Depth-Recursive Tomography Along the Eger Rift Using the S01 Profile Refraction Data: Tested at the KTB Super Drilling Hole, Structural Interpretation Supported by Magnetic, Gravity and Petrophysical Data

Abstract

The refraction data from the SUDETES 2003 experiment were used for high-resolution tomography along the profile S01. The S01 profile crosses the zone Erbendorf-Vohenstrauss (ZEV) near the KTB site, then follows the SW–NE oriented Eger Rift in the middle part and continues toward the NE across the Elbe zone and the Sudetic structures as far as the Trans-European Suture Zone. To get the best resolution in the velocity image only the first arrivals of Pg waves with minimum picking errors were used. The previous depth-recursive tomographic method, based on Claerbout’s imaging principle, has been adapted to perform the linearized inversions in iterative mode. This innovative DRTG method (Depth-Recursive Tomography on Grid) uses a regular system of refraction rays covering uniformly the mapped domain. The DRTG iterations yielded a fine-grid velocity model with a required level of RMS travel-time fit and the model roughness. The travel-time residuals, assessed at single depth levels, were used to derive the statistical lateral resolution of “lens-shaped” velocity anomalies. Thus, for the 95% confidence level and 5% anomalies, one can resolve their lateral sizes from 15 to 40 km at the depths from 0 to 20 km. The DRTG tomography succeeded in resolving a significant low-velocity zone (LVZ) bound to the Franconian lineament nearby the KTB site. It is shown that the next optimization of the model best updated during the DRTG iterations tends to a minimum-feature model with sweeping out any LVZs. The velocities derived by the depth-recursive tomography relate to the horizontal directions of wave propagation rather than to the vertical. This was proved at the KTB site where pronounced anisotropic behavior of a steeply tilted metamorphic rock complex of the ZEV unit has been previously determined. Involving a ~7% anisotropy observed for the “slow” axis of symmetry oriented coincidentally in the horizontal SW–NE direction of the S01 profile, the DRTG velocity model agrees fairly well with the log velocities at the KTB site. Comparison with the reflectivity map obtained on the reflection seismic profile KTB8502 confirmed the validity of DRTG velocity model at maximum depths of ~16 km. The DRTG tomography enabled us to follow the relationship of major geological units of Bohemian Massif as they manifested in the obtained P-wave velocity image down to 15 km. Although the contact of Saxothuringian and the Teplá-Barrandian Unit (TBU) is collateral with the S01 profile direction, several major tectonic zones are rather perpendicular to the Variscan strike and so fairly imaged in the S01 cross-section. They exhibit a weak velocity gradient of sub-horizontal directions within the middle crust. In particular, the Moldanubian and TBU contact beneath the Western Krušné hory/Erzgebirge Pluton, the buried contact of the Lusatia unit and the TBU within the Elbe fault zone were identified. The maxima on the 6,100 ms⁻¹ isovelocity in the middle crust delimitated the known ultrabasic Erbendorf complex and implied also two next ultrabasic massifs beneath the Doupovské hory and the České středohoří volcanic complexes. The intermediate mid-crustal P-wave velocity lows are interpreted as granitic bodies. The presented geological model is suggested in agreement with available gravity, aeromagnetic and petrophysical data.

Appendix A: Depth Recursive Tomography

1.1 A1: Linearization and Depth-Recursive Concept

We present here the 3-D extension of the grid depth-recursive method proposed originally in (Novotný 1981) only for 2-D case and 1-D starting model.

Assuming certain proximity of the starting slowness model s ₀(x, y, z) and searched models s(x, y, z) = s ₀(x, y, z) + Δs(x, y, z), expressed using the L2 norm as

$$ \left\| {s(x,y,z) - s_{0} (x,y,z)} \right\| = \left\| {\Updelta s(x,y,z)} \right\| \approx 0, $$

(4)

the following well-known linear relation between the time misfit Δt and the unknown slowness model correction Δs is valid

$$ \Updelta t(G_{m} ) \approx \int\limits_{{G_{m} }} {\Updelta s(x,y,z){\text{d}}g} . $$

(5)

Integration is performed along a particular ray G _m computed for the starting model s ₀ and Δt(G _m ) denotes the time difference of observed and calculated arrival time for the ray path G _m . Thus, in a grid model representation, the Eq. 5 relates the time misfits to the searched slowness corrections that are to be assigned to all grid cells hit by this ray. The linearization errors bounded to the approximation (5) are necessary to reduce using more than one iteration with updated starting velocity models.

To obtain a depth-recursive solution for the unknown slowness corrections let us proceed consecutively in the depth steps given by the chosen grid sampling z ₀, z ₁, z ₂, …. Assume in the i-th recursive step that the grid slowness corrections Δs(x _j , y _k , z) are established at all the grid nodes down to z ≤ z _i. Consider a refraction ray G _m crossing the plane z = z _i. Integrate (5) separately in the two depth domains, below and above z = z _i. The integration of (5) in the upper domain z ≤ z _i can be performed along G _m utilizing the already known nodal corrections Δs (see close grid nodes in the illustration of G _m refraction rays bottoming at the z _i+1 level in Fig. 15):

$$ \Updelta t(G_{m} ) - \int\limits_{{z_{0} \le z \le z_{i} }} {\Updelta s(x,y,z)d{\text{g}}} = \int\limits_{z\, > zi} {\Updelta s(x,y,z){\text{d}}g.} $$

(6)

Fig. 15

The grid rays generated for the nodes at the z _i+1 grid level

The left-hand side term contains the integration across the known part of model and virtually means the Δt misfit extrapolated downward to the depth level z = z _i . The integration involves the both branches of the refraction ray path G _m cut by the horizontal grid plane z = z _i—see Fig. 15. For the downward-extrapolated time misfit Δt we will use the notation

$$ \Updelta t(G_{m} ,z_{o} ,z_{i} ) = \Updelta t(G_{m} ) - \int\limits_{{z_{0} \le z \le z_{i} }} {\Updelta s(x,y,z){\text{d}}g.} $$

(7)

A reasonable choice and ordering of refraction rays used for calculating the unknown tomographic corrections with the recursive computation scheme (6) is described in following.

The ray paths corresponding to surface-to-surface refraction arrivals have at least one bottom point with the horizontal ray direction. A natural specification of imaging rays is to define at least one ray G _m for each grid point m assuming that the ray propagates at the node in a horizontal direction with a chosen azimuthal angle. The choice of azimuthal angles may be important if we have areal data and the anisotropic behaviour of 3-D medium is studied. In the case of 2-D data, the angle is given by the azimuth of seismic line. We suppose a fine -grid model to cope with the peculiarities of the observed travel-time data.

The refraction rays G _m (called grid rays) may be simply computed by any shooting ray-tracing procedure as two ray branches initiated in a particular grid point m with two horizontal, mutually opposite, directions. The emergence points E ₁ and E ₂ of a ray G _m , if they exist, define its hypothetical source and receiver points with the corresponding offset and midpoint position (see illustration in Fig. 15).

Naturally, not all grid rays reach the surface or have the emergence in the range of available observations. As well, their trajectories differ during iterations in dependence on the starting model used. However, the concept of grid rays calculated in the each iteration and at all model grid nodes means that the closest correspondence between the observed and model travel times is searched. The proposed new term “tomoscopic rays” for grid rays reflects their investigative character—no all tomoscopic rays serve to imaging.

In one step of depth recursion, the grid rays G _m bottoming at all grid nodes z = z _i+1 are required. Knowing their emergence points determining the offsets and midpoint positions, the observed travel times can be assigned. For this, a continuous numerical representation of observed travel times should be used and certain criteria eliminating irregular rays beyond the range of available travel times must be applied. The subset of “successful” tomoscopic rays G _m with successfully interpolated field travel times are then used to determinate the tomographic corrections at the z _i+1 grid level in the way as follows.

In agreement with our depth recursive concept, in the i + 1 step we have determined the tomographic corrections in the domain z ≤ z _i (see the close nodes in the Fig. 15). Let us derive the equations for tomographic corrections Δs at next plane z = z _i+1. For this the raypaths $G^{i + 1}_{m}$ bottoming at the grid level z _i+1 will be engaged. Assume the index m = 1, 2, …, M defines in an unambiguous way the x and y positions (j _m , k _m ) of their bottom points, i.e. grid nodes, in the grid plane z = z _i+1. Using the notation (7) the Eq. 6 takes then the form

$$ \Updelta t\left( {G^{i + 1}_{m} ,\,z_{o} ,\,z_{i} } \right) = \int\limits_{{z_{i} \le z \le z_{i + 1} }} {\Updelta s(x,y,z)\,{\text{d}}g_{m} .} $$

(8)

Now, the downward extrapolation $\Updelta t\left( {G^{i + 1}_{m} ,z_{o} ,z_{i} } \right)$ relates to the ray $G^{i + 1}_{m}$ bottoming at the z _i+1 grid level under investigation. Note that integration in (8) involves just the deepest bottoming part of the $G^{i + 1}_{m}$ raypath between the z _i and z _i+1 levels. Expand the Δs function among the grid nodes using the linear spline functions,

$$ \Updelta s(x,y,z) = \sum\limits_{j,k,l} {\Updelta s(x_{j} ,y_{k} ,z_{l} ) X_{j} (x)Y_{k} (y)Z_{l} (z)} $$

(9)

where the spikes, e.g., Z _l (z), are defined as (Fig. 16)

$$ Z_{{l}} (z) = \left\{ {\begin{array}{*{20}c} {(z - z_{{{{l}} - 1}} )/(z_{{l}} -z_{{{{l}} - 1}} ),} & {{\text{for }}z_{{{{l}} - 1}} < z \le z_{{l}} ,} \\ {(z_{{{{l}} + 1}} - z)/(z_{{{{l}} + 1}} -z_{{l}} ),} & {{\text{for }}z_{{l}} \le z < z_{{{{l}} + 1}} ,} \\ 0 & {{\text{elsewhere}}.} \\ \end{array} } \right. $$

(10)

Fig. 16

The linear spikes used in the depth-recursive tomography

By the use of expansion (9) in the relation (8) and after evaluating all ray integrals m = 1,2,…,M., a system of linear equations for unknown grid values Δs(x _j ,y _k ,z _i+1) in the plane z _i+1 is clearly obtained. In more detail, substituting (9) with Z _l (z) expressed from (10) into integral (8) and writing down the non-zero terms for l = i and l = i + 1, one gets

$$ \Updelta t\left( {G ^{i + 1}_{m} ,z_{o} ,z_{i} } \right) = C_{m} + \sum\limits_{j,\,k} {D_{\text{jkm}} \,\Updelta s(x_{j} ,y_{k} ,z_{i + 1} ),} $$

(11)

where

$$ C_{m} = (z_{i + 1} - z_{i} )^{ - 1} \sum\limits_{j,k} {\Updelta s(x_{j} ,y_{k,} z_{i} )} \int\limits_{{z_{i} \le z \le z_{i + 1} }} {X_{j} (x)Y_{k} (y)(z_{i + 1} - z){\text{d}}g_{m} } , $$

(12)

$$ D_{\text{jkm}} = \left( {z_{i + 1} -z_{i} } \right)^{ - 1} \int\limits_{{z_{i} \le z \le z_{i + 1} }} {X_{j} (x)Y_{k} (y)\left( {z - z_{i} } \right){\text{d}}g_{m} } . $$

(13)

j, k run over all grid points of the planes z = z _i and z = z _i+1. Non-zero contributions to integrations (12) and (13) can only come from the grid points closest to the raypath $G^{i + 1}_{m}$. The vector C _m and the interpolation matrices D _jkm correspond subsequently (m = 1,2,…,M) to the imaged (j _m , k _m ) grid points at the grid plane z = z _i+1. The computation of $\Updelta t\left( {G^{i + 1}_{m} ,z_{o} ,z_{i} } \right)$ according to (7) and (11) involves all the slowness differences Δs(x _j , y _k , z _l ) mapped out in the previous depth recursions l = 1,2,…, i:

$$ \Updelta t\left( {G^{i + 1}_{m} ,z_{o} ,z_{i} } \right) = \Updelta t\left( {G^{ i + 1}_{m} } \right) - \sum\limits_{j,k,l} {\Updelta s(x_{j} ,y_{k} ,z_{l} )} \int\limits_{{z_{0} \le z \le z_{i} }} { X_{j} (x)Y_{k} (y)Z_{l} (z){\text{d}}g_{m} } . $$

(14)

In the presented numerical implementation, the integrations in (11–14) are performed in the time step used for the ray path storage.

1.2 A2: Direct Inversion Method

A reasonable computational scheme for obtaining the unknown Δs(x _j , y _k , z _i+1) in (11) depends on the extent and also on the character of input data available. The simplest scheme can be derived for a grid step ∆z sufficiently small that allows for neglecting the x and y interpolation near the bottom point at the z = z _i+1 plane. Then, a common fixed value Δs _m (z _i+1) may be used in the sum (11) instead of Δs(x _j, y _k, z _i+1) and so

$$ \Updelta t\left( {G ^{i + 1}_{m} ,z_{o} ,z_{i} } \right) \approx C_{m} + \Updelta s_{m} (z_{i + 1} )\sum\limits_{j,k} { D_{\text{jkm}} } $$

(15)

or

$$ \Updelta s_{m} (z_{i + 1} ) \approx {{\left[ {\Updelta t\left( {G ^{i + 1}_{m} ,z_{o} ,z_{i} } \right) - C_{m} } \right]} \mathord{\left/ {\vphantom {{\left[ {\Updelta t\left( {G ^{i + 1}_{m} ,z_{o} ,z_{i} } \right) - C_{m} } \right]} {A_{m} }}} \right. \kern-\nulldelimiterspace} {A_{m} }} $$

(16)

with A _m denoting the sum

$$ A_{m} = \sum\limits_{j,k} {D_{\text{jkm}} } $$

(17)

According to (16), one Δs _m (z _i+1) correction can be derived from one downward-extrapolated value of Δt for the raypath $G^{i + 1}_{m}$. This property implied the name of the method as “Direct Inversion Method” (DIME). Thus, starting with i = 1,2,… the Eq. 16 can be used for recursive deriving of all tomographic corrections Δs(x _j , y _k , z _i+1) for updating the next slowness model

$$ s(x_{j} ,y_{k} ,z_{i + 1}) = s_{o} (x_{j} ,y_{k} ,z_{i + 1}) + \Updelta s(x_{j} ,y_{k} ,z_{i + 1}) $$

(18)

in the solution domain.

A new numerical implementation of refraction inversion, based on the above relations, was performed and named as the DRTG method (Depth-Recursive Tomography in Grid). Using the formula (16) is advantageous for imaging “unstable” domains of velocity model thanks to its good localisation property: one downward-extrapolated time difference $\Updelta t\left( {G^{i + 1}_{m} ,z_{o} ,z_{i} } \right)$ due to one input refraction arrival determines just one grid slowness correction.

As follows from the above derivation, the integration in (8) involves just the deepest bottoming part of imaging rays. Thus, the DRTG method establishes the velocities of engaged elastic waves rather for horizontal directions than for vertical ones that can substantially differ in the case of anisotropic behaviour of studied medium.