Errors due to the anisotropic-common-ray approximation of the coupling ray theory

The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common-ray approximation applied. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray) approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order perturbations of travel time.     

Prevailing-frequency approximation of the coupling ray theory along the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately uniaxial

The behaviour of the actual polarization of an electromagnetic wave or elastic S–wave is described by the coupling ray theory, which represents the generalization of both the zero–order isotropic and anisotropic ray theories and provides continuous transition between them. The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. In a generally anisotropic or bianisotropic medium, the actual wave paths may be approximated by the anisotropic–ray–theory rays if these rays behave reasonably. In an approximately uniaxial (approximately transversely isotropic) anisotropic medium, we can define and trace the SH (ordinary) and SV (extraordinary) reference rays, and use them as reference rays for the prevailing–frequency approximation of the coupling ray theory. In both cases, i.e. for the anisotropic–ray–theory rays or the SH and SV reference rays, we have two sets of reference rays. We thus obtain two arrivals along each reference ray of the first set and have to select the correct one. Analogously, we obtain two arrivals along each reference ray of the second set and have to select the correct one. In this paper, we suggest the way of selecting the correct arrivals. We then demonstrate the accuracy of the resulting prevailing–frequency approximation of the coupling ray theory using elastic S waves along the SH and SV reference rays in four different approximately uniaxial (approximately transversely isotropic) velocity models.     

Prevailing-frequency approximation of the coupling ray theory for electromagnetic waves or elastic S waves

The coupling–ray–theory tensor Green function for electromagnetic waves or elastic S waves is frequency dependent, and is usually calculated for many frequencies. This frequency dependence represents no problem in calculating the Green function, but may represent a great problem in storing the Green function at the nodes of dense grids, typical for applications such as the Born approximation. This paper is devoted to the approximation of the coupling–ray–theory tensor Green function, which practically eliminates this frequency dependence within a reasonably broad frequency band.     

Effects of 1-D versus 3-D velocity models on moment tensor inversion in the Dobrá Voda area in the Little Carpathians region,Slovakia

Retrieving the parameters of a seismic source from seismograms involves deconvolving the response of the medium from seismic records. Thus, in general, source parameters are determined from both seismograms and the Green functions describing the properties of the medium in which the earthquake focus is buried. The quality of each of these two datasets is equally significant for the successful determination of source characteristics. As a rule, both sets are subject to contamination by effects that decrease the resolution of the source parameters. Seismic records are generally contaminated by noise that appears as a spurious signal unrelated to the source. Since an improper model of the medium is quite often employed, due to poor knowledge of the seismic velocity of the area under study, and since the hypocentre may be mislocated, the Green functions are not without fault. Thus, structures not modelled by Green functions are assigned to the source, distorting the source mechanism. To demonstrate these effects, we performed a synthetic case study by simulating seismic observations in the Dobrá Voda area of the Little Carpathians region of Slovakia. Simplified 1-D and 3-D laterally inhomogeneous structural models were constructed, and synthetic data were calculated using the 3-D model. Both models were employed during a moment tensor inversion. The synthetic data were contaminated by random noise up to 10 and 20 % of the maximum signal amplitude. We compared the influence of these two effects on retrieving moment tensors, and determined that a poor structural model can be compensated for by high-quality data; and that, in a similar manner, a lack of data can be compensated for by a detailed model of the medium. For examples, five local events from the Dobrá Voda area were processed.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

Two-point ray tracing in 3-D

Algorithm for determination of all two-point rays of a given elementary wave by means of the shooting method is presented. The algorithm is designed for general 3-D models composed of inhomogeneous geological blocks separated by curved interfaces. It is independent of the initial conditions for rays and of the initial-value ray tracer. The algorithm described has been coded in Fortran 77, using subroutine packages MODEL and CRT for model specification and for initial-value ray tracing.     

Importance of borehole deviation surveys for monitoring of hydraulic fracturing treatments

During seismic monitoring of hydraulic fracturing treatment, it is very common to ignore the deviations of the monitoring or treatment wells from their assumed positions. For example, a well is assumed to be perfectly vertical, but in fact, it deviates from verticality. This can lead to significant errors in the observed azimuth and other parameters of the monitored fracture‐system geometry derived from microseismic event locations. For common hydraulic fracturing geometries, a 2° deviation uncertainty on the positions of the monitoring or treatment well survey can cause a more than 20° uncertainty of the inverted fracture azimuths. Furthermore, if the positions of both the injection point and the receiver array are not known accurately and the velocity model is adjusted to locate perforations on the assumed positions, several‐millisecond discrepancies between measured and modeled SH‐P traveltime differences may appear along the receiver array. These traveltime discrepancies may then be misinterpreted as an effect of anisotropy, and the use of such anisotropic model may lead to the mislocation of the detected fracture system. The uncertainty of the relative positions between the monitoring and treatment wells can have a cumulative, nonlinear effect on inverted fracture parameters. We show that incorporation of borehole deviation surveys allows reasonably accurate positioning of the microseismic events. In this study, we concentrate on the effects of horizontal uncertainties of receiver and perforation positions. Understanding them is sufficient for treatment of vertical wells, and also necessary for horizontal wells.