-value frequency decrease closer to the limb. The variation tendency does not change with travel distance, but the variation magnitude is approximately proportional to travel distance. For comparison, the flow-induced travel-time shifts show little frequency dependence. These observational properties provide more clues on the nature of the CtoL effect, and also possibly lead to new ways of effect-removal for a more robust determination of the deep meridional flow.

A Comprehensive Method to Measure Solar Meridional Circulation and the Center-to-limb Effect Using Time–Distance Helioseismology

Abstract Meridional circulation is a crucial component of the Sun’s internal dynamics, but its inference in the deep interior is complicated by a systematic center-to-limb effect in helioseismic measurement techniques. Previously, an empirical method, removing travel-time shifts measured for the east–west traveling waves in the equatorial area from those measured for the north–south traveling waves in the central meridian area, was used, but its validity and accuracy need to be assessed. Here, we develop a new method to separate the center-to-limb effect, , and the meridional-flow-induced travel-time shifts, , in a more robust way. Using 7-year observations of the Solar Dynamics Observatory/Helioseismic and Magnetic Imager, we exhaustively measure the travel-time shifts between two surface locations along the solar disk’s radial direction for all azimuthal angles and all skip distances. The measured travel-time shifts are a linear combination of and , which can be disentangled by solving the linear equation set. The is found to be isotropic relative to the azimuthal angle, and the is then inverted for the meridional circulation. Our inversion results show a three-layer flow structure, with equatorward flow found between about 0.82 and 0.91 R ☉ for low latitude areas and between about 0.85 and 0.91 R ☉ for higher latitude areas. Poleward flows are found below and above the equatorward flow zones, indicating a double-cell circulation in each hemisphere.

Simultaneous impedance inversion and interpretation for an offshore turbiditic reservoir

By using simultaneous impedance inversion, we obtained P- and S-wave impedance ([Formula: see text] and [Formula: see text]) volumes from angle stacks at a siliciclastic turbidite oil reservoir offshore northwest Australia. The ultimate goal was to interpret these elastic variables for fluid, porosity, and mineralogy. This is why an essential part of our workflow was finding the appropriate rock-physics model based on well data. The model-corrected S-wave velocity [Formula: see text] in the wells was used as an input to impedance inversion. The inversion parameters were optimized in small vertical sections around two wells to obtain the best possible match between the seismic impedances and the upscaled impedances measured at the wells. Special attention was paid to the seismically derived [Formula: see text] ratio because we relied on this parameter for hydrocarbon identification. Even after performing crosscorrelation between the angle stacks to correct for two-way traveltime shifts to align the stacks, these stacks did not indicate a coherent amplitude variation with angle (AVA) dependence. To deal with this common problem, we corrected the mid and far stacks by using the near and ultrafar stacks as anchoring points for fitting a [Formula: see text] AVA curve. This choice allowed us to match the seismically derived [Formula: see text] ratio with that predicted by the rock-physics model in the reservoir. Finally, the rock-physics model was used to interpret these [Formula: see text] and [Formula: see text] for the fluid, porosity, and mineralogy. The new paradigm in our inversion/interpretation workflow is that the ultimate quality control of the inversion is in an accurate deterministic match between the seismically derived petrophysical variables and the corresponding upscaled depth curves at the wells. Our interpretation is very sensitive to the inversion results, especially the [Formula: see text] ratio. Despite this fact, we were able to obtain accurate estimates of porosity and clay content in the reservoir and around it.

Helioseismic Holography of Simulated Sunspots: dependence of the travel time on magnetic field strength and Wilson depression.

Improving methods for determining the subsurface structure of sunspots from their seismic signature requires a better understanding of the interaction of waves with magnetic field concentrations. We aim to quantify the impact of changes in the internal structure of sunspots on local helioseismic signals. We have numerically simulated the propagation of a stochastic wave field through sunspot models with different properties, accounting for changes in the Wilson depression between 250 and 550 km and in the photospheric umbral magnetic field between 1500 and 3500 G. The results show that travel-time shifts at frequencies above approximately 3.50 mHz (depending on the phase-speed filter) are insensitive to the magnetic field strength. The travel time of these waves is determined exclusively by the Wilson depression and sound-speed perturbation. The travel time of waves with lower frequencies is affected by the direct effect of the magnetic field, although photospheric field strengths below 1500 G do not leave a significant trace on the travel-time measurements. These results could potentially be used to develop simplified travel-time inversion methods.

Experimental Validation of a Fast Forward Model for Guided Wave Tomography of Pipe Elbows.

Ultrasonic guided wave tomography (GWT) methods for the detection of corrosion and erosion damage in straight pipe sections are now well advanced. However, successful application of GWT to pipe bends has not yet been demonstrated due to the computational burden associated with the complex forward model required to simulate guided wave propagation through the bend. In a previous paper [Brath et al., IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 61, pp. 815-829, 2014], we have shown that the speed of the forward model can be increased by replacing the 3-D pipe bend with a 2-D rectangular domain in which guided wave propagation is formulated based on an artificially inhomogeneous and elliptically anisotropic (INELAN) acoustic model. This paper provides further experimental validation of the INLEAN model by studying the traveltime shifts caused by the introduction of shallow defects on the elbow of a pipe bend. Comparison between experiments and simulations confirms that a defect can be modeled as a phase velocity perturbation to the INLEAN velocity field with accuracy that is within the experimental error of the measurements. In addition, it is found that the sensitivity of traveltime measurements to the presence of damage decreases as the damage position moves from the interior side of the bend (intrados) to the exterior one (extrados). This effect is due to the nonuniform ray coverage obtainable when transmitting the guided wave signals with one ring array of sources on one side of the elbow and receiving with a second array on the other side.

The imprint of crustal density heterogeneities on regional seismic wave propagation

Abstract. Density heterogeneities are the source of mass transport in the Earth. However, the 3-D density structure remains poorly constrained because travel times of seismic waves are only weakly sensitive to density. Inspired by recent developments in seismic waveform tomography, we investigate whether the visibility of 3-D density heterogeneities may be improved by inverting not only travel times of specific seismic phases but complete seismograms.As a first step in this direction, we perform numerical experiments to estimate the effect of 3-D crustal density heterogeneities on regional seismic wave propagation. While a finite number of numerical experiments may not capture the full range of possible scenarios, our results still indicate that realistic crustal density variations may lead to travel-time shifts of up to ∼ 1 s and amplitude variations of several tens of percent over propagation distances of ∼ 1000 km. Both amplitude and travel-time variations increase with increasing epicentral distance and increasing medium complexity, i.e. decreasing correlation length of the heterogeneities. They are practically negligible when the correlation length of the heterogeneities is much larger than the wavelength. However, when the correlation length approaches the wavelength, density-induced waveform perturbations become prominent. Recent regional-scale full-waveform inversions that resolve structure at the scale of a wavelength already reach this regime.Our numerical experiments suggest that waveform perturbations induced by realistic crustal density variations can be observed in high-quality regional seismic data. While density-induced travel-time differences will often be small, amplitude variations exceeding ±10 % are comparable to those induced by 3-D velocity structure and attenuation. While these results certainly encourage more research on the development of 3-D density tomography, they also suggest that current full-waveform inversions that use amplitude information may be biased due to the neglect of 3-D variations in density.

Analysis of travel time and mode choice shift for non-work stops in commuting: case study of Beijing, China

This paper investigates travel time attributed to non-work stops in multi-purpose commuting trips. Travel time is explained by socio-demographics and spatial attributes, mode shift is also included to analyze the extra travel time, and its trade-off with activity time for four types of non-work stops—eating out, shopping, leisure/social activities, and family/personal/other. Data come from the “Daily Activity and Travel Survey of Beijing 2012”. Descriptive analysis shows that almost 20% of the multi-purpose commuting trips include a mode shift toward a more motorized transport mode than their direct counterparts. Regression results indicate that extra travel time due to detours is significantly related to the activity durations. Regardless of the activity type, longer work duration reduces the travel time. Furthermore, the interaction terms between personal/trip/spatial factors and activities durations show the impacts differ across the activity types. Gender difference is found only for eating out, which suggests that male commuters travel longer for the same amount of activity time. Concerning spatial factors, a greater mix of facilities near workplaces helps to reduce the extra travel time invested for a time unit of shopping and family/personal/other activities. Compared with public transit users, active mode users have shorter travel time for eating out. Timing, work duration and commuting duration as time-budget-related variables show negative impacts on the extra travel time for eating-out, shopping and family/personal/other activities. However, mode shift does not show any significant impact as the hypothesis proposed. Instead of rebalancing the trade-off between travel and activity time, commuters may relocate activity with an expanded opportunity choice set given by a higher mobility level.

HELIOSEISMIC HOLOGRAPHY OF SIMULATED SUNSPOTS: MAGNETIC AND THERMAL CONTRIBUTIONS TO TRAVEL TIMES.

Wave propagation through sunspots involves conversion between waves of acoustic and magnetic character. In addition, the thermal structure of sunspots is very different than that of the quiet Sun. As a consequence, the interpretation of local helioseismic measurements of sunspots has long been a challenge. With the aim of understanding these measurements, we carry out numerical simulations of wave propagation through sunspots. Helioseismic holography measurements made from the resulting simulated wavefields show qualitative agreement with observations of real sunspots. We use additional numerical experiments to determine, separately, the influence of the thermal structure of the sunspot and the direct effect of the sunspot magnetic field. We use the ray approximation to show that the travel-time shifts in the thermal (non-magnetic) sunspot model are primarily produced by changes in the wave path due to the Wilson depression rather than variations in the wave speed. This shows that inversions for the subsurface structure of sunspots must account for local changes in the density. In some ranges of horizontal phase speed and frequency there is agreement (within the noise level in the simulations) between the travel times measured in the full magnetic sunspot model and the thermal model. If this conclusion proves to be robust for a wide range of models, it would suggest a path toward inversions for sunspot structure.

Education

``We are on line of position 157-337. Will repeat this message on 6210 kilocycles. We are running north and south.” --Amelia Earhart on 2 July 1937 in her last transmission to Coast Guard cutter Itasca This was the last verified transmission from Amelia Earhart and her navigator Fred Noonan as they were crossing over the Pacific Ocean from Lae to Howland Island in an attempt to fly around the world along an equatorial route. Their disappearance, despite the most exhaustive maritime search in history, made them two of the most famous missing people in the world. Arguably, had their wreckage been rapidly discovered, the artifacts would have led to a clear understanding of what happened to them, and their loss would be much less notable in aviation history. Then and now, we accept failure as long as we learn from it. It never sits well with the public when the best technology, sound reasoning, and concerted effort fails to answer questions like, “Where are they?'' and “What happened to them?” Almost eighty years later, we are considerably more advanced. Long distance commercial airline travel is incredibly safe. Crew members can communicate with an air traffic control facility at any time. Using sophisticated satellite, inertial, radar, and VHF navigation equipment, airline crew and ground stations can determine an aircraft's precise position relatively easily. In fact, commercial aircraft have features which automatically communicate with ground stations at regular intervals. And yet in March of 2014, a state-of-the-art commercial jetliner carrying 239 souls disappeared over the Pacific, and once again, despite the best efforts of several nations, the crash site has yet to be discovered. Similar to the disappearance of Amelia Earhart almost 80 years ago, determining the most likely trajectory and final position is a complex mathematical problem. This Education section features a self-contained module by John Zweck that walks through the mathematics underlying the search for the crash site of MH370. Since MH370 flew for more than six hours after the last known communication between the pilot and air traffic control, the primary evidence in the search is the pings from the automated Aircraft Communications Addressing and Reporting System. Since these pings travel to a satellite and then to a ground station, analysis of ping travel time and Doppler shift provides some information about the aircraft position and speed. The article contains a very understandable derivation that maps the aircraft trajectory on a sphere and satellite positions to ping duration and Doppler shift. Each model for the aircraft's trajectory needs to be reconciled with actual data to determine model parameters, ultimately leading to a classic optimization problem. Using the ideas in this article together with official data from MH370, one can compute something very close to the priority search area used by the Australian Transportation Safety Board, the organization that is coordinating the underwater search. In the time since this article was received, wreckage from MH370 has washed ashore on Réunion Island and later other scattered locations, consistent with Lagrangian trajectories from the priority search area. The underwater search still continues and hopefully will be more successful than the search for Earhart's missing Lockheed Electra 10E, bringing closure to the families of the victims and to a public that still wonders what happened. This article is self-contained and appropriate for an advanced undergraduate course or a project. One interesting feature of the official inquiry into the MH370 disappearance is that the investigation team publicly released the data and their reasoning. This allowed many groups to explore the data and verify the official determination of the most likely trajectory. In addition to the scientific and public relations benefits to this approach, the release of the data had the unforeseen benefit of allowing exciting problems like these to quickly enter the classroom and inspire students with compelling problems.

Effects of granulation on the visibility of solar oscillations

Context. The interaction of solar oscillations with near surface convection is poorly understood. These interactions are likely the cause of several problems in helio- and astero-seismology, including the so-called surface effect and apparently unphysical travel time shifts as a function of center to limb distance. There is thus a clear need for further theoretical understanding and observational tests. Aims. The aim is to determine how the observed modes are affected by the convection. Methods. I use HMI velocity and intensity images to construct k-$\omega$ diagrams showing how the oscillation amplitude and phase depend on the local granulation intensity. Results. There is a clear and significant dependence of the observed properties of the oscillations on the local convection state.

Localized temporal variation of Earth’s inner-core boundary from high-quality waveform doublets

The topography of the inner core is crucial to understand its growth process and interaction with the geodynamo. With the accuracy of teleseismic waveform doublets in determining the travel-time shifts between PKPcd and PKPdf inner-core phases, we examined the temporal change of the inner-core boundary sampled by new earthquake doublets that occurred in the Western Pacific and those previously found from the South Sandwich Islands. The receiver stations are those within the distance range of 128°–142° from the hypocenters of the waveform doublets. Our results suggest that temporal changes in PKPcd-PKPdf differential times are very subtle except some isolated regions under central America and Africa. The observations may indicate localized topography of the inner core and/or the inner-core boundary as transient slurry in isolated regions.

Inversion of multicomponent seismic time shifts for reservoir pressure and length: a feasibility study

Pressure drops associated with reservoir production generate excess stress and strain that cause travel-time shifts of reflected waves. Here,we invert time shifts of P-, S-, and PS-waves measured between baseline and monitor surveys for pressure reduction and reservoir length. The inversion results can be used to estimate compaction-induced stress and strain changes around the reservoir. We implement a hybrid inversion algorithm that incorporates elements of gradient, global/genetic, and nearest neighbour methods and permits exploration of the parameter space while simultaneously following local misfit gradients. Our synthetic examples indicate that optimal estimates of reservoir pressure from P-wave data can be obtained using the reflections from the reservoir top. For S-waves, time shifts from the top of the reservoir can be accurately inverted for pressure if the noise level is low. However, if noise contamination is significant, it is preferable to use S-wave data (or combined shifts of all three modes) from reflectors beneath the reservoir. Joint wave type inversions demonstrate improvements over any single pure mode. Reservoir length can be estimated using the time shifts of any mode from the reservoir top or deeper reflectors. We also evaluate the differences between the actual strain field and those corresponding to the best-case inversion results obtained using P- and S-wave data. Another series of tests addresses the inversion of the time shifts for the pressure drops in two-compartment reservoirs, as well as for the associated strain field. Numerical testing shows that a potentially serious source of error in the inversion is a distortion in the strain-sensitivity coefficients, which govern the magnitude of stiffness changes. This feasibility study suggests which wave types and reflector locations may provide the most accurate estimates of reservoir parameters from compaction-induced time shifts.

Analysis of time-lapse travel-time and amplitude changes to assess reservoir compartmentalization

Fluid depletion within a compacting reservoir can lead to significant stress and strain changes and potentially severe geomechanical issues, both inside and outside the reservoir. We extend previous research of time-lapse seismic interpretation by incorporating synthetic near-offset and full-offset common-midpoint reflection data using anisotropic ray tracing to investigate uncertainties in time-lapse seismic observations. The time-lapse seismic simulations use dynamic elasticity models built from hydro-geomechanical simulation output and a stress-dependent rock physics model. The reservoir model is a conceptual two-fault graben reservoir, where we allow the fault fluid-flow transmissibility to vary from high to low to simulate non-compartmentalized and compartmentalized reservoirs, respectively. The results indicate time-lapse seismic amplitude changes and travel-time shifts can be used to qualitatively identify reservoir compartmentalization. Due to the high repeatability and good quality of the time-lapse synthetic dataset, the estimated travel-time shifts and amplitude changes for near-offset data match the true model subsurface changes with minimal errors. A 1D velocity–strain relation was used to estimate the vertical velocity change for the reservoir bottom interface by applying zero-offset time shifts from both the near-offset and full-offset measurements. For near-offset data, the estimated P-wave velocity changes were within 10% of the true value. However, for full-offset data, time-lapse attributes are quantitatively reliable using standard time-lapse seismic methods when an updated velocity model is used rather than the baseline model.

Time-Distance Seismology and the Solar Transition Region

Time-Distance ‘travel time’ perturbations (as inferred from wave phase) are calculated relative to the quiet-Sun as a function of wave orientation and field inclination in a uniform inclined magnetic field. Modelling indicates that the chromosphere-corona Transition Region (TR) profoundly alters travel times at inclinations from the vertical θ for which the ramp-reduced acoustic cutoff frequency ωccosθ is similar to the wave frequency ω. At smaller inclinations phase shifts are much smaller as the waves are largely reflected before reaching the TR. At larger inclinations, the shifts resume their quiet-Sun values, although with some resonant oscillatory behaviour. Changing the height of the TR in the model atmosphere has some effect, but the thickness and temperature jump do not change the results substantially. There is a strong correspondence between travel-time shifts and the Alfven flux that emerges at the top of the modelled region as a result of fast/Alfven mode conversion. We confirm that the TR transmission coefficient for Alfven waves generated by mode conversion in the chromosphere is far larger (typically 30 % or more) than for Alfven waves injected from the photosphere.

Least-squares migration in the presence of velocity errors

Seismic migration requires an accurate background velocity model that correctly predicts the kinematics of wave propagation in the true subsurface. Least-squares migration, which seeks the inverse rather than the adjoint of a forward modeling operator, is especially sensitive to errors in this background model. This can result in traveltime differences between predicted and observed data that lead to incoherent and defocused migration images. We have developed an alternative misfit function for use in least-squares migration that measures amplitude differences between predicted and observed data, i.e., differences after correcting for nonzero traveltime shifts between predicted and observed data. We demonstrated on synthetic and field data that, when the background velocity model is incorrect, the use of this misfit function results in better focused migration images. Results suggest that our method best enhances image focusing when differences between predicted and observed data can be explained by traveltime shifts.

EVALUATION OF THE CAPABILITY OF LOCAL HELIOSEISMOLOGY TO DISCERN BETWEEN MONOLITHIC AND SPAGHETTI SUNSPOT MODELS.

The helioseismic properties of the wave scattering generated by monolithic and spaghetti sunspots are analyzed by means of numerical simulations. In these computations, an incident f- or p1-mode travels through the sunspot model, which produces absorption and phase shift of the waves. The scattering is studied by inspecting the wavefield, computing travel-time shifts, and performing Fourier-Hankel analysis. The comparison between the results obtained for both sunspot models reveals that the differences in the absorption coefficient can be detected above noise level. The spaghetti model produces a steep increase of the phase shift with the degree of the mode at short wavelengths, while mode mixing is more efficient for the monolithic model. These results provide a clue for what to look for in solar observations to discern the constitution of sunspots between the proposed monolithic and spaghetti models.

Using a pseudo-steady-state flow equation and 4D seismic traveltime shifts for estimation of pressure and saturation changes

Pressure and saturation changes cause a change in seismic P-wave velocity resulting in a shift in 4D traveltime. A discrimination between both effects is important to enhance reservoir management decisions. We explored the possibility of linking the fields of reservoir engineering and 4D seismic. A simple inversion scheme based on a reservoir engineering flow equation for pore pressure predictions combined with the Hertz-Mindlin and Gassmann rock-physics models to discriminate pressure and saturation effects from 4D time shifts was presented. To account for no-flow reservoir boundaries, the pseudo-steady-state flow equation giving the pressure distribution was extended with a superpositioning method. During the inversion, three parameters were determined by a nonlinear, least-squares fitting method: the thickness of the hydrocarbon column, the pressure at the well position, and its decay rate away from the well. We successfully tested our approach on a synthetic and a field data case, where the [Formula: see text] injection well at the Snøhvit field, Norway, served as a good example to demonstrate our method. We observed a good correlation between simulation and inversion in our synthetic study. Our theory was limited to homogeneous reservoir conditions and produces spatially low frequent outputs. To apply the model in complex fields, such as at Snøhvit, we introduced a spatially varying Mindlin-exponent over the field and used the theory as an heuristic model assuming that a variation of the Mindlin-exponent takes a change in pressure sensitivity into account. This assumption provided the possibility to indirectly include heterogeneities in grain sorting and porosity variation and improved the model significantly. We see the advantage of our method in its fast and direct implementation to study first-order effects of pressure and saturation behavior on time-lapse seismic data using a simple inversion algorithm instead of computationally intensive simulations.

The Sagnac effect in conformal Weyl gravity

We investigate the Sagnac effect, representing the difference in travel time or phase shift observed for relativistic matter or light beams counter-propagating in a rotating interferometer. This is done for the Schwarzschild-like and Kerr-like vacuum solutions of conformal gravity which describe the field around a massive static and rotating object respectively, in the fourth order theory. To do this we employ the formal analogy between the Sagnac and Aharonov–Bohm effects that has been used earlier to study this effect in flat and curved spacetimes in general relativity. In particular we show that the linear term in the static metric of conformal gravity has a diminishing effect on the Sagnac time delay. Moreover using the expression for the Sagnac effect in the rotating Kerr-like spacetime, we show that the detection of the Weyl gravity contributions to the Sagnac time delay in the case of the Earth, lies beyond the capability of current experiments.

VERIFICATION OF THE HELIOSEISMOLOGY TRAVEL-TIME MEASUREMENT TECHNIQUE AND THE INVERSION PROCEDURE FOR SOUND SPEED USING ARTIFICIAL DATA

We performed three-dimensional numerical simulations of the solar surface acoustic wave field for the quiet Sun and for three models with different localized sound-speed perturbations in the interior with deep, shallow, and two-layer structures. We used the simulated data generated by two solar acoustics codes that employ the same standard solar model as a background model, but utilize different integration techniques and different models of stochastic wave excitation. Acoustic travel times were measured using a time-distance helioseismology technique, and compared with predictions from ray theory frequently used for helioseismic travel-time inversions. It is found that the measured travel-time shifts agree well with the helioseismic theory for sound-speed perturbations, and for the measurement procedure with and without phase-speed filtering of the oscillation signals. This testing verifies the whole measuring-filtering-inversion procedure for static sound-speed anomalies with small amplitude inside the Sun outside regions of strong magnetic field. It is shown that the phase-speed filtering, frequently used to extract specific wave packets and improve the signal-to-noise ratio, does not introduce significant systematic errors. Results of the sound-speed inversion procedure show good agreement with the perturbation models in all cases. Due to its smoothing nature, the inversion procedure may overestimate sound-speed variations in regions with sharp gradients of the sound-speed profile.