One-dimensional Velocity Structure Research Articles

Estimation of shallow structure along the hinagu fault by applying seismic interferometry to DAS observations conducted along national route 3 in Kumamoto prefecture, Japan

Distributed acoustic sensing (DAS) is a newly developed geophysical observation method and has attracted wide attention in seismology for realizing ultra-high-density observations. DAS uses fiber-optic cables and measures the strain at every point along the cable. This advantage renders DAS an effective tool for investigating near-surface geotechnical properties. Near fault zones, it is important to obtain detailed geotechnical information in advance because of the potential for significant damage in an earthquake. In this study, we recorded continuous ground motion for approximately 1 month using a 40 km-long fiber-optic communication cable running under National Route 3 in Kumamoto Prefecture, Japan. The cross-correlation function (CCF) was calculated using ambient noise, and three-station interferometry was applied to improve the signal-to-noise ratio of the CCF. Using the reconstructed CCF between channels, we calculated the dispersion curves by conducting multichannel surface wave analysis and estimated the one-dimensional velocity structure of each section from the fundamental modes of the dispersion curves. We obtained the detailed shallow S-wave velocity structure to a depth of 180 m along the Hinagu Fault for approximately 2.5 km. The obtained velocity structure showed that the low-velocity region increased abruptly with depth from the center to the latter half of the analyzed section. This velocity change occurs when the national highway running parallel to the fault gradually leaves the fault, suggesting a structural change from solid volcanic layers to thick shallow sedimentary layers derived from the Yatsushiro Plain.Graphical abstract

A 3D computational model for ground motion simulation in Peninsular India

Due to the gradual and constant accumulation of seismic energy, Peninsular India (PI) is typically considered seismically stable with low to moderate seismicity. The seismic studies in Peninsular India always resorted to synthetic ground motion simulations, because of the limited instrumentation and hence lack of recorded data. In the absence of a well-defined medium model for PI, the usual practice is to use simple site proxies or one-dimensional velocity structures for ground motion simulations. However, the region consists of multi-scale geometric complexities, significant topography, and sedimentary basins and is surrounded by deep oceans. Thus, the radiated seismic wave field in the region is influenced by the medium properties and in the absence of a well-defined tomography model the reliable estimation of seismic hazard is a challenging problem in PI. Therefore, the seismic wave propagation in PI can be investigated using numerical simulation with reliable 3D computational model for PI, incorporating the knowledge of the underlying Earth structure. Hence, the present study attempts to develop a sophisticated three-dimensional (3D) medium model of Peninsular India for physics-based ground motion simulations for regional earthquakes. This is aided by the availability of one-dimensional (1D) velocity models and the crustal structure from the receiver function analysis which provides valuable insight into the variation of material properties in the region. In the present study, >100 s of 1D velocity profiles are collected from various literature, which is then grouped under 23 different geological regions identified in PI (as per GSI (2000)). The averaged material properties are assigned per each geological region and the information on sediment depths, basin geometry, topography, and bathymetry are incorporated. We use the spectral element method (SEM) to calibrate our 3D computational model by simulating synthetic seismograms and comparing them to recorded ground motions for two past earthquakes: the 2001 Mw 7.6 Bhuj earthquake and the 1997 Mw 5.8 Jabalpur earthquake. Further, the seismic waveforms at the near field of 2001 Mw 7.6 Bhuj event are simulated using a refined regional model. The spatial variability of associated seismic intensities and peak ground velocity (PGV) amplification are investigated. In addition, a study of the impact of model depth truncation and sphericity on ground motion is also conducted. The implemented medium model is the first of its kind for Peninsular India and can reliably be used in seismic wave propagation studies in the region. The simulated outcomes from the model are of engineering importance as these results can be used for seismic hazard assessment of the region.

Active and passive seismic methods for site characterization in Nuweiba, Gulf of Aqaba, Egypt

Nuweiba is located on the western coast of the Gulf of Aqaba (GoA), which represents the southern part of the Gulf of Aqaba - Dead Sea transform fault (GoA-DST). The GoA is characterized by high seismic activity and is the source of the November 22, 1995, Nuweiba earthquake, with a moment magnitude of Mw 7.3. This seismic unrest caused extensive damage in the city of Nuweiba due to the strong shaking caused by the earthquake. In addition, soil amplification triggered by the thick soft-sediments of the Wadi Watir delta beneath the city amplified the destructive power of the event. Therefore, an assessment of the effect of the Wadi Watir sediments on the ground motion at Nuweiba is very important for future construction, seismic hazard assessment, and seismic risk mitigation. Accordingly, the goal of the current study is to define the amplification characteristics of the Wadi Watir deposits using microtremor and surface wave data sets. The horizontal-to-vertical spectral ratio (HVSR) technique is applied to the microtremor data and Multichannel Analysis of Surface Waves (MASW) is applied to the surface wave data. The data are processed to obtain the distribution of the fundamental resonance frequency (F0), the depth of the geophysical bedrock using joint inversion of the Rayleigh wave dispersion curve and ellipticity peak, the shear wave velocity structure, the time-averaged shear-wave velocity in the topmost 30 m (Vs30), and the shear wave transfer function. The results of the HVSR show that F0 in the northern part of the city is 4.2–5.2 Hz and 0.5–0.8 Hz in the middle of the city. Bedrock depth is found to be in the range of 27 and 199 m. The one-dimensional velocity structure obtained from the MASW measurements has resolved two sedimentary layers within the upper 15 m. The Vs30 ranges between 207 and 556 m/s, according to the Vs30 the soil in the study area is categorized into soil classes B and C according to Eurocode 8. The obtained shear wave transfer functions show that the highest amplification factors are observed in the middle part of the study area.

An Algorithm of Acoustic Emission Location for Complex Composite Structure

Acoustic emission (AE) is widely used in engineering and rock mechanics. The algorithm of AE location based on homogeneous medium or single velocity structure is confronted with lower accuracy when it is applied to the actual working conditions that are prevailing complicated and heterogeneous. In this paper, an AE location algorithm based on complex composite structure (CCS) is proposed via carrying out the following studies: (1) A new travel time calculation scheme suitable for CCS with step-like velocity change is proposed based on an optimized shortest path algorithm. By doing this, a reasonable ray path that is only deflected at the interface is obtained to improve the travel time accuracy. The time complexity of the new algorithm is O(nlog2n). (2) The availability of the new algorithm is verified via a theoretical analysis under a one-dimensional velocity structure as well as an AE experiment using a complex structure under artificial excitation. (3) The AE location during the failure of a simulated CCS in the laboratory indicates that the new algorithm can effectively calculate the travel time and ray path of the sample.

One-dimensional velocity structure modeling of the Earth's crust in the northwestern Dinarides

Abstract. The studied area of the northwestern (NW) Dinarides is located in the northeastern (NE) corner of the Adriatic microplate and is bordered by the Adriatic foreland, the Southern Alps, and the Pannonian basin. Its complex crustal structure is the result of interactions among different tectonic units, the most important of which are the Eurasian plate and the Adriatic microplate. Despite numerous seismic studies in this tectonically complex area, there is still a need for a detailed, small-scale study focusing mainly on the upper, brittle part of the crust. In this work, we investigated the velocity structure of the crust with one-dimensional (1-D) simultaneous hypocenter–velocity inversion using routinely picked P- and S-wave arrival times. Most of the models computed in the combined P and S inversion converged to a stable solution in the depth range between 0 and 26 km. We further evaluated the inversion results with hypocenter shift tests, high- and low-velocity tests, and relocations. This helped us to select the best performing velocity model for the entire study area. Based on these results and the seismicity distribution, we divided the study area into three subregions, reselected earthquakes and stations, and performed the combined P and S inversion for each subregion separately to gain better insight into the crustal structure. In the eastern subregion, the P velocities in the upper 8 km of the crust are lower compared to the regional velocities and the velocities of the other two subregions. The P velocities between 8 and 23 km depth are otherwise very similar for all three models. Conversely, the S velocities between 2 and 23 km depth are highest in the eastern subregion. The NW and southwestern (SW) subregions are very similar in terms of the crustal structure between 0 and 23 km depth, with slightly higher P velocities and lower S velocities in the SW subregion. High vP/vS values were obtained for the layers between 0 and 4 km depth. Below that, no major deviations of vP/vS in the regional model from the value of 1.73 are observed, but in each subregion we can clearly distinguish two zones separated by a decrease in vP/vS at 16 km depth. Compared to the model currently used by the Slovenian Environment Agency to locate earthquakes, the obtained velocity models show higher velocities and agree very well with some of the previous studies. In addition to the general structural implications and the potential to improve the results of seismic tomography, the new 1-D P and S velocity models can also be used for reliable routine earthquake location and for detecting systematic travel time errors in seismological bulletins.

Comparison of crustal structures in the fossil spreading center of South China Sea basins and the tectonic significance

This study focuses on the magmatic activities and seamounts in the fossil spreading center of three sub-basins in the South China Sea (SCS). The northwest sub-basin (NWSB), southwest sub-basin (SWSB), and east sub-basin (ESB) are of great importance for understanding the tectonic evolution and post-spreading magmatic activities of the SCS. Deep crustal structures beneath the spreading center of these basins have a significant bearing on the magmatic process. For this study, on the basis of a previous velocity model, we used two-dimensional ray-tracing methods and high-quality ocean-bottom seismometer (OBS) data collected in 2006 to image the NWSB deep P-wave velocity structures. We then performed travel-time analysis and estimated uncertainties of the final model parameters to prove that the velocity model was convincing. Finally, we compared the one-dimensional velocity structures among OBS profiles 2006-p1 in the NWSB, G8G0 in the ESB, and T1 in the SWSB. Our study of crustal structure yielded the following results: (1) the NWSB is confirmed to be oceanic crust compared with typical Atlantic oceanic crust. The crustal velocity is 5.7–7.0 km/s, whereas the crustal thickness is 6–8 km in the basin (90–130 km). There is no high-velocity layer. The Mohorovicic discontinuity (Moho) is distributed symmetrically along the spreading axis and gradually increases from the basin center to the edge. The depth of the Moho is 11–12 km in the basin (55–70 km) and changes dramatically to 16 km depth under the north margin (170–200 km) and the Zhongsha Islands (0–55 km). The depth of the Moho in all three sub-basins increases by 1 km in the spreading center. This suggests that crustal structure in the SCS is greatly influenced by magmatic activities. The Shuangfeng seamount, with a 6.5-km-thick crust, located in the center of the spreading ridge, is a post-spreading volcanic seamount. The 6.4-km/s velocity contour beneath the Shuangfeng seamount gradually rises toward the northwest part, and the velocity at the seamount bottom is lower than 7.4 km/s. Therefore, we speculate that the magma underplating does not exist. Consistent with multichannel seismic date, we conclude that the Shuangfeng seamount was mainly formed by an intrusive process: The magma intruded along the former extinct spreading axis. The Longnan seamount in the SWSB and the Jixiang seamount in the ESB were mainly formed by an eruption process, supposedly the result of the buoyant decompression melting mechanism. (2) The detailed velocity model of the NWSB shows that the oceanic crust exhibits obvious across-axis velocity variations. The lateral variations in crustal thickness and velocity are also found in the ESB and SWSB. This shows that the previously formed oceanic crust in all three SCS sub-basins are reformed by post-spreading magma. The difference in crustal velocity of the NWSB indicates that the magma supply varied in space during or even after the seafloor spreading. The OBS 2006-p1 velocity model indicates that the velocity contours gradually rise from the southeast to northwest (140–160 km) direction. The crust layer may have experienced asymmetric basalt magma flow, and the northwest part of the basin may have received more magma than the southeast part during the process of seafloor spreading. Low crustal velocities were found in the SWSB and ESB but not in the NWSB. This indicates an inadequate supply of magma in the NWSB. Compared with the other two sub-basins, the Cenozoic tectonic activities in the NWSB were less active. The residual magma might have been consumed after the NWSB spreading. Either there are no magma chambers in the NWSB spreading center, or the magma chambers are too small to detect.

One-Dimensional Velocity Structure of the Crust in Fujian, Southeast China

12095 P-wave phase data were selected in this study from four NW-oriented and four NE-oriented explosion sounding survey lines conducted along the coast and midland of Southeast China during the years 2010 and 2012. The 1-D crust P-wave velocity model was obtained in the continental margin of Southeast China (Fujian Province) using the travel time residual as the threshold and performing linear iterative inversion. This crust model includes 5 layers with the velocities being 5.04, 5.44, 6.06, 6.16, 6.39 km s^(-1), respectively, with the bottom depths being 0.23, 2.82, 6.44, 18.81, 30.42 km, respectively, and the uppermost mantle velocity being 8.08 km s^(-1). Compared with previous work the four P-wave phase data could effectively reflect the shallow and deep crust characteristics. The joint inversion method involves both the velocity and depth. The results in this paper could therefore be more reasonable and applicable than previous findings, with fairly good control in both the shallow and deep crusts. These findings have practical significance for compiling the earthquake travel time table and precisely locating earthquakes in this area. This work also provides an accurate preliminary model for subsequent 2-D and 3-D velocity structure inversions in the Southeast China (Fujian Province) continental margin.

Rupture Process of the 1964 MJMA 7.5 Niigata Earthquake Estimated from Regional Strong-Motion Records

The source rupture process of the 1964 Niigata earthquake ( M JMA 7.5), Japan, is inferred in detail from regional strong-motion data recorded with low-gain mechanical displacement seismographs. A fault-plane model with high-angle dipping to the west is assumed as a result of reinvestigating the aftershock distribution, and the thickness of the fault zone is determined from recent high-resolution seismicity in the source region. To develop appropriate Green’s functions, a one-dimensional velocity structure model for each station is constructed by modeling the waveforms from recent small earthquakes in the source region. A multiple-time-window technique is applied to invert the observed waveforms for the spatial and temporal slip distribution. The obtained source model indicates large slips mainly in the northern part of the fault plane and secondary slips in the vicinity of the hypocenter and slightly expanding to the south. The total moment release reached ![Graphic][1] ( M w 7.6) and the rupture velocity of the first time-window sweep was 1.5 km/s. The static stress drop was estimated to be 8.9 MPa by assuming a circular fault model. The crustal deformation estimated from the slip distribution agrees well with the tsunami-generating area, the sea-bottom movement, and the uplift measurement at Awashima Island, located above the fault plane. [1]: /embed/inline-graphic-1.gif

Fine fracture structures in the geothermal region of Hakone volcano, revealed by well-resolved earthquake hypocenters and focal mechanisms

We investigated precise hypocentral distribution and the mechanisms of small earthquakes in the geothermal region of Hakone volcano, central Japan, where swarm activity has been frequently observed. Earthquake swarms were remarkably prevalent in 2001 and 2006, and were accompanied by crustal deformation. We first determined initial hypocenters, using station corrections and one-dimensional velocity structures as estimated by the joint hypocenter determination method. Then, we applied the double-difference method to relocate the initial hypocenters using the differential arrival time obtained by both manual picking and waveform cross-correlation analysis. Subsequently, we determined the focal mechanisms from the absolute P- and SH-wave amplitudes and P-wave polarities. From the relocated hypocenter distribution, we found that most swarm earthquakes are distributed on thin plane-like zones with width/length of 100 m to 1 km. We found that these plane-like hypocentral distributions range from the E–W to N–S strikes and are consistent with one of the nodal planes of the focal mechanism. It is likely that most swarm earthquakes occurred on pre-existing fracture planes, shaping the plane-like hypocenter distributions. Previous studies suggested that a highly permeable fracture system has developed in the fault interaction area, and these highly permeable fractures affect the occurrence of swarm earthquakes and surface geothermal activity. Since Hakone volcano is located in the interaction area of the active Tanna and Hirayama faults, it is suggested that the fracture planes revealed by the relocated hypocenter distribution have been developed by the activity of the two faults. The fracture planes in the caldera might be channels for the hydrothermal water of hot springs in the caldera of Hakone volcano which are rich in sodium chloride and considered to be composed of fluid heated by a deep-seated magma. The strike of the fracture plane on which the swarm earthquakes of the 2001 activity occurred is inconsistent with the orientation of open crack models estimated from geodetic data. It is therefore possible that the hydrothermal fluid aseismically intruded the open cracks during the 2001 activity.

10.1007/s11486-008-1008-z

The deep structure of the upper mantle is determined from data on phase velocities of Love and Rayleigh waves measured by a differential method on traces between two stations in central Western Europe. One-dimensional velocity structures are first constructed from data of each pair of stations, after which two-dimensional distributions of SH and SV velocities are calculated by the method of two-dimensional tomography from S wave velocities at fixed depths. The results are presented in the form of 2-D vertical structures of the average S wave velocity (S = (SV + SH)/2) constructed along profiles crossing the region in directions of the best resolution. The main structural features are a higher velocity zone at depths of 60–80 km in the area (48°–50°N, 9°–11°E) and a lower velocity zone in the western part of the region at depths of 100–150 km, probably extending farther beyond the studied area.

Source Rupture Process of the 2004 Chuetsu, Mid-Niigata Prefecture, Japan, Earthquake Inferred from Waveform Inversion with Dense Strong-Motion Data

Abstract The kinematic source rupture process of the 2004 Chuetsu, mid-Niigata prefecture, Japan, earthquake, is estimated from strong-motion data by the linear waveform inversion method. In order to develop appropriate Green’s functions, one-dimensional velocity structure models for each station are constructed by modeling the aftershock waveforms. The estimated top of the bedrock is deeper at the western side of the fault and relatively shallow at the eastern side. This tendency coincides with other investigations of seismic tomography and microtremor array observations. The obtained source model shows large slips in the vicinity of the hypocenter. The reverse slipping rupture originated from the deeper part of the fault and propagated toward the up-dip and southwest directions. The near-surface slip is small. Two tests are demonstrated to check the stability of the obtained source model. The first test examines the effects of the number of available stations on the solution, and the second test examines how the difference in the target waveform type affects the solution. The number of stations affects the variation of slip amount, and more than 12 stations appear to be sufficient to obtain a stable solution in this case. The difference in the target wave type in the data set does not significantly affect the solution if the number of stations is sufficient and the Green’s functions are well calibrated. The model obtained using calibrated velocity structure models shows clear image of high-slip area compared to the model obtained using a single velocity structure model.

Upper mantle structure beneath central Western Europe from data on phase velocities of surface waves

The deep structure of the upper mantle is determined from data on phase velocities of Love and Rayleigh waves measured by a differential method on traces between two stations in central Western Europe. One-dimensional velocity structures are first constructed from data of each pair of stations, after which two-dimensional distributions of SH and SV velocities are calculated by the method of two-dimensional tomography from S wave velocities at fixed depths. The results are presented in the form of 2-D vertical structures of the average S wave velocity (S = (SV + SH)/2) constructed along profiles crossing the region in directions of the best resolution. The main structural features are a higher velocity zone at depths of 60–80 km in the area (48°–50°N, 9°–11°E) and a lower velocity zone in the western part of the region at depths of 100–150 km, probably extending farther beyond the studied area.

A Review of Receiver Function Techniques for Estimation of One-Dimensional Velocity Structures

Two receiver function techniques both broadly used by seismologists to estimate one-dimensional shear wave velocity structures of the crust and upper mantle beneath seismic stations have been evaluated. One employs a deconvolution filtering, which is directly accomplished in the time domain (Vinnik, 1977; Kosarev et al., 1987). The other is completed through a source-equalization, which is performed by spectra division in the frequency domain (Langston, 1977; Owens et al., 1984). In this study, the performance of these techniques is examined from two sets of synthetic seismograms that are computed from two one-dimensional models by using Thomson and Haskell’s method (Haskell, 1962). The results suggest that both techniques can mostly recover the assumed models very well when the energy of multiples (or crustal reverberations) in the direction of P-wave propagation is minor. If the model is more complicated or the multiple energy becomes stronger, however, the results from Vinnik-Kosarev technique appear to be better than the other in both modeling the converted P-SV phases and inverting the structures. Combining the results of the synthetic tests and theoretic comparisons, it is concluded that the differences resulting from both techniques are primarily caused by the processes through different domains. Besides, both techniques consistently indicate that the inversion results are dependent upon the incident angles. For the nearly vertical incidence, neither technique could resolve the models very well.

Preliminary study of crust-upper mantle structure of the Tibetan Plateau by using broadband teleseismic body waveforms

As part of a joint Sino-U.S. research project to study the deep structure of the Tibetan Plateau, 11 broadband digital seismic recorders were deployed on the Plateau for one year of passive seismic recording. In this report we use teleseimic P waveforms to study the seismic velocity structure of crust and upper mantle under three stations by receiver function inversion. The receiver function is obtained by first rotating two horizontal components of seismic records into radial and tangential components and then deconvolving the vertical component from them. The receiver function depends only on the structure near the station because the source and path effects have been removed by the deconvolution. To suppress noise, receiver functions calculated from events clustered in a small range of back-azimuths and epicentral distances are stacked. Using a matrix formalism describing the propagation of elastic waves in laterally homogeneous stratified medium, a synthetic receiver function and differential receiver functions for the parameters in each layer can be calculated to establish a linearized inversion for one-dimensional velocity structure. Preliminary results of three stations, Wen-quan, Golmud and Xigatze (Coded as WNDO, TUNL and XIGA), located in central, northern and southern Plateau are given in this paper. The receiver functions of all three stations show clear P-S converted phases. The time delays of these converted phases relative to direct P arrivals are: WNDO 7.9s (for NE direction) and 8.3s (for SE direction), TUNL 8.2s, XIGA 9.0s. Such long time delays indicate the great thickness of crust under the Plateau. The differences between receiver function of these three station shows the tectonic difference between southern and north-central Plateau. The waveforms of the receiver functions for WNDO and TUNL are very simple, while the receiver function of XIGA has an additional midcrustal converted phase. The S wave velocity structures at these three stations are estimated from inversions of the receiver function. The crustal shear wave velocities at WNDO and TUNL are vertically homogeneous, with value between 3.5–3.6 km/s down to Moho. This value in the lower crust is lower than the normal value for the lower crust of continents, which is consistent with the observed strong Sn attenuation in this region. The velocity structure at XIGA shows a velocity discontinuity at depth of 20 km and high velocity value of 4.0 km/s in the midcrust between 20–30 km depth. Similar results are obtained from a DSS profile in southern Tibet. The velocity under XIGA decreases below a depth of 30 km, reaching the lowest value of 3.2 km/s between 50–55 km. depth. This may imply that the Indian crust underthrusts the low part of Tibetan crust in the southern Plateau, forming a “double crust”. The crustal thickness at each of these sites is: WNDO, 68 km; TUNL, 70 km; XI-GA, 80 km.

Derivation of analytic generalised inverses for transmission+reflection tomography

A methodology is presented for deriving analytic generalised inverses for arbitrary combinations of transmission and reflection traveltime data. These inverses are valid for negligible ray bending and one-dimensional velocity structure. The following conclusions are drawn from this analysis. (1) The slowness variance of a layer decreases with an increase in the number and length of rays which terminate in it. (2) The slowness variance associated with offset transmission VSP data will always be less than that associated with VSP reflection data. (3) The slowness covariance matrix for VSP transmission data is tridiagonal, i.e. the slowness errors in the ith layer only couple to those from the (i+1)th and (i-1)th layers. On the other hand, reflection VSP data will introduce a mutual coupling term between the slowness errors in the reflecting layer and the surface layer. (4) The slowness variance associated with transmission+reflection VSP data will generally be less than that from reflection or transmission VSP data. Moreover, the slowness covariance matrix for transmission+reflection data will be fully populated, meaning that there will be global coupling of slowness errors. However, if the segment lengths of reflecting rays are somewhat similar then the covariance matrix will be essentially tridiagonal. It is concluded from this analysis that the combination of reflection data with transmission data should most often improve the accuracy of slowness reconstruction. This assumes that the reflection traveltime errors are no more than twice that of the transmission traveltime errors.

Frequency dependence of the quality factor in the upper crust: a deep seismic sounding approach

Ray amplitudes are computed in a one-dimensional velocity structure where the quality factor Q varies continuously with depth. An iterative process is then proposed to derive the quality factor distribution in the upper crust from deep sounding data. Results for compressional waves in the French Massif Central and for a signal frequency close to 20 Hz show that Q increases in a rather linear way from about 40 in surface up to 600 at 7 km depth. Q seems to be higher in the Central Alps: 180 in surface, 1600 at 5 km depth (at 20 Hz). Using proper signal processing, the frequency dependence of Q is finally investigated in the frequency range 10–25 Hz. The results indicate a dependence of the form Q=Q0fα, where α=0.25 ± 0.1.

A Backus--Gilbert approach to inversion of travel-time data for three-dimensional velocity structure

Summary. Considerable effort has recently been directed towards constructing three-dimensional velocity models to fit seismic traveltime data. Most studies have involved parameterizing the model by a limited number of boxes within which the velocity is assumed constant. An alternative viewpoint discussed in this paper is that the structure is an unknown function which can be viewed through a smoothing window. For a limited data set, the resolution length scale of this window will never be zero. Although formulation of the inverse problem for one-dimensional velocity structure is a straightforward application of the ideas of Backus and Gilbert, it does not generalize directly to three dimensions. This is because rays in three dimensions do not sample the structure in the volume between the rays and thus the data kernels are not suitable for approximating a delta function whose peak does not lie at the intersection of many rays. We solve this problem by replacing the delta function by an ideal averaging volume with sufficiently large dimensions that it is always intersected by rays. Non-square-integrable singularities in the data kernels are quelled by integration. The mean-square misfit between the transformed ideal averaging volume and its approximation (the actual averaging volume) constructed from the quelled data kernels leads to two natural measures of the resolution length scale of the actual averaging volume. For a simple synthetic example, one of these resolution measures converges rapidly to the scale of the ideal averaging volume as the size of the ideal volume increases. The other measure converges more slowly. We discuss strategy for finding the minimum permissible size of the ideal volume for which the average in the actual volume is a reliable estimate of the average in the ideal volume. In our synthetic example, this always requires in excess of 25 rays penetrating the averaging volume, but this result is not general and may be pessimistic for many real problems. For inaccurate data, the evaluation of the trade-off of resolution versus the variance of the smoothed model can proceed by relaxing the fit of the approximate averaging volume to the ideal volume in a manner exactly analogous to that described by Backus and Gilbert. Another strategy with considerable computational advantage which may sometimes be useful is to increase the size of the ideal averaging volume above its permissible minimum without relaxing the fit.