Observed Surface Displacements Research Articles

Evaluation of process-induced residual stresses in orthogonal 3D woven composites via nonlinear finite element modeling validated by hole drilling experiments

Effects of matrix material constitutive model and presence of defects on development of manufacturing induced residual stresses are studied using Finite Element (FE) analysis in an orthogonal 3D woven composite. The residual stresses are assumed to arise from a difference in the coefficients of thermal expansion between the reinforcement and the matrix. Numerical results are compared with experimentally observed surface displacements resulting from blind hole drilling and measured via electronic speckle pattern interferometry. It is demonstrated that for accurate reproduction of the residual stress fields using FE modeling, viscoelastic matrix properties should be used and possible presence of cracks at the tow/matrix interface should not be neglected.

Making the most of the Mogi model: Size matters

Magma movements are almost universally associated with volcanic deformation. The Mogi (1958) and McTigue (1987) models link observed surface displacements to behaviour within inaccessible magmatic plumbing systems. Mogi and McTigue models are well-used due to their computational simplicity and ease of application, but both models are limited by their assumptions about the deformation source and its embedding domain. Domain assumptions, including elasticity, homogeneity, and flat topography, have been previously described and corrected for. Whilst recognising the limits of these models, their frequent use in the literature requires an objective assessment of their utility against more sophisticated Finite Element (FE) models, their operational limits (radius-to-depth ratio, ε) and their relative merits in the light of limited field data. Here, we relax the source assumption of a small ε. We simulate volcanic deformation using Mogi, McTigue and FE models - the latter unrestricted by ε - to validate the maximum ε for which the analytical models can be applied, and to compare analytical and FE interpretations of deformation data from Kīlauea Volcano, Hawai'i. We find that analytical and FE models correspond for deformation sources with a range of ε that is wider than previously suggested limits. The differences between simulated surface displacements (forward modelling) and estimated deformation source parameters (inverse modelling) are less than 5% when ε < 0.37 (Mogi) or ε < 0.59 (McTigue). Misfits between analytical and FE models depend on whether radial or vertical displacements are considered simultaneously or independently, and on the values of source radius and depth - not only their ratio, as was assumed previously. There is little or no difference between best-fitting source parameters inferred using Mogi, McTigue and FE models at Kīlauea Volcano, despite the high ε of the system geometry, but sometimes poor correspondences between model results and GNSS observations. Our results demonstrate that Mogi and McTigue models can be applied to volcanoes with a wider range of magma reservoir radii and depths than was hitherto supposed, but previously-established corrections for domain simplifications are necessary to accurately interpret volcanic deformation.

Tsunami Induced by the Strike‐Slip Fault of the 2018 Palu Earthquake (Mw = 7.5), Sulawesi Island, Indonesia

AbstractAn unusual devastating tsunami occurred on September 28, 2018 after a strike‐slip faulting earthquake in Sulawesi, Indonesia. The induced tsunami struck Palu city with ∼4‐m flow depth. We performed two analyses to investigate the source of the tsunami. We first conducted the teleseismic source inversion and obtained the overall slip distribution of the strike‐slip fault. Our tsunami simulation from the coseismic deformation of the seismically estimated strike‐slip faulting produced a tsunami comparable to the leading part of the observation at Pantoloan. In order to reconstruct the detailed slip distribution on the fault plane, we then jointly utilized the tsunami waveform and Synthetic Aperture Radar (SAR) data. Because of the lack of SAR data in the bay, the tsunami data is necessary to constrain the offshore slip distribution, which directly induces the tsunami. The inverted source model shows a strike‐slip fault which consists of three segments extending from the epicenter to the south of 1.4°S with two bends and two asperities around Palu city. The joint inversion model accurately reconstructs the observed surface displacements and the leading part of the tsunami waveform. Our result exhibits the significant contribution of the strike‐slip faulting to the tsunami, but it also suggests additional tsunami sources, such as landslides, for the high inundations near Palu bay. The result also indicates that regional devastating tsunamis can be generated by an onshore strike‐slip fault with localized large dip slip.

Tidal Influence on Seismic Activity During the 2011–2013 El Hierro Volcanic Unrest

AbstractThe El Hierro volcanic unrest started in July 2011, with an increase in observed seismicity rates and surface deformation. After the initial onset, hypocenters migrated southward through September 2011, culminating in a submarine eruption beginning on October 10, 2011 and finishing in February 2012. The seismic activity continued, with remarkable periods of unrest through 2012 and 2013. The most significant episodes of seismic activity during this unrest are related to magma migration at depth. In this work, we compute tidal stress for each earthquake, at its hypocenter depth, and assign them a tidal stress phase angle. We have found statistically significant correlations between the occurrence of earthquakes and tidal stress phase angles, corresponding mainly to increasing tidal stress change rates. We found primarily that the magnitude of vertical and E‐W horizontal tidal stress values and their changing rates with time were correlated with earthquake occurrence times. We also found that there is no correlation between tides and seismicity at times with no observed surface displacements, suggesting that tidal modulation might be related to overpressure during migration of magma. Tidal modulation changes with depth and the influence of ocean‐loading tides is stronger than the influence of solid Earth tides. Our results support the hypothesis that tidal stress may modulate the seismicity during volcanic unrest, particularly during shallow depth magma migration.

Complex Surface Displacements above the Storage Cavern Field at Epe, NW-Germany, Observed by Multi-Temporal SAR-Interferometry

The storage cavern field at Epe has been brined out of a salt deposit belonging to the lower Rhine salt flat, which extends under the surface of the North German lowlands and part of the Netherlands. Cavern convergence and operational pressure changes cause surface displacements that have been studied for this work with the help of SAR interferometry (InSAR) using distributed and persistent scatterers. Vertical and East-West movements have been determined based on Sentinel-1 data from ascending and descending orbit. Simple geophysical modeling is used to support InSAR processing and helps to interpret the observations. In particular, an approach is presented that allows to relate the deposit pressures with the observed surface displacements. Seasonal movements occurring over a fen situated over the western part of the storage site further complicate the analysis. Findings are validated with ground truth from levelling and groundwater level measurements.

Half‐Graben Rupture Geometry of the 30 October 2016 MW6.6 Mt. Vettore‐Mt. Bove Earthquake, Central Italy

AbstractExisting models for the rupture geometry and slip distribution associated with the 30 October MW6.6 Mt. Vettore‐Mt. Bove earthquake in central Italy show significant dissimilarities. Indeed, due to the quite complicated observed deformation pattern, the activation of a complex multifault structure during a single seismic event was invoked. In this study, we explore different rupture scenarios and we develop a robust model of the rupture process of the 30 October earthquake, designed from new field observations, aftershock distribution, and static coseismic offsets including new near‐field survey‐mode global position system measurements, regional Global Positioning System observations, Interferometric Synthetic Aperture Radar interferograms, and static displacements derived from strong‐motion stations. Our preferred best fit model involves the simultaneous rupture of the master Mt. Vettore‐Mt. Bove normal fault and of at least two secondary antithetic faults (as they significantly contributed to the total deformation field), which overall describe a “simple conceptual” half‐graben normal fault system and whose arrangement fits the geological, seismological, and coseismic evidence of surface faulting. Notably, our model fits the geometry of seismogenic structures defined prior to the 2016–2017 seismic sequence by field Quaternary geological observations. In addition, no significant coseismic slip on faults alternative to the master and antithetic faults is necessary to explain the observed surface displacements during the 30 October Mt. Vettore‐Mt. Bove earthquake.

UAV Remote Sensing Surveillance of a Mine Tailings Impoundment in Sub-Arctic Conditions

Mining typically involves extensive areas where environmental monitoring is spatially sporadic. New remote sensing techniques and platforms such as Structure from Motion (SfM) and unmanned aerial vehicles (UAVs) may offer one solution for more comprehensive and spatially continuous measurements. We conducted UAV campaigns in three consecutive summers (2015–2017) at a sub-Arctic mining site where production was temporarily suspended. The aim was to monitor a 0.5 km2 tailings impoundment and measure potential subsidence of tailings. SfM photogrammetry was used to produce yearly topographical models of the tailings surface, which allowed the amount of surface displacement between years to be tracked. Ground checkpoints surveyed in stable areas of the impoundment were utilized in assessing the vertical accuracy of the models. Observed surface displacements were linked to a combination of erosion, tailings settlement, and possible compaction of the peat layer underlying the tailings. The accuracy obtained indicated that UAV-assisted monitoring of tailings impoundments is sufficiently accurate for supporting impoundment management operations and for tracking surface displacements in the decimeter range.

Graben formation and dike arrest during the 2009 Harrat Lunayyir dike intrusion in Saudi Arabia: Insights from InSAR, stress calculations and analog experiments

AbstractDetailed spatial and temporal accounts of propagating dikes from crustal deformation data, including their interplay with faulting, are rare, leaving many questions about how this interplay affects graben formation and the arrest of dikes unanswered. Here we use interferometric synthetic aperture radar (InSAR) observations, stress calculations, and analog experiments to investigate the interaction between an intruding dike and normal faulting during the 2009 Harrat Lunayyir dike intrusion in western Saudi Arabia. We generated five displacement maps from InSAR data to unravel the temporal evolution of deformation covering the majority of the intrusion. We find that the observed surface displacements can be modeled by a ~2 m thick dike with an upper tip ~2 km below the surface on 16 May 2009, 4 weeks after the onset of seismic unrest. In the following three days, the dike propagated to within ~1 km of the surface with graben‐bounding normal faulting dominating the near‐field deformation. The volume of the dike doubled between mid‐May and mid‐June. We carried out analog experiments that indicate that the wedge‐shaped graben grew outward with the faulting style changing progressively from normal faulting to oblique. Coulomb failure stress change calculations show that the intruding dike caused two zones of shallow horizontal tension on both sides of the dike, producing two zones of fissuring and normal faulting at the surface. In return, the faulting provoked compression around the upper tip of the dike, holding back its vertical propagation.

How well do surface slip measurements track slip at depth in large strike-slip earthquakes? The importance of fault structural maturity in controlling on-fault slip versus off-fault surface deformation

Comparisons of observed surface displacements with geodetically inferred slip at depth in six large (Mw⩾7.1) strike-slip earthquakes reveal a correlation between the structural maturity of the fault and the ratio of deep slip to surface slip that occurs on localized zones of surface rupture. Specifically, structurally immature faults (⩽25 km of total displacement) manifest only ∼50–60% of total slip on narrow fault surface traces versus ∼85–95% for structurally mature faults (⩾85 km total slip). The same pattern holds when structurally simple parts of a surface rupture are analyzed separately from parts that exhibit obvious structural complexity (e.g., discontinuous or multiple traces or significant dip slip). These results imply that geologic measurements of surface slip along structurally immature faults are likely to significantly underestimate the true slip at depth in large earthquakes. This observation has implications for a number of important problems, including determination of fault slip rates, which are based on surface offsets; earthquake probability assessments, which are based on geologic fault slip rates; comparisons of geologic and geodetic rate data in the search for strain transients; the structural evolution of fault zones; estimation of paleo-earthquake magnitudes based on geomorphic offsets; analyses of the relative importance of faulting vs. distributed deformation in accommodating relative plate motions in continental crust; the design and construction of infrastructure built near active faults; and possibly for the prediction of strong ground motions, which may at least partially depend on the degree of slip localization to the surface.

Recent surface displacements in the Upper Rhine Graben — Preliminary results from geodetic networks

Datasets of the GNSS Upper Rhine Graben Network (GURN) and the national levelling networks in Germany, France and Switzerland are investigated with respect to current surface displacements in the Upper Rhine Graben (URG) area. GURN consists of about 80 permanent GNSS (Global Navigation Satellite Systems) stations. The terrestrial levelling network comprises 1st and 2nd order levelling lines that have been remeasured at intervals of roughly 25 years, starting in 1922. Compared to earlier studies national institutions and private companies made available raw data, allowing for consistent solutions for the URG region. We focussed on the southern and eastern parts of the investigation area. Our preliminary results show that the levelling and GNSS datasets are sensitive to resolve small surface displacement rates down to an order of magnitude of 0.2 mm/a and 0.4 mm/a, respectively. The observed horizontal velocity components for a test region south of Strasbourg, obtained from GNSS coordinate time series, vary around 0.5 mm/a. The results are in general agreement with interseismic strain built-up in a sinistral strike–slip regime. Since the accuracy of the GNSS derived vertical component is insufficient, data of precise levelling networks is used to determine vertical displacement rates. More than 75% of the vertical rates obtained from a kinematic adjustment of 1st order levelling lines in the eastern part of URG vary between − 0.2 mm/a and + 0.2 mm/a, indicating that this region behaves stable. Higher rates up to 0.5 mm/a in a limited region south of Freiburg are in general agreement with active faulting. We conclude that both networks deliver stable results that reflect real surface movements in the URG area. We note, however, that geodetically observed surface displacements generally result from a superposition of different effects, and that a separation in tectonic and non-tectonic processes needs additional information and expertise.

Postseismic motion after the 2001 MW 7.8 Kokoxili earthquake in Tibet observed by InSAR time series

On November 14th 2001, a Mw 7.8 earthquake occurred in the Kokoxili region of northern Tibet. The earthquake ruptured more than 400 km along the western part of the Kunlun fault with a maximum of 8 m left‐lateral slip. In this paper, we use a multitemporal Interferometric SAR (InSAR) time series technique to map the postseismic motion following the large Kokoxili event. SAR data from Envisat descending orbits along five adjacent tracks covering almost the entire ruptured fault length are used to calculate the displacement time series for a period between 2 and 6 years after the earthquake. A peak‐to‐trough signal of 8 cm in the radar line of sight is observed during the period between 2003 and 2008. Two different mechanisms are employed to explain the observed surface displacements, namely afterslip and viscoelastic relaxation. The observations inverted for afterslip on and below the coseismic rupture plane shows that the maximum slip in the afterslip model is 0.6 m. The position of the maximum postseismic slip is located in the middle of two relatively high coseismic slip patches, which suggests that afterslip is a plausible mechanism. Models of viscoelastic stress relaxation in a Maxwell half‐space give a best fitting viscosity for the mid‐to‐lower crust of 2–5 × 1019 Pa s, and the principal postseismic relaxation process is due to viscous flow in the lower crust to upper mantle. However, the InSAR observations are incapable of distinguishing between localized (afterslip) and distributed (viscoelastic relaxation) deformation. And the lowest misfits are produced by mixed models of viscoelastic relaxation in the mantle below 70 km and afterslip in the crust. Modeling of viscoelastic relaxation in a Maxwell half‐space, and also a mixed mechanism model, enables us to place an effective viscosity of 2 × 1019 Pa s on the lower crust to mantle of northern Tibet.

Surface form memory in NiTi shape memory alloys by laser shock indentation

An indentation-planarization method for NiTi shape memory alloys has been developed that produces a robust surface topographical memory effect that we call "surface form memory", or SFM. Surface form memory entails reversible transitions between one surface form (flat) and another (say, wavy) that occur on changing temperature. These transitions are cyclically stable and exhibit very high mechanical energy density. Our previous study has demonstrated SFM transitions in NiTi alloys derived from quasistatic (i.e., low strain rate) spherical indents, as well as other geometries. Here, we report on experiments using confined laser ablation to indent a similar martensitic NiTi substrate, but in the dynamical regime (very high strain rate). As in the quasistatic case, subsurface plastic strain gradients are created via martensite twinning reactions, and later by dislocation-mediated slip. The resulting defects and stress fields support the two-way shape memory effect underlying SFM. In the dynamical case however, relative cyclic two-way displacements are found to be significantly larger, when normalized to the initial indent depth, than is the case with quasistatic indentation. This confers certain processing and boundary condition advantages. Analysis of the shock dynamics is found to be consistent with the observed surface displacements.

Complex landslide behaviour and structural control: a three-dimensional conceptual model of Åknes rockslide, Norway

AbstractÅknes is an active complex large rockslide of approximately 30–40 Mm3 located within the Proterozoic gneisses of western Norway. The observed surface displacements indicate that this rockslide is divided into several blocks moving in different directions at velocities of between 3 and 10 cm year−1. Because of regional safety issues and economic interests this rockslide has been extensively monitored since 2004.The understanding of the deformation mechanism is crucial for the implementation of a viable monitoring system. Detailed field investigations and the analysis of a digital elevation model (DEM) indicate that the movements and the block geometry are controlled by the main schistosity (S1) in gneisses, folds, joints and regional faults. Such complex slope deformations use pre-existing structures, but also result in new failure surfaces and deformation zones, like preferential rupture in fold-hinge zones.Our interpretation provides a consistent conceptual three-dimensional (3D) model for the movements measured by various methods that is crucial for numerical stability modelling. In addition, this reinterpretation of the morphology confirms that in the past several rockslides occurred from the Åknes slope. They may be related to scars propagating along the vertical foliation in folds hinges. Finally, a model of the evolution of the Åknes slope is presented.

Automated fault model discretization for inversions for coseismic slip distributions

Geoscientists increasingly rely on coseismic slip distributions inferred from geodetic observations to drive sophisticated models of the seismic cycle. To date, little work has been done on optimizing the parameterization of these fault models so that they reflect the resolving power of observed surface displacements. The locations of noisy surface displacement observations are often widely scattered far from features we wish to analyze in the subsurface and result in highly variable model resolution with depth. The few attempts to produce variably discretized fault planes are arduously constructed by hand and may not correctly reflect the ability of the data to resolve slip features. Motivated by the increasing size of geodetic data pools and the need for distributed slip models that accurately represent features the data can resolve, we present a fully automated algorithm that iteratively adjusts the sizes of dislocations in a fault model. We use the concept of smoothing scales, derived from the model resolution matrix, to resize dislocations so that each dislocation is sized appropriately given the area over which slip in that region of the fault would be smoothed. We present a series of synthetic tests that utilize both sparse and dense data sets, and we compare our variably discretized inversions to traditional regularly discretized inversions. We also use our approach to invert for slip from geodetic observations of the 2004 Mw 6.0 Parkfield, California, earthquake and the 1995 Mw 8.1 Antofagasta, Chile, earthquake.

On the use of quasi-static deformation to understand reservoir fluid flow

Deformation above a producing reservoir provides a valuable source of information concerning fluid flow and flow properties. Quasi-static deformation occurs when the displacements are so slow that we may neglect inertial terms in the equations of motion. We present a method for inferring reservoir volume change and flow properties, such as permeability, from observations of quasi-static deformation. Such displacements may represent surface deformation such as tilt, leveling, interferometric synthetic aperture radar (InSAR), or bathymetry observations or subsurface deformation, as inferred from time-lapse seismic surveys. In our approach, the equation for fluid flow in a deforming reservoir provides a mapping from estimated fractional volume changes to reservoir permeability variations. If the reservoir behaves poroelastically over the interval of interest, all the steps in this approach are linear. Thus, the inference of reservoir permeability from deformation data becomes a linear inverse problem. In an application to the Wilmington oil field in California, we find that observed surface displacements, obtained by leveling and InSAR, are indeed compatible with measured reservoir volume fluxes. We find that the permeability variations in certain layers coincide with fault-block boundaries suggesting that, in some cases, faults are controlling fluid flow at depth.

Cross‐matching with interpreted warping of 3D streamer and 3D ocean‐bottom‐cable data at Valhall for time‐lapse assessment

ABSTRACTLegacy streamer data and newer 3D ocean‐bottom‐cable data are cross‐matched and analysed for time‐lapse analysis of geomechanical changes due to production in the Valhall Field. The issues relating to time‐lapse analysis using two such distinctly different data sets are addressed to provide an optimal cross‐matching workflow that includes 3D warping. Additionally an assessment of the differences between the imaging using single‐azimuth streamer and multi‐azimuth ocean‐bottom‐cable data is provided. The 3D warping utilized in the cross‐matching procedure is sensitive to acquisition and processing differences but is also found to provide valuable insight into the geometrical changes that occur in the subsurface due to production. As such, this work also provides a demonstration of the use of high‐resolution 3D interpreted warping to resolve the 3D heterogeneity of the compaction and subsidence. This is an important tool for Valhall, and possibly other fields, where compaction and subsidence (and monitoring thereof) are key factors in the reservoir management since the predominant observed production‐induced changes are compaction of the soft, high‐porosity chalk reservoir, due to pore‐pressure reduction, and the resultant overburden subsidence. Such reservoir compaction could have significant implications for production by changing permeabilities and production rates. Furthermore the subsidence effects could impact upon subsea installations and well‐bore stability. Geomechanical studies that have previously been used to model such subsidence and compaction are only constrained by observed surface displacements and measured reservoir pressure changes, with the geological overburden being largely neglected. The approaches suggested herein provide the potential for monitoring and assessment in three dimensions, including the probable heterogeneity and shearing, that is needed for full understanding of reservoir compaction and the resultant effects on the overburden to, for example, mitigate well‐bore failures.

Slip distribution of the 1997 Umbria‐Marche earthquake sequence: Joint inversion of GPS and synthetic aperture radar interferometry data

In September–October 1997 a seismic sequence struck the southern part of the Umbria‐Marche region in central Italy. The three largest earthquakes took place on 26 September (0033 UT, Mw 5.7; 09:40 GU, Mw 6.0) near Colfiorito, and on 14 October (Mw 5.6) near Sellano. We present a slip model for the main sequence over two separate fault planes. We use previously published Global Positioning System (GPS) and differential interferometric synthetic aperture radar (DInSAR) data as well as new DInSAR data covering the Sellano event as input to a simulated annealing solution for the distribution of fault slip. The combination of both types of geodetic data as well as the combination of SAR interferograms from both ascending and descending satellite tracks allows us to make a more complete geodetically based assessment of the fault slip for these earthquakes. For the solutions presented here we required the slip distribution to match the observed surface displacements while maintaining a total scalar seismic moment approximately equal to the seismologically inferred moments for this sequence. We find the slip was concentrated in a zone at 4–5.5 km depth with a shallower tail extending to near the surface at the northern end of the Colfiorito sequence. For both the second Colfiorito event and the Sellano event, individual patches were calculated with maximum slip of 2 m, implying stress drops as high as 20 MPa. Coulomb stress calculations show that the coseismic slip distribution for the Colfiorito events facilitated slip of the adjacent Sellano earthquake.

Geodetically Observed Surface Displacements of the 1999 Chi-Chi Earthquake near Southern Termination of the Chelungpu Fault

Using the Global Positioning System (GPS) we investigated coseismic deformation in southwestern Taiwan for the Rayli (7/7/1998, ML =6.2), Chi-Chi (9/21/1999, ML=7.3) and Chiayi (10/22/1999, ML =6.4) earthquakes based on data collected from 1996 to 2001. The GPS network includes 130 campaign-surveyed, 32 local tracking stations and 13 IGS (International GPS Service for Geodynamics) sites. The geodetic data were processed with GAMIT/GLOBK software. The results indicate that coseismic displacement vectors show opposite moving directions across the surface trace of the Chelungpu-Tachienshan-Chukou thrust fault. In general, the hanging-wall moves toward NW (or W, SW)as opposed to NE (or E, SE) for the footwall. The average amount of displacement is larger in the hanging-wall than in the footwall, with maximum displacement appearing at sties near the mainshock. An array of NW -SE trending left-lateral tear faults distributed in a zone of tens of kilometers wide are interpreted as transfer faults accommodating SW -directed, right-lateral thrusting in the vicinity of southern termination of the Chelungpu fault.

Geodetically observed surface displacements of the 1999 Chi-Chi, Taiwan, Earthquake

The 21 September 1999 Chi-Chi, Taiwan, earthquake of magnitude MW = 7.6 (ML = 7.3) severely deformed the Earth’s crust in the central Taiwan region. The earthquake created an 85-km-long surface rupture along the Chelungpu fault. The epicenter was located at 23.85°N, 120.81°E, near the southern end of the rupture zone. Three-dimensional displacements of 285 geodetic control stations were determined in this study from Global Positioning System (GPS) observations collected before and after the earthquake. The detailed surface displacement field shows that individual stations are vertically uplifted by up to 4 m and displaced horizontally by up to 9 m, with the largest displacement occurring near the northern end of the ruptured thrust fault. The azimuth of the surface displacement field is approximately parallel to the direction of tectonic convergence of the Eurasian and Philippine Sea plates. The maximum three-dimensional displacement of 9.9 m is among the largest fault movements ever measured for modern earthquakes.

THE NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS CAN BE PERFORMED EITHER BY TIME EXTRAPOLATION OR BY DEPTH EXTRAPOLATION1

ABSTRACTThe classical aim of non‐linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained.Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some a priori values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and a priori initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non‐linear inversion using forward‐backward time propagations now works, this non‐linear inversion using downward‐upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).