No abstract

Testing and specification of seismic and earthquake-engineering sensors and recorders has been marked by significant variations in procedures and selected parameters. These variations cause difficulty in comparing such specifications and test results.

We present the results of applying autocorrelogram migration to inverse vertical seismic‐while‐drilling (IVSPWD) profile data collected in the Austin Chalk formation. The seismic reflections were excited by a drill bit moving along a horizontal well at a depth of 2800 m. The data were recorded by a receiver array on the surface. There were 609 twenty‐second traces recorded at each of 10 three‐component stations. After preprocessing, the traces were autocorrelated and migrated. Two imaging conditions were examined. The ghost‐reflection imaging condition produced a reflectivity section that agreed with a nearby common‐depth‐point section. The migration section obtained with a primary‐reflection imaging condition produced a reflectivity section inferior in quality to that of the ghost image. A possible explanation is that the drill‐bit location was not precisely known, which can be shown to induce weak errors in the ghost‐imaging condition but stronger errors in the primary‐reflection imaging condition. Ghost migration images reflections not only below the drill bit, but also above the drill bit. This investigation is one of the first examples of successfully imaging the earth's reflectivity section from horizontal drill‐bit data, and it offers a potentially useful method for seismic imaging from drill‐bit data.

We present the equations for migrating IVSPWD (inverse vertical seismic profile while drilling) and common-midpoint-point (CMP) autocorrelograms. These equations generalize the 1-D autocorrelation imaging methods of Katz and Claerbout to 2-D and 3-D media, and also provide a formal mathematical justification for inverting the reflectivity distribution from autocorrelograms. The autocorrelogram imaging conditions are designed to migrate either the primary reflection energy or the free-surface ghost reflections. The main advantage in migrating autocorrelograms is that the source wavelet does not need to be known, which is the case for seismic data generated by a rotating drill-bit or for vibroseis data with a corrupted pilot signal. Another advantage is that the source and receiver static problems are mitigated by autocorrelation migration. The key limitation is that autocorrelation of traces produces undesirable coherent noises, which are denoted as "virtual multiples". Similar to "physical 1 multiples", such noise can, in principle, be partially suppressed by filtering and stacking of migration images obtained from many different shot gathers. Results with both synthetic and field data validate this conjecture, and show that autocorrelogram migration can be a better alternative to standard migration when the source signal is not adequately known.

We give a systematic formulation and a rigorous justification of a perturbation technique for the computation of the eigenvalues and eigenfunctions of Love waves (and toroidal oscillations by an appropriate change for variables) in an anelastic medium with a constitutive law modelling geophysical media of current interest such as the Kelvin-Voigt Solid, the Maxwell Solid, the Standard Linear Solid, and the Standard Linear Solid with a continuous spectrum of relaxation times. We develop expressions relating the eigenvalues of eigenfunctions for Love waves in a continuously varying vertically stratified anelastic half-space to the corresponding elastic eigenvalues and eigenfunctions. Analytically, our correspondence principle has the form of a regular perturbation expansion in terms of a parameter E for both the eigenvalues and eigenfunctions. The identification of E is motivated by the dissipativity principle of viscoelasticity theory. Moreover, we show that our correspondence principle applies respectively only in the high and low frequency range for the Maxwell and Kelvin-Voigt Solids. Outside of the applicable range of frequencies, our correspondence principle yields no useful information. For the family of Standard Linear Solids it is uniformly applicable for all non-zero frequencies.We also derive an explicit formula to estimate the radius of convergence of our perturbation expansions. This estimate of the radius of convergence for each eigenvalue and eigenfunction is functionally defined by the constitutive model for the anelastic medium. The estimate is frequency dependent and depends on the separation distance between the eigenvalue and the remainder of the spectrum of the corresponding elastic problem.The stress-strain constitutive relationship for an anelastic medium is derived in Christensen (1971), Leitman & Fisher (1973), or Pipkin (1972. It is given by is the classical infinitesimal strain tensor. Gijkl(x, t ) has the representationWe will restrict our discussion to isotropic media. In this case (Christensen 1971), Gjjkz(x, t ) = 7 3 { G z (~, t ) -G I (X, t ) ) 6ij6kl' % G I ( x , t)(&ikajZ + 6iZbjk)( 2.4) where G l ( x , t ) is called the shear relaxation function, G z ( x , t ) is called the longitudinal relaxation function, andh j j is the Kronecker symbol. The relaxation functions Gi, i = 1, 2, are represented by Gi(t) = Gi(0) + IOt Gl(7)dT i = 1 , 2 , (2-5) where the real constants G i ( 0 ) are called the instantaneous moduli. They are related to the elastic response of the medium. If G,(O) = 0, i = 1 , 2 , then the medium has no instantaneous elastic response. If Gi(-) = lim G,(t) exist, then Gi(m) are called the equilibrium moduli. The momentum equation (2.1) and the constitutive relation (2.2) form a complete system of equations when supplemented with appropriate initial and boundary conditions (Aki & Richards 1980). Next, we derive the equation for Love waves in a continuously varying vertically stratified anelastic half-space. Let i = 1 , 2 , t-+ -5 2 = { ( x , z ) : -m < x < m , z 2 O } ...