Monte Carlo Inversion Research Articles

Collection and three‐dimensional modeling of GPS and tilt data at Merapi volcano, Java

We study here the deformations associated with the November 1996 to March 1997 eruption period at Mount Merapi (Central Java), one of the most active volcanoes in Indonesia. This activity period includes a vertical explosion on January 17 and an increase of the lava dome volume by about 3×106 m3. Two Global Positioning System (GPS) campaigns have been carried out on a six‐benchmark network at the beginning and at the end of the period. Relative displacements with respect to the reference point show an average subsidence of 6.5 cm. A multicomponent tilt station installed on the southeast flank, 3 km from the summit, recorded a tilt of 11.1 ± 0.7 μrad in the tangential direction and 0.9 ± 0.4 μrad in the radial direction. These data are interpreted using a three‐dimensional (3‐D) elastic model based on the mixed boundary element method and a near‐neighbor Monte Carlo inversion. Interpretation of tilt data requires an accurate mesh for discretizing the 3‐D topography. The final result supports a horizontal elliptic magma source located 8.5 ± 0.4 km below the summit and 2 ± 0.4 km to the east of it. In particular, the data cannot be consistent with the location of a magma chamber determined from seismic activity analysis (i.e., 2 km below the summit). The computed depth depends strongly on the source shape and cannot be constrained properly because of the small amount of data. The computed deflation of 11 ± 2×106 m3 is about 3 times larger than the observed increase in the lava dome volume. This difference is attributed to rock avalanches and pyroclastic flows on the flanks of the volcano.

Nonlinear refraction and reflection travel time tomography

We develop a rapid nonlinear travel time tomography method that simultaneously inverts refraction and reflection travel times on a regular velocity grid. For travel time and ray path calculations, we apply a wave front method employing graph theory. The first‐arrival refraction travel times are calculated on the basis of cell velocities, and the later refraction and reflection travel times are computed using both cell velocities and given interfaces. We solve a regularized nonlinear inverse problem. A Laplacian operator is applied to regularize the model parameters (cell slownesses and reflector geometry) so that the inverse problem is valid for a continuum. The travel times are also regularized such that we invert travel time curves rather than travel time points. A conjugate gradient method is applied to minimize the nonlinear objective function. After obtaining a solution, we perform nonlinear Monte Carlo inversions for uncertainty analysis and compute the posterior model covariance. In numerical experiments, we demonstrate that combining the first arrival refraction travel times with later reflection travel times can better reconstruct the velocity field as well as the reflector geometry. This combination is particularly important for modeling crustal structures where large velocity variations occur in the upper crust. We apply this approach to model the crustal structure of the California Borderland using ocean bottom seismometer and land data collected during the Los Angeles Region Seismic Experiment along two marine survey lines. Details of our image include a high‐velocity zone under the Catalina Ridge, but a smooth gradient zone between Catalina Ridge and San Clemente Ridge. The Moho depth is about 22 km with lateral variations.

Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique

The absorption and transport scattering coefficients of caucasian and negroid dermis, subdermal fat and muscle have been measured for all wavelengths between 620 and 1000 nm. Samples of tissue 2 mm thick were measured ex vivo to determine their reflectance and transmittance. A Monte Carlo model of the measurement system and light transport in tissue was then used to recover the optical coefficients. The sample reflectance and transmittance were measured using a single integrating sphere `comparison' method. This has the advantage over conventional double-sphere techniques in that no corrections are required for sphere properties, and so measurements sufficiently accurate to recover the absorption coefficient reliably could be made. The optical properties of caucasian dermis were found to be approximately twice those of the underlying fat layer. At 633 nm, the mean optical properties over 12 samples were and for absorption coefficient and and for transport scattering coefficient for caucasian dermis and the underlying fat layer respectively. The transport scattering coefficient for all biological samples showed a monotonic decrease with increasing wavelength. The method was calibrated using solid tissue phantoms and by comparison with a temporally resolved technique.

Crustal structure of the Mid‐Atlantic Ridge south of the Kane Fracture Zone from seafloor and sea surface gravity data

Seafloor and sea surface gravity data are inverted together to construct a model for the near‐axis crustal structure of a slow spreading ridge. The seafloor data set offers two main advantages: it allows us to recover shorter‐wavelengths signal and to constrain the value of a potential field at two different levels. The model we propose here would not have been derived from sea surface data alone. It is based on a dense sea surface gravity coverage and on 121 sea bottom gravity measurements collected in the Mid‐Atlantic Ridge at Kane (MARK) area, during the Hydrosnake (1988) and Gravinaute (1993) cruises. The primary goal of the seafloor surveys was to test for the presence of a magma reservoir beneath the axial neovolcanic ridge. First, a forward two‐dimensional (2‐D) model of the crustal structure across the axis is fit to observed gravity anomalies, using constraints from geological and structural observations. Bouguer anomalies computed from sea bottom measurements and downward continuation of sea surface measurements both constrain the forward modeling. This forward model is the starting point of a 2‐D Monte Carlo inversion of seafloor and sea surface data. In addition to the crustal thickness variations along‐axis, our data document the amplitude variations of the crustal thickness and/or its density in the across‐axis direction. The model resulting from our inversion exhibits several features of the crustal structure in the MARK area: (1) The presence of a low‐density (Δρ =−300±50 kg/m3) body beneath the neovolcanic ridge is suggested and could correspond to a magma chamber, or more probably to a highly hydrothermally fissured zone. (2) Both long‐ and short‐wavelength gravity signals exhibit a difference between the western and eastern sides of the axial domain: the mean value and the amplitude of Bouguer anomalies are higher on the western part. This difference suggests that axial processes, in this area, are very asymmetric. (3) Abyssal hills are not associated with a single gravity signature: for instance, on the west side of the axis, one of the explored hills has no Bouguer anomaly and is interpreted as a neovolcanic ridge, whereas the others are associated with a shifted Bouguer anomaly high and are interpreted as having thinner magmatic crust. (4) The last feature of the crustal fabric we document here is the asymmetric emplacement of some deep rocks outcrops. In the MARK area, we find that “Pink Hill,” a topographic high where serpentinized peridotites are outcropping, is much more serpentinized on its east flank, toward the axial valley, than on its west flank. Alteration occurring mainly by fluid circulation through faulted zones, the asymmetric serpentinization suggests that deep‐origin rocks have outcropped by means of a main fault zone and are not emplaced by diapirism.

One-Dimensional Model Study of the PNG Dataset Using Site-Independent Groom-Bailey Decomposition.

Conventional magnetotelluric (MT) analyses of the Papua New Guinea (PNG) dataset reveal that it is a very good two-dimensional (2D) dataset with relatively high S/N ratios up to 10 2 s. However, traditional Swift's angles are unstable at some sites and differ slightly site to site. This implies that weak but significant distortions are present at all sites. Hence, the regional 2D strike angle was sought by minimizing the global X 2 misfits derived from the site- and frequency-independent Groom-Bailey (GB) decomposition at seven selected sites. The other three sites were eliminated by X 2 -test. The strike angle was determined to be 60° west of geographic north. One-dimensional (1D) Monte-Carlo inversion was performed using the Berdichevsky averages of the decomposed 2D principal tensor elements. The 2D model simply based on the 1D inversions was able to reproduce the observed impedances, except for the systematically high values of TM phase in the period range of 10 -1 to 10 s. This indicates that a 1D approximation of the electrical conductivity structure beneath this region is reasonably valid. The 1D models indicated the existence of a thin (up to 2.5 km) resistive layer just below the surface, suggesting the 'single' Darai limestone.

Slope of the geoid spectrum and constraints on mantle viscosity stratification

Spectral analysis of a recently obtained high resolution tomographic model, describing the top 1200 km of the mantle, shows a power‐law dependence on the degree, for degrees greater than around 10. The spectrum of recent geoid models is also found to decay in a linear fashion with degree on a log‐log plot. We have employed the logarithmic slope of the geoid between degrees 10 and 25 as a constraint on the viscosity structure of the top 1200 km of the mantle. The constraint of fitting the geoid slope represents a new and independent approach to the determination of the upper‐mantle viscosity structure. From conducting over one million runs in a Monte Carlo inversion, we have found that there are basically three families of viscosity which can fit the geoid slope. They are (1) with a viscosity hill between 660 and 1000 km, (2) with a weak viscosity increase at 660 km, and (3) with a significant viscosity increase at depths between 820 and 1000 km. Below 1000 km the viscosity of the lower‐mantle for all 3 families is larger than that in the upper mantle. These results corroborate the complexity of the mantle viscosity profile between 660 and 1200 km, which would have important ramifications on flows between the upper and lower mantle.

Pseudofermion observables for static heavy meson decay constants on the lattice

A method based on the Monte Carlo inversion of the Dirac operator on the lattice provides low noise results for the correlations entering the definition of the heavy meson decay constant in the static limit. The method is complementary to the usual method of smeared sources, avoids the systematic error arising from optimizing the size of the smearing volume ans is more efficient for the values of lattice parameters that we have explored.

Estimating Lithospheric Properties at Atla Regio, Venus

Magellan spherical harmonic gravity and topography models are used to estimate lithospheric properties at Atla Regio, Venus, a proposed hotspot with dynamic support from mantle plume(s). Global spherical harmonic and local representations of the gravity field share common properties in the Atla region in terms of their spectral behavior over a wavelength band from ∼2100 to ∼700 km. The estimated free-air admittance spectrum displays a rather featureless long-wavelength portion followed by a sharp rise at wavelengths shorter than about 1000 km. This sharp rise requires significant flexural support of short-wavelength structures. The Bouguer coherence also displays a sharp drop in this wavelength band, indicating a finite flexural rigidity of the lithosphere. A simple model for lithospheric loading from above and below is introduced (D. W. Forsyth, 1985, J. Geophys. Res. 90, 12,623-12,632) with four parameters: f, the ratio of bottom loading to top loading; Z m , crustal thickness; Z l, depth to bottom loading source; and T e, elastic lithosphere thickness. No model is found that fits simultaneously the estimated free-air admittance and Bouguer coherence across the entire spectrum. A dual-mode compensation model is introduced in which the shorter wavelengths (λ ⪉ 1000 km) might be explained best by a predominance of top loading by the large shield volcanoes Maat Mons, Ozza Mons, and Sapas Mons, and the longer wavelengths (λ ⪊ 1500 km) might be explained best by a deep depth of compensation, possibly representing bottom loading by a dynamic source. A Monte Carlo inversion technique is introduced to thoroughly search out the four-space of the model parameters and to examine parameter correlations in the solutions. It is shown that simultaneous inversion on admittance and Bouguer coherence in the short-wavelength band constrains the elastic thickness to a range of approximately 40 to 50 km. Introduction of the free-air coherence as an observable in the inversion narrows the estimates of the other three parameters ( f = 0.10 ± 0.01, Z m = 30 ± 13 km, Z l = 7 ± 5 km) and provides a solution T e = 45 ± 3 km. In the long-wavelength band, Z l is found to be about 150 km, and this is the only parameter that can be estimated robustly. Crustal thickness, Z m , is unbounded, and the solutions are degenerate in that there is no discrimination between top and bottom loading. Acceptable elastic thicknesses range from 0 to 140 km. The elastic thickness results from both wavelength bands are corroborated in the spatial domain by examining residual standard deviations for a range of T e values. The results found here are consistent with the results of Smrekar (1994) for wavelengths longer than 600 km. Values of Z l are consistent with other analyses that exclude wavelengths longer than about 2,000 km. Inclusion of wavelengths longer than about 3000 km leads to compensation depths of approximately 200 km. Moment-curvature relationships based on yield strength envelopes are used to constrain temperature gradient. Average curvature is obtained from the flexurally deformed Moho surface by Fourier transform techniques. Surface temperature gradients are obtained in the range 7-10 K km -1, depending on the strain rate adopted. The relationship between elastic thickness and temperature gradient is calibrated by comparing elastic thickness estimates of two East African hotspots with measured heat flow. The excess heat flow at the East African hotspots over the African continental mean, if applicable to Venus, can be used to show that the globally averaged heat flow on Venus must be much lower than the Earth-scaled value predicts. A thick (∼350-km) ambient thermal lithosphere, as proposed by A. B. Kucinskas and D. L. Turcotte in this issue, and interpretable from the long-wavelength depth of compensation at Atla Regio, provides an acceptable background temperature gradient. When it is added to the excess heat flow or temperature gradient, the predicted heat flow at Atla Regio is consistent with the moment-curvature results. The thermal lithosphere beneath Atla Regio may be about 100 km thick and this answer can be obtained by two independent methods: from the ambient thermal lithosphere thickness and simple elevation scaling (result dependent on long gravity wavelengths) and from the elastic thickness estimate converted to temperature gradient (result dependent on short gravity wavelengths). Either Venus is considerably deficient in heat sources relative to Earth, or the thermal lithosphere is overthickened in response to an earlier episode of significant heat loss from the planet.

Efficient methods for Monte Carlo inversion of quark matrices

We present a Monte Carlo method for estimating linear combinations of inverse quark matrix entries. For conventional Monte Carlo inversion, Z 2 noise yields unbiased, minimum variance estimates. An enhanced version of the method is introduced, which has the potential to substantially reduce variance, particularly when there is rapid falloff with increasing separation between lattice points. Simulation results for one and two dimensions are presented, which confirm the promise of the enhanced version.

Nonlinear velocity inversion by a two‐step Monte Carlo method.

Seismic reflection data contain information on small‐scale impedance variations and a smooth reference velocity model. Given a reference velocity model, the reflectors can be obtained by linearized migration‐inversion. If the reference velocity is incorrect, the reflectors obtained by inverting different subsets of the data will be incoherent. We propose to use the coherency of these images to invert for the background velocity distribution. We have developed a two‐step iterative inversion method in which we separate the retrieval of small‐scale variations of the seismic velocity from the longer‐period reference velocity model. Given an initial background velocity model, we use a waveform misfit‐functional for the inversion of small‐scale velocity variations. For this linear step we use the linearized migration‐inversion method based on ray theory that we have recently developed with Lambaré and Virieux. The reference velocity model is then updated by a Monte Carlo inversion method. For the nonlinear inversion of the velocity background, we introduce an objective functional that measures the coherency of the short wavelength components obtained by inverting different common shot gathers at the same locations. The nonlinear functional is calculated directly in migrated data space to avoid expensive numerical forward modeling by finite differences or ray theory. Our method is somewhat similar to an iterative migration velocity analysis, but we do an automatic search for relatively large‐scale 1-D reference velocity models. We apply the nonlinear inversion method to a marine data set from the North Sea and also show that nonlinear inversion can be applied to realistic scale data sets to obtain a laterally heterogeneous velocity model with a reasonable amount of computer time.

Viscoelastic responses of a hard transition zone: effects on postglacial uplifts and rotational signatures

Recent studies from laboratory (S. Karato, 1989) and Monte Carlo inversion (Y. Ricard et al., 1989) have argued that the transition zone between 400 and 670 km depth may be stiffer than the lower mantle and the upper mantle above it. Using a five-layer viscoelastic spherical model, we have calculated the transient displacements of postglacial rebound, the induced polar motions and the temporal variations of the geopotential up to degree eight of the zonal coefficients. We have compared two models, (a) one with two viscoelastic layers separated at 670 km, and (b) the other with three layers in which a hard garnet layer lies between the upper and lower mantle. Forward modelling shows that it may be possible to discern the presence of a hard garnet layer with a viscosity of at least ten times greater than the upper mantle, on the basis of uplift data near the center of the former Laurentide ice-sheet and from polar wander and j_. 2 data. Temporal variations of higher gravity harmonics, such as j_. 6 and j_. 8 , can potentially place even tighter constraints on the rheological properties of the hard transition zone. A lower mantle viscosity between 2 and 4 × 10 22 Pa s is generally preferred in models with a garnet layer which may be as large as 50 times more viscous than the upper mantle.

Mantle heterogeneities, geoid, and plate motion: A Monte Carlo inversion

Seismic tomography in both the upper and the lower mantle, as well as subducting oceanic slabs defined by seismicity, has been translated into density heterogeneities to generate models of mantle circulation. These models can predict both the surface velocities and the geoid, which can be compared with plate tectonics and gravity data. A given model is specified by 6 parameters related to the viscosities of 3 mantle layers and the absolute amplitudes of density variations in the upper and lower mantle as well as in the slabs. The values of these parameters are chosen at random within an acceptable range. Each model is submitted to an appropriate test comparing observations with predictions. The results of the most successful models selected by this Monte Carlo inversion are displayed. They yield preferred mantle viscosity structures exhibiting large variations at depth. With a physical interface between upper and lower mantle, i.e., with the possibility for the circulation to penetrate the 650 km discontinuity, two classes of viscosity profiles stand out. The first one implies a regular increase of the viscosity in the sublithospheric mantle, with reasonable values for the density parameters. The second one is unexpected in the sense that it predicts a very stiff bottom for the upper mantle. It also requires vanishingly small amplitudes for the upper mantle density heterogeneities defined by tomography, which would thus have to be of lithological rather than thermal origin. With a chemical interface at 650 km the outcome is very similar: the same two classes of viscosity structures do yield a satisfactory geoid prediction. However only the class of models with a stiff layer at midmantle depths predicts acceptable surface velocities. Altogether the best models out of some 60,000 which have been tested only explain one third of the geoid and two thirds of the surface divergence for spherical harmonic degrees 1 to 6. Nevertheless the main features of these two observed patterns are present in the computed maps, and 4 out of 6 correlation coefficients lie close to the 90% confidence level. This is true for the geoid as well as for the surface divergence of the displacement velocity. However, as the internal viscosity structure has been assumed to have spherical symmetry, the rotational component of the surface velocities cannot be predicted.

Palaeomagnetism of late Precambrian-Cambrian volcanics and intrusives from the Armorican Massif, France

Summary. Results are presented for a magnetotelluric and magnetovariational study in northern Scotland. For nine of the 20 sites at which observations were made the results of the analysis are presented fully together with one-dimensional (1-D) Monte-Carlo inversions for representative sites from the Lewisian Foreland, the North-west Highlands, the Great Glen region and the Grampian Highlands. For the remaining sites, the results are summarized. The 1-D models of all sites situated in a traverse extending from the Lewisian Foreland to the neighbourhood of the Highland Boundary Fault have been used as the basis for preliminary 2-D modelling. Maps of magnetic field induction vectors and transfer functions for an inducing field perpendicular to the average geological strike are presented for periods of 23-819s. The 2-D conductivity models were constrained by both the magnetotelluric and magnetovariational responses. The period range of observation in this study was too limited to resolve structural differences in the upper crust but adequate to reveal a moderately good conductor at upper mantle/lower crustal depths in the Caledonian metamorphic zone. In contrast the whole of the crust in the Lewisian Foreland is highly resistive and there is no evidence of an intermediate conducting layer between the crust and mantle in this region.

On the electrical crust--mantle structure in Fennoscandia: no Moho, and the asthenosphere revealed?

Summary. An interpretation of the geomagnetic inductive response function, C(ω, 0), observed at Kiruna in northern Sweden, is herein undertaken. The bounds of acceptable solutions are initially discovered by a Monte-Carlo random search procedure, and the best-fitting solutions are examined by the application of linear theory to the problem. The data are shown to have a higher degree of internal consistency than that described by the estimated variances of each datum. A further Monte-Carlo inversion of the variance- reduced data set gives solutions with well defined model parameters. The two major features of the models are: (1) a small, or non-existent, electrical conductivity variation across the seismic Moho boundary, and (2) the unequivocal existence of an electrical asthenosphere, under the Fenno- scandian shield, beginning at a depth of between 155–185 km, and of 60km minimum thickness. Both of these observations have seismic counterparts. Finally, possible mantle temperature profiles are deduced which depend on the assumptions and laboratory data employed.

Magnetovariational and Magnetotelluric Investigations in S. Scotland

A two-dimensional array of 20 Gough-Reitzel magnetometers was operated over S. Scotland in 1973 and in 1974-5 magnetotelluric and magnetovariational observations in the period range 10-10, 000s were made in the same region. In this paper, the analyses of the magnetic data from both studies are presented in the form of induction vectors and hypothetical event contours. They suggest that the lateral variations in electrical conductivity structure associated with the Eskdalemuir anomaly are more complex than suggested by earlier studies. A marked discontinuity in electrical structure is apparent in a narrow belt parallel to and south of the S. Uplands fault. This belt is associated with a major gravity anomaly and with steep gradients in the seismic profile at crustal depths. Another discontinuity is indicated near the North-umberland Basin. Representative examples of the magnetotelluric analysis and of one-dimensional Monte Carlo inversion of the M-T data are presented for the three regions separated by these discontinuities. They show that the conducting zone associated with the Eskdalemuir anomaly is at a depth greater than 24km, while on either side of this region, there are good conductors within crustal depths.

A multi-station magnetotelluric study in southern Scotland -- II. Monte-Carlo inversion of the data and its geophysical and tectonic implications

Summary A Monte-Carlo inversion procedure is developed and applied to magnetotelluric data from six locations, two of which are in the Midland Valley of Scotland, three in the Southern Uplands, and one in northern England. The method is described in full in respect of one of the six locations to illustrate both the importance of satisfying the phase as well as the amplitude data and the effect of model acceptance level. The electrical resistivity profiles resulting from application of the method indicate that; (a) there is a conducting zone under the Midland Valley at a depth no greater than 12 km, (b) the crust under the Southerri Uplands is mainly resistive, (c) there is a conductor at a depth greater than 24 km in this region, and (d) under northern England there is probably a very highly conducting region very close to the surface. A brief discussion of the possible geophysical and tectonic significance of these models follows.

Great-circle Rayleigh wave attenuation and group velocity, Part IV: Regionalization and pure-path models for shear velocity and attenuation

Pure-path averages for group velocities and specific attenuation have been calculated from individual observations and from path averages for two regionalizations; one original to this study and the other previously devised by Wu. Both are based on four upper-mantle provinces: ocean basin, continent, island arc and mid-ocean ridge. Pure-path group velocities and specific attenuation have also been calculated for combinations of regions and provide well separated regional measurements for such composite regions. Shear-velocity models for pure and combined regions have been derived by a controlled Monte Carlo inversion procedure and indicates that a low-velocity zone is required beneath the oceans, but is not required beneath continents. Models have been produced for pure and combined ocean, ocean-ridge, continent and continent-arc provinces. Q −1 R determined from pure-path average group velocities and attenuation coefficients has been regionalized successfully for 2- and 3-region combinations. The resulting pure-path Q −1 R for continents is much lower than that for ocean basins and ocean-ridge provinces. Inversion of Q −1 R for ocean-ridge provinces shows that the average Q β for the upper 200 km of these regions is between 85 and 100.

Great-circle Rayleigh wave attenuation and group velocity, Part III: Inversion of global average group velocities and attenuation coefficients

Average shear-velocity models for the upper mantle have been derived by controlled Monte Carlo inversion of global average Rayleigh wave group velocity (GAGV) data for periods between 50 and 300 seconds. GAGV data have been corrected for attenuative dispersion using a method based on the theory of Liu, Anderson and Kanamori. Two types of model bounds have been used with one- or two-layer low-velocity zones beginning at depths of 70 and 100 km. All models fitting GAGV data within one standard deviation have low-velocity zones in the 100–200 km depth range. Models with low-velocity zones beginning at 70 km, as well as 100 km, fit GAGV data within one standard deviation, so the average thickness of the lithosphere (taken as the depth to the top of the low-velocity zone) cannot be determined with precision. Global average models for shear-wave attenuation ( Q −1 β ) have been derived from global average Rayleigh wave attenuation coefficients for periods between 50 and 300 s and average shear-velocity models. Zones of high Q −1 β coincide with the low-velocity zones of all shear-velocity models, however, models with low-velocity zones beginning at a depth of 70 km have the highest-attenuation layer in the lower half of the low-velocity zone. Resolution kernels for these attenuation models show that parameters for layers shallower than the lower part of the low-velocity-high-attenuation zone are strongly coupled but are distinct from the lower part of this zone. This suggests that the deeper part of the low-velocity-high-attenuation zone is the most mobile part of the zone or that on the average, the top of the zone is deeper than 70 km. The average Q β of the lithosphere, low-velocity zone, and sub-low-velocity layer (asthenosphere) are approximately 200, 85–110 and 170–200, respectively.

Resolution Analysis for Discrete Systems

summary. Treatments of geophysical inverse problems have tended to polarize into approaches intended to generate models either described by piecewise continuous functions or with some prior discretization. The two approaches are here developed in parallel, and the ideas of a trade-off between the anticipated error and the attainable level of detail in the model estimate are extended to the discrete case, either with even or uneven discretization. An alternative approach to specifying the potential resolution of a model is to establish upper and lower bounds on parameter values. Linear programming methods are extended to determine bounds which allow for subjective limits on parameter values. For a non-linear system the possible resolution may be investigated by estimation procedures based on the full set of successful solutions obtained by Monte-Carlo inversion.

Upper and Lower Mantle Shear Velocity Modelling by Monte Carlo Inversion

Summary Non-uniqueness bounds for shear velocity in the upper and lower mantle have been derived by Monte Carlo inversion, with constraints imposed by the data of Nuttli, Robinson & Kovach, and Hales & Roberts. The depth to the lower major discontinuity in the upper mantle is estimated to be 690 ± 25 km. Depths calculated for the corresponding discontinuity for compressional velocity are shallower than this, mainly as a result of differences in the interpretation of later arrivals in the travel-time data. Models of Hales & Roberts and Robinson & Kovach fall within our bounds for the lower mantle but the Fairborn mean model is excluded for depths greater than 2200 km.