Slowness-domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre-critical slowness match

INTRODUCTION

A review of the existing studies about moveout approximations for homogeneous and layered orthorhombic models is provided in Part I of this work; therefore, we only mention here the most relevant works for the current study. For long offsets in azimuthally isotropic layered media (e.g. VTI), Fomel and Stovas (2010) and Stovas and Fomel (2012) proposed the ‘generalized moveout approximations’ in both offset and slowness domains, respectively. This method requires information from an additional long-offset (wide-angle) ray. Alternative formulae, which are valid for an extended offset/slowness range, up to the case of nearly horizontal rays in multi-layer VTI media, have been presented in the offset domain by Ravve and Koren (2017) and in the slowness domain by Koren and Ravve (2018). In Part I of this work, we have extended the theory presented by Koren and Ravve (2018) from VTI to orthorhombic layered models, where, for each slowness azimuth, the additional wide-angle ray was considered to be the critical ray (critical slowness match). It was assumed that the model includes a dominant high-velocity layer, and that for nearly critical rays, most of the contributions to the offset and traveltime are due to the propagation in the high-velocity layer. In this work, Part II, the assumption of a dominant high-velocity layer is removed, and instead of the critical ray, a long-offset pre-critical ray is used. We derive the azimuthally varying slowness-domain kinematical characteristics (radial and transverse offset components, intercept time and traveltime) for compressional and converted waves, where the detailed derivations are described within the appendices. In Appendices A and B, we derive all partial derivatives of the intercept time, needed for the pre-critical slowness match, including the mixed derivatives, and the higher derivatives. In Appendix C, we discuss the limit case of the pre-critical slowness match, and compare it with the critical slowness match.

Using the same synthetic models as in Part I, we demonstrated that the accuracy of the pre-critical slowness match is higher than the accuracy of the critical slowness match within the given bounded slowness range.

Although this two-part paper is devoted to the forward computation of the kinematical characteristics for layered orthorhombic media, we have added a section describing the main role of the kinematical characteristics and the corresponding set of effective parameters within the seismic inversion workflow.

APPROXIMATIONS FOR THE KINEMATICAL CHARACTERISTICS

The approximation formulae for the critical slowness match presented in Part I and the pre-critical slowness match to be used in this part are identical. The azimuthally dependent coefficients are different and will be derived in this work. The expressions for the slowness-domain kinematical characteristics presented in Part I are given as:

• For the radial offset,
  (1)
• For the intercept time,
  (2)
• For the traveltime,
  (3)
• For the transverse offset,
  (4)

where are derivatives of coefficients with respect to the slowness azimuth ψ_slw, are the non-normalized and normalized horizontal slowness, respectively, is the normalized complementary slowness,

(5)

and is the critical slowness at a given slowness azimuth ψ_slw.

COEFFICIENTS FOR THE PRE-CRITICAL SLOWNESS MATCH

The resolving set of five equations needed in order to find the five azimuthally dependent coefficients for the pre-critical slowness match differs from that for the critical slowness match. The first two equations are still related to the normal-incidence ray, whose second- and fourth-order normal moveout (NMO) velocities, , can be computed for any pure-mode or converted wave type. Hence, like in Part I, the first two equations are given by

(6a)

(6b)

The three other equations are related to the pre-critical slowness match, and they are derived from the information given by the additional pre-critical traced ray. For 1D layered models, ray tracing actually means solving the vertical slowness equation A3 in Part I for the given horizontal slowness components, . The vertical slowness in a given layer is used to compute the contribution of that layer into the intercept time. The partial derivatives, , are then used to compute the contributions of that layer into the two offset components and the traveltime.

Considering the total two-way intercept time τ and its first and second derivatives with respect to the normalized horizontal slowness , the three additional conditions are

(7a)

(7b)

(7c)

Thus, for a given matching horizontal slowness magnitude and azimuth ψ_slw, the values of the intercept time and its derivatives with respect to the normalized horizontal slowness are specified.

The first derivative of the intercept time with respect to the horizontal slowness is given by

(8)

which is an obvious formula, because

(9)

Next we compute the second derivative of the intercept time,

(10)

We introduce parameters and for the normalized and non-normalized pre-critical matching horizontal slowness, respectively,

(11)

where is the normalized complementary matching slowness. Combining equations 2, (6), 8 and 10 and introducing the matching horizontal slowness instead of an arbitrary horizontal slowness, we obtain the resolving linear set. The five equations can be arranged in the following matrix form:

(12)

The 5 × 5 matrix on the left-hand side of equation 12 reads

(13)

and the right-hand side vector reads

(14)

The horizontal slowness in equation 14 corresponds to the pre-critical matching horizontal slowness . Solving the linear set of five equations for the given fixed magnitude and azimuth of the horizontal slowness (pre-critical slowness match), we find the unknown azimuthally dependent coefficients . We note that only the right-hand side vector J depends on the slowness azimuth. Its components have units of distance. The matrix components are independent of the azimuth. These components are normalized (unitless). The unknown variables also have units of distance. Once the approximation coefficients are obtained, the horizontal slowness in the approximation formulae (1)–(3) can accept any value, not to exceed the matching limit, . For the normalized horizontal slowness above the matching value, the errors of the approximations increase rapidly and catastrophically.

DERIVATIVES OF THE COEFFICIENTS FOR PRE-CRITICAL SLOWNESS MATCH

The five coefficients of the approximations for the radial offset, intercept time and traveltime depend on the slowness azimuth. However, in order to obtain the transverse offset component, we also need the azimuthal derivatives of these coefficients, . First, azimuthal derivatives can be computed from the equations for the second- and fourth-order normal moveout (NMO) velocities,

(15)

Next, we differentiate equations 2, 8 and 10 (for the intercept time τ and its derivatives , ) with respect to the slowness azimuth ψ_slw, keeping the normalized horizontal slowness constant. Constant also means constant normalized complementary slowness , because .

The azimuthal derivative of the intercept time reads

(16)

The mixed second derivative reads

(17)

The mixed third derivative reads

(18)

Combining equations 15–18, we obtain the linear set,

(19)

where the 5 × 5 matrix is the same as in equation 12, and its components are listed in equation 13. The right-hand side vector is different,

(20)

The normalized horizontal slowness in equation 20 corresponds to the pre-critical matching value, . Once the approximation coefficients are obtained, the horizontal slowness in the approximation formula (4) can accept any value within the range of . We emphasize that when computing the azimuthal derivatives, keeping the normalized horizontal slowness constant is not the same as keeping the horizontal slowness constant, because the normalizing factor – the critical slowness – depends on the slowness azimuth. It follows from equation 20 that three such derivatives are needed,

(21)

Note that the notation for the constant is related to the derivative with respect to the slowness azimuth only. The relationships between the azimuthal derivatives keeping constant and those keeping constant are given in Appendix A. Thus, we first compute regular partial derivatives of the intercept time (either with respect to the horizontal slowness magnitude, keeping the slowness azimuth constant, or vice versa). Next, we apply the relationships of Appendix A to establish the values for equation 21.

We invert matrix in equation 13, and then combining equations 12 and 19, we obtain

(22)

where and , , are components of the right-hand side vectors in equation sets (12) and (19), respectively (listed in equations 14 and 20).

DERIVATIVES OF THE INTERCEPT TIME

For the constant properties of each layer, the two-way intercept time can be presented as a sum of intercept times for the individual layers,

(23)

where i is the layer index, is the two-way intercept time of layer i, is the layer thickness and n is the number of layers in the model. Parameters and represent the vertical slowness of the ith layer for the incident and reflected waves (both are positive values). They depend on the wave mode, and, therefore, they are identical for pure-mode waves and different for converted waves. Thus, to obtain the derivatives of the intercept time, we just compute the derivatives of the slowness surface (the derivatives of the vertical slowness with respect to the magnitude and azimuth of the horizontal slowness), and stack these derivatives with the corresponding weights; each weight is the layer thickness . The derivatives of the slowness surface, for all wave types propagating within the layered orthorhombic media, are obtained analytically by implicit differentiation of the Christoffel polynomial equation, as described in Appendix B.

SYNTHETIC EXAMPLES

To test the accuracy of the pre-critical slowness match for the kinematical characteristics (the radial and transverse offset, intercept time and traveltime), we compute the relative errors of the approximations comparing them with the values computed by numerical ray tracing. We consider the same homogeneous and layered orthorhombic models as used in Part I, studying compressional waves and two types of converted waves, P–S1 and P–S2. The elastic and geometric properties of the layers for the two models are listed in Tables 1 and 2 in Part I. The error plots and maximum error tables demonstrate higher accuracy of the pre-critical slowness match that decreases as the matching slowness increases.

Table 1. Relative errors of the approximations for the single-layer model

	Wave Type		Radial Offset	Transverse	Intercept Time	Traveltime
0.86	P–P	3.905
	P–S1	2.554
	P–S2	2.572
0.90	P–P	4.887
	P–S1	3.070
	P–S2	3.092
0.92	P–P	5.617
	P–S1	3.447
	P–S2	3.470
0.95	P–P	7.399
	P–S1	4.356
	P–S2	4.382
0.97	P–P	9.808
	P–S1	5.573
	P–S2	5.600
1	P–P	∞
	P–S1	∞
	P–S2	∞

Table 2. Relative errors of the approximations for the multi-layer model

	Wave Type		Radial Offset	Transverse	Intercept Time	Traveltime
0.90	P–P	2.202
	P–S1	1.438
	P–S2	1.509
0.92	P–P	2.373
	P–S1	1.532
	P–S2	1.603
0.95	P–P	2.743
	P–S1	1.729
	P–S2	1.802
0.98	P–P	3.589
	P–S1	2.164
	P–S2	2.238
0.99	P–P	4.448
	P–S1	2.598
	P–S2	2.672
0.995	P–P	5.620
	P–S1	3.186
	P–S2	3.261
0.998	P–P	7.910
	P–S1	4.332
	P–S2	4.407
0.999	P–P	10.48
	P–S1	5.615
	P–S2	5.691
1	P–P	∞
	P–S1	∞
	P–S2	∞

Accuracy of the homogeneous model

Figures 1, 2 and 3 demonstrate the accuracy of the approximations for compressional waves, for the normalized matching horizontal slowness values , respectively. Figures 4, 5 and 6 demonstrate the accuracy for P–S1 converted waves for the same matching horizontal slowness, and Figs 7, 8 and 9 for P–S2 converted waves. The relative errors are plotted for all slowness azimuths and for all magnitudes of the horizontal slowness, up to the matching values.

Accuracy of the layered model

Figures 10, 11 and 12 demonstrate the accuracy of the approximations for compressional waves, for the normalized matching horizontal slowness values , respectively. Figures 13, 14 and 15 demonstrate the accuracy for P–S1 converted waves for the same matching horizontal slowness, and Figs 16, 17 and 18 for P–S2 converted waves. For the pre-critical slowness match, the same tendency holds as for the critical slowness match. The intercept time approximation is more accurate than the approximations for the radial offset and the traveltime, because the integration operator smoothens off the errors/inaccuracies of the function under the integral operator.

Tables 1 and 2 summarize the maximum relative errors for all magnitudes and azimuths of the horizontal slowness for the homogeneous and layered orthorhombic models, respectively. The first column in these tables shows the matching pre-critical slowness. Critical slowness match is considered a particular case. The second column is the wave type.

The normalized matching horizontal slowness is a suitable computational parameter; however, the offset-to-depth ratio is more informative. Given a fixed normalized or non-normalized horizontal slowness ( or ), the offset-to-depth ratio in an orthorhombic model becomes azimuthally dependent. However, the exact value for each azimuth is not that essential, and therefore, we computed the average value for all slowness azimuths (column 3). The incidence and reflection angles are also azimuthally dependent for constant .

Note that the normalized slowness matching values (0.999, 0.998 and 0.995) applied for the layered model were intentionally chosen much closer to the critical slowness than the matching values that we used for the homogeneous model (0.97, 0.95 and 0.92). For a multi-layer model, when approaches 1, the offset-to-depth ratio increases with a smaller rate than for a single layer. The reason is the low-velocity layers, whose contribution to the offset (and traveltime) remains finite as the normalized slowness approaches 1. Columns 4–7 show the maximum relative errors for the four kinematical characteristics. The greatest errors are indicated in bold font and correspond to the critical slowness match.

The accuracy of the approximations for the pre-critical slowness match is much higher than that for the critical slowness match, even in the cases when the matching slowness differs from the critical value by a couple of per cent (single-layer model) or even a small fraction of a per cent (multi-layer model).

DISCUSSION: THE ROLE OF KINEMATIC CHARACTERISTICS IN SEISMIC INVERSION

We believe that one should not underestimate the importance of the derived global and local effective parameters that provide explicit approximation formulae for the kinematical characteristics. Even within the frame of waveform- or ray-based inversions, the effective parameters provide valuable information (e.g. Zhang and Alkhalifah 2017). The seismic method, with all the available data, still cannot uniquely determine the complexity of the subsurface model, especially in areas characterized by considerable heterogeneity and anisotropy effects. The average nature of the effective parameters plays an important role in setting the framework of the model to be derived and the corresponding physical constraints to be applied.

CONCLUSIONS

Considering compressional and converted waves, we propose accurate, azimuthally dependent slowness-domain approximations for the kinematical characteristics: the radial and transverse offset components, intercept time and traveltime for horizontally layered orthorhombic models. The approximations are valid up to a given wide angle range. The approximations are based on merging the series expansion coefficients of the intercept time for two (virtual) rays: the normal-incidence ray and an additional azimuthally dependent wide-angle ray. In Part I, the wide-angle ray was considered the azimuthally varying critical ray, where we showed the effectiveness of this method in cases where the subsurface models include dominant high-velocity layers. In Part II, the requirement for a dominant high-velocity layer is removed, and the azimuthally varying wide-angle ray is associated with a finite long offset related to a given pre-critical slowness. The general form of the approximated formulae is the same; however, the coefficients are different. Within the given pre-critical slowness range, the accuracy of the pre-critical slowness match presented in Part II is higher than the accuracy obtained by the critical slowness match presented in Part I.