ALBERT

Description: This paper presents a new method of computing internal displacement, stress, strain, and gravitational changes caused by a point dislocation in a spherical Earth model. Specifically, the asymptotic solutions of the radial functions are introduced. The conventional method expresses the deformation field as an infinite series of spherical harmonics, and it cannot avoid the problem of the series not converging near the dislocation. The proposed method using asymptotic solutions can overcome this problem and compute the deformation field even near the dislocation. This paper focuses on deformations in a homogeneous sphere to elucidate the problem and solve it with simplicity. The proposed method is used to compute the volumetric strains caused by four independent dislocation types: vertical strike-slip, vertical dip-slip, horizontal tensile fracturing and vertical tensile fracturing. The effect of sphericity on the deformation field is also investigated by comparing the computational results with those for a homogeneous semi-infinite medium. The discrepancy between the results of the homogeneous sphere and those of the half-space reached up to 15–20 per cent at an epicentral distance of 2 $^\circ$ –5°. In particular, large differences were observed in the following cases: (i) the dislocation type is tensile fracturing, (ii) the depth of the source is large and (iii) the strain is measured at a large depth (for any source depth).

Description: The 2015 Gorkha earthquake ( M w  7.8) occurred by thrust faulting on a ~150 km long and ~70 km wide, locked downdip segment of the Main Himalayan Thrust (MHT), causing the Himalaya to slip SSW over the Indian Plate, and was followed by major-to-moderate aftershocks. Back projection of teleseismic P -wave and inversion of teleseismic body waves provide constraints on the geometry and kinematics of the main-shock rupture and source mechanism of aftershocks. The main-shock initiated ~80 km west of Katmandu, close to the locking line on the MHT and propagated eastwards along ~117 $\rm ^{\circ }$ azimuth for a duration of ~70 s, with varying rupture velocity on a heterogeneous fault surface. The main-shock has been modelled using four subevents, propagating from west-to-east. The first subevent (0–20 s) ruptured at a velocity of ~3.5 km s – 1 on a ~6 $\rm ^{\circ }$ N dipping flat segment of the MHT with thrust motion. The second subevent (20–35 s) ruptured a ~18 $\rm ^{\circ }$ W dipping lateral ramp on the MHT in oblique thrust motion. The rupture velocity dropped from 3.5 km s – 1 to 2.5 km s – 1 , as a result of updip propagation of the rupture. The third subevent (35–50 s) ruptured a ~7 $\rm ^{\circ }$ N dipping, eastward flat segment of the MHT with thrust motion and resulted in the largest amplitude arrivals at teleseismic distances. The fourth subevent (50–70 s) occurred by left-lateral strike-slip motion on a steeply dipping transverse fault, at high angle to the MHT and arrested the eastward propagation of the main-shock rupture. Eastward stress build-up following the main-shock resulted in the largest aftershock ( M w  7.3), which occurred on the MHT, immediately east of the main-shock rupture. Source mechanisms of moderate aftershocks reveal stress adjustment at the edges of the main-shock fault, flexural faulting on top of the downgoing Indian Plate and extensional faulting in the hanging wall of the MHT.

Description: Earthquake faults, and all frictional surfaces, establish contact through asperities. A detailed knowledge of how asperities form will enable a better understanding of the manner in which they communicate during foreshock failure sequences that are observed, leading to the larger main shock. We present results of experiments where a pressure sensitive film was used to map, size and measure the magnitudes of the normal stresses at asperities along a seismogenic section of a laboratory simulated fault. We measured seismicity acoustically and foreshocks were found to be the result of localized asperity failure during the nucleation phase of gross fault rupture. Since surface roughness plays an important role in how asperities are formed, two Hurst exponents were measured to characterize a highly worn interface using roughness profiles: (i) long wavelength estimates ( H ~ 0.45) and (ii) short wavelength estimates ( H ~ 0.8–1.2). The short wavelength roughness estimates were computed at the scale of single asperity junction points. Macroscopically, the number of asperities and real contact area increased with additional application of normal force while the mean normal stress remained constant. The ratio of real to nominal contact area was low – ranging from 0.02 〈 A r / A 0 〈 0.05—predicting that the asperities should be elastically independent of each other. Results from the pressure sensitive film showed that asperities were closely spaced and could not be treated as mechanically independent. Larger asperities carried both higher levels of average normal stress and higher levels of normal stress heterogeneity than smaller ones. Using linear stability theorem, the critical slip distance on foreshocking asperities was estimated to be d 0 ~ 0.65–3 μm. The critical slip distance d 0 was ~1.8–11.5 per cent of the premonitory slip needed to initiate gross fault rupture of the interface (20–40 μm) and the overall slip necessary to initiate gross fault rupture was on the order of the average asperity diameter (52 μm). Foreshocks may be due to a change in the critical slip distance, at localized sections of the fault, caused by the two distinct roughness profiles measured at short and long length scales.

Description: An approach for approximate direct quadratic non-linear inversion in two-parameter (density and bulk modulus) heterogeneous acoustic media is being presented and discussed in this paper. The approach consists of two parts: the first is a linear generalized Radon transform (GRT) migration procedure based on the weighted true-amplitude summation of pre-stack seismic scattered data that is adapted to a virtually arbitrary observing system, and the second is a non-iterative quadratic inversion operation, produced from the explicit expression of amplitude radiation pattern that is acting on the migrated data. This ensures the asymptotic inversion can continue to simultaneously locate the discontinuities and reconstruct the size of the discontinuities in the perturbation parameters describing the acoustic media. We identify that the amplitude radiation pattern is the binary quadratic combination of the parameters in the process of formulating non-linear inverse scattering problems based on second-order Born approximation. The coefficients of the quadratic terms are computed by appropriately handling the double scattering effects. These added quadratic terms provide a better amplitude correction for the parameters inversion. Through numerical tests, we show that for strong perturbations, the errors of the linear inversion are significant and unacceptable. In contrast, the quadratic non-linear inversion can give fairly accurate inversion results and keep almost the same computational complexity as conventional GRT liner inversion.

Description: We revisit the problem of coseismic rupture of the 2008 M w 7.9 Wenchuan earthquake. Precise determination of the fault structure and slip distribution provides critical information about the mechanical behaviour of the fault system and earthquake rupture. We use all the geodetic data available, craft a more realistic Earth structure and fault model compared to previous studies, and employ a nonlinear inversion scheme to optimally solve for the fault geometry and slip distribution. Compared to a homogeneous elastic half-space model and laterally uniform layered models, adopting separate layered elastic structure models on both sides of the Beichuan fault significantly improved data fitting. Our results reveal that: (1) The Beichuan fault is listric in shape, with near surface fault dip angles increasing from ~36° at the southwest end to ~83° at the northeast end of the rupture. (2) The fault rupture style changes from predominantly thrust at the southwest end to dextral at the northeast end of the fault rupture. (3) Fault slip peaks near the surface for most parts of the fault, with ~8.4 m thrust and ~5 m dextral slip near Hongkou and ~6 m thrust and ~8.4 m dextral slip near Beichuan, respectively. (4) The peak slips are located around fault geometric complexities, suggesting that earthquake style and rupture propagation were determined by fault zone geometric barriers. Such barriers exist primarily along restraining left stepping discontinuities of the dextral-compressional fault system. (5) The seismic moment released on the fault above 20 km depth is 8.2 x 10 21  N m, corresponding to an M w 7.9 event. The seismic moments released on the local slip concentrations are equivalent to events of M w 7.5 at Yingxiu-Hongkou, M w 7.3 at Beichuan-Pingtong, M w 7.2 near Qingping, M w 7.1 near Qingchuan, and M w 6.7 near Nanba, respectively. (6) The fault geometry and kinematics are consistent with a model in which crustal deformation at the eastern margin of the Tibetan plateau is decoupled by differential motion across a decollement in the mid crust, above which deformation is dominated by brittle reverse faulting and below which deformation occurs by viscous horizontal shortening and vertical thickening.

Description: In this paper, an underground experiment at the Äspö Hard Rock Laboratory (HRL) is described. Main goal is optimizing geothermal heat exchange in crystalline rock mass at depth by multistage hydraulic fracturing with minimal impact on the environment, that is, seismic events. For this, three arrays with acoustic emission, microseismicity and electromagnetic sensors are installed mapping hydraulic fracture initiation and growth. Fractures are driven by three different water injection schemes (continuous, progressive and pulse pressurization). After a brief review of hydraulic fracture operations in crystalline rock mass at mine scale, the site geology and the stress conditions at Äspö HRL are described. Then, the continuous, single-flow rate and alternative, multiple-flow rate fracture breakdown tests in a horizontal borehole at depth level 410 m are described together with the monitoring networks and sensitivity. Monitoring results include the primary catalogue of acoustic emission hypocentres obtained from four hydraulic fractures with the in situ trigger and localizing network. The continuous versus alternative water injection schemes are discussed in terms of the fracture breakdown pressure, the fracture pattern from impression packer result and the monitoring at the arrays. An example of multistage hydraulic fracturing with several phases of opening and closing of fracture walls is evaluated using data from acoustic emissions, seismic broad-band recordings and electromagnetic signal response. Based on our limited amount of in situ tests (six) and evaluation of three tests in Ävrö granodiorite, in the multiple-flow rate test with progressively increasing target pressure, the acoustic emission activity starts at a later stage in the fracturing process compared to the conventional fracturing case with continuous water injection. In tendency, also the total number and magnitude of acoustic events are found to be smaller in the progressive treatment with frequent phases of depressurization.

Description: We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave propagation modelling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the subvolume containing the perturbations or scatterers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modelled numerically, while in the surrounding volume, where the medium isn't perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhof–Helmholtz integrals that naturally adapt to the numerical scheme employed for modelling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered ‘exact’ as the total numerical error is comparable to that of the method employed for modelling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localized sensitivity kernels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarized in a dedicated section.

Description: This study considers the propagation of harmonic plane waves in a double-porosity dual-permeability solid saturated with single viscous fluid. Christoffel system is obtained to explain the existence of three longitudinal waves and a transverse wave in the medium considered. Each wave is identified with a complex velocity, which is resolved for inhomogeneous propagation to calculate the phase velocity and attenuation of the wave. Pore-fluid pressures are expressed in terms of velocities of solid particles corresponding to the propagation of three longitudinal waves. Then, transfer rate of pore-fluid between two porosities induced by each longitudinal wave is calculated as a function of its complex velocity. Numerical example is solved to study the dispersion in phase velocity and attenuation for each of the four waves. Effects of pore-fluid viscosity, wave-inhomogeneity and composition of double porosity on inhomogeneous propagation are analysed graphically. Transfer rate of pore-fluid, induced by each of the three longitudinal waves, is calculated as a periodic waveform. Variations in the fluid-flow profile are exhibited for different values of pore-fluid viscosity, skeleton permeability, wave-frequency and wave-inhomogeneity.

Description: Obtaining new seismic responses from existing recordings is generally referred to as seismic interferometry (SI). Conventionally, the SI responses are retrieved by simple crosscorrelation of recordings made by separate receivers: one of the receivers acts as a ‘virtual source’ whose response is retrieved at the other receivers. When SI is applied to recordings of ambient seismic noise, mostly surface waves are retrieved. The newly retrieved surface wave responses can be used to extract receiver-receiver phase velocities. These phase velocities often serve as input parameters for tomographic inverse problems. Another application of SI exploits the temporal stability of the multiply scattered arrivals of the newly retrieved surface wave responses. Temporal variations in the stability and/or arrival time of these multiply scattered arrivals can often be linked to temporally varying parameters such as hydrocarbon production and precipitation. For all applications, however, the accuracy of the retrieved responses is paramount. Correct response retrieval relies on a uniform illumination of the receivers: irregularities in the illumination pattern degrade the accuracy of the newly retrieved responses. In practice, the illumination pattern is often far from uniform. In that case, simple crosscorrelation of separate receiver recordings only yields an estimate of the actual, correct virtual-source response. Reformulating the theory underlying SI by crosscorrelation as a multidimensional deconvolution (MDD) process, allows this estimate to be improved. SI by MDD corrects for the non-uniform illumination pattern by means of a so-called point-spread function (PSF), which captures the irregularities in the illumination pattern. Deconvolution by this PSF removes the imprint of the irregularities on the responses obtained through simple crosscorrelation. We apply SI by MDD to surface wave data recorded by the Malargüe seismic array in western Argentina. The aperture of the array is approximately 60 km and it is located on a plateau just east of the Andean mountain range. The array has a T-shape: the receivers along one of the two lines act as virtual sources whose responses are recorded by the receivers along the other (perpendicular) line. We select time windows dominated by surface wave noise travelling in a favourable direction, that is, traversing the line of virtual sources before arriving at the receivers at which we aim to retrieve the virtual-source responses. These time windows are selected through a frequency-dependent slowness analysis along the two receiver lines. From the selected time windows, estimates of virtual-source responses are retrieved by means of crosscorrelations. Similarly, crosscorrelations between the positions of the virtual sources are computed to build the PSF. We use the PSF to deconvolve the effect of illumination irregularities and the source function from the virtual-source responses retrieved by crosscorrelation. The combined effect of time-window selection and MDD results in more accurate and temporally stable surface wave responses.

Description: We develop a general theory for interferometry by correlation that (i) properly accounts for heterogeneously distributed sources of continuous or transient nature, (ii) fully incorporates any type of linear and nonlinear processing, such as one-bit normalization, spectral whitening and phase-weighted stacking, (iii) operates for any type of medium, including 3-D elastic, heterogeneous and attenuating media, (iv) enables the exploitation of complete correlation waveforms, including seemingly unphysical arrivals, and (v) unifies the earthquake-based two-station method and ambient noise correlations. Our central theme is not to equate interferometry with Green function retrieval, and to extract information directly from processed interstation correlations, regardless of their relation to the Green function. We demonstrate that processing transforms the actual wavefield sources and actual wave propagation physics into effective sources and effective wave propagation. This transformation is uniquely determined by the processing applied to the observed data, and can be easily computed. The effective forward model, that links effective sources and propagation to synthetic interstation correlations, may not be perfect. A forward modelling error, induced by processing, describes the extent to which processed correlations can actually be interpreted as proper correlations, that is, as resulting from some effective source and some effective wave propagation. The magnitude of the forward modelling error is controlled by the processing scheme and the temporal variability of the sources. Applying adjoint techniques to the effective forward model, we derive finite-frequency Fréchet kernels for the sources of the wavefield and Earth structure, that should be inverted jointly. The structure kernels depend on the sources of the wavefield and the processing scheme applied to the raw data. Therefore, both must be taken into account correctly in order to make accurate inferences on Earth structure. Not making any restrictive assumptions on the nature of the wavefield sources, our theory can be applied to earthquake and ambient noise data, either separately or combined. This allows us (i) to locate earthquakes using interstation correlations and without knowledge of the origin time, (ii) to unify the earthquake-based two-station method and noise correlations without the need to exclude either of the two data types, and (iii) to eliminate the requirement to remove earthquake signals from noise recordings prior to the computation of correlation functions. In addition to the basic theory for acoustic wavefields, we present numerical examples for 2-D media, an extension to the most general viscoelastic case, and a method for the design of optimal processing schemes that eliminate the forward modelling error completely. This work is intended to provide a comprehensive theoretical foundation of full-waveform interferometry by correlation, and to suggest improvements to current passive monitoring methods.