Acta Geophysica vol. 60, no. 1, Feb. 2012, pp. 59-75

DOI: 10.2478/s11600-011-0058-5

________________________________________________ © 2011 Institute of Geophysics, Polish Academy of Sciences

The Structure of the Crust in TESZ Area by Kriging Interpolation

Mariusz MAJDAŃSKI

Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland e-mail: mmajd@igf.edu.pl

A b s t r a c t

A precise 3D model of the crust is necessary to start any tectonic or geodynamic interpretation. It is also essential for seismic interpretations of structures lying below as well as for correct analysis of shallow struc-tures using reflection seismics. During the last decades, a number of wide-angle refraction experiments were performed on the territory of central and eastern Europe (POLONAISE’97, CELEBRATION 2000, SUDETES 2003), resulting in many high quality 2D models. It is an in-teresting and complicated transition zone between Precambrian and Pa-laeozoic Platforms. This paper presents 3D model of the velocity distribution in the crust and upper mantle interpolated from 2D models of the structure along 33 profiles. The obtained model extends to a depth of 50 km and accurately describes the main features of the crustal structures of Poland and surrounding areas. Different interpolation techniques (Kriging, linear) are compared to assure maximum precision. The final model with estimated uncertainty is an interesting reference of the area for other studies.

Key words: crustal structure, TESZ, interpolation, Kriging.

1. INTRODUCTION AND MOTIVATION The Trans-European Suture Zone (TESZ) localized in central Europe is one of the most interesting tectonic structures in Europe. It is a boundary be-tween the old Precambrian East European Craton (EEC) and younger struc-tures accreted to the Palaeozoic Platform. The tectonic situation is even more

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complicated by the Carpathians front attacking it from the south, as pre-sented in Fig. 1. Several seismic experiments were focused on explaining this complicated structure, resulting in many detailed models of P-wave

Fig. 1. Schematic geological map of the TESZ area. The yellow rectangular marks the area of interest. It is a complicated zone with significantly different types of the crust: old Precambrian in the NE, younger Palaeozoic in SW, and the Carpathian front in the south. The small map below marks the above area on the map of Europe. PB – Pannonian basin, TB – Transylvania basin, HCM – Holy Cross Mountains, MM – Małoposka Massif, USB – Upper Silesian basin, and LT – Lublin trough.

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velocity distributions along seismic lines. The complication of the structure is so large that close profiles might have a different number of layers. Because of that, it is difficult to create a precise 3D model for such a large area.

One possibility to obtain a reliable 3D model is to use interpolation and estimate velocity field in a volume. Simple algorithms like linear interpola-tion can interpolate values between profiles, but they can neither extrapolate them nor estimate uncertainties of the results. Linear interpolation can also introduce sharp changes at the edges of modeled area. These nuisances can be neglected by more advanced geostatistical techniques like Kriging (Isaaks and Srivastava 1989, Clark 2001). These techniques, based on the assump-tion of smooth changes in analyzed data, can estimate values at any point in space and give an estimation of uncertainty. They are also fast and easy to implement.

The final 3D model, even without sharp velocity contrasts at boundaries, can describe the main structures in the crustal if initial data were densely sampled. With uncertainty estimation it can be used as a reference model for other studies, like receiver function, teleseismic tomography, potential fields or even full waveform inversion.

2. DATA SET In the study area that spreads from 47°N to 55°N and from 13°E to 25°E we have a large number of good quality seismic profiles from experiments POLONAISE’97 (Guterch et al. 1999), CELEBRATION 2000 (Guterch et al. 2003) and SUDETES 2003 (Grad et al. 2003b). Altogether, there were 33 profiles taken for the analysis: from POLONAISE’97 – P1 (Jensen et al. 1999), P2 (Janik et al. 2002), P3 (Środa et al. 1999), P4 (Grad et al. 2003a), P5 (Czuba et al. 2002); from SUDETES 2003 – S01 (Grad et al. 2008), S02, S03, and S06 (Majdański et al. 2006), S04 (Hrubcová et al. 2010), S05 (per-sonal communication with T. Janik); from CELEBRATION 2000 – CEL01 (Środa et al. 2006), CEL02 (Malinowski et al. 2005), CEL03 (Janik et al. 2005), CEL05 (Grad et al. 2006), CEL09 (Hrubcová et al. 2005), CEL10 (Hrubcová et al. 2008), CEL04, CEL06, CEL11, CEL12, CEL13, CEL14, CEL21, CEL22, and CEL23 (Janik et al. 2009). There were also other single profiles included in the analysis: LT-2, LT-4, LT-5, and LT-7 (Grad et al. 2005), ALP01 (Brückl et al. 2007), Eurobridge (EURO-BRIDGE’95 Work-ing Group 2001), and PANCAKE (personal communication with T. Janik). The map showing profiles included in this analysis is presented in Fig. 2. For each of the above models, that were modelled using raytracing technique, P-wave velocity distribution was sampled every 10 km along the profile and every 1 km with depth. Similar sampling technique and estimation of geo-graphical coordinates for those points are explained with more details by

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Fig. 2. Location of deep seismic sounding wide-angle refraction profiles in the study area. Each colour marks different experiment: POLONAISE’97 (P), CELEBRA-TION 2000 (C), SUDETES 2003 (S), LT, and single profile from ALP2002, Euro-bridge 96, and PANCAKE. Profiles are also marked with labels. The points every 10 km along profiles where 1D velocity models (with values on every 1 km depth) were sampled are indicated by black dots. The rectangular marks the area as in Fig. 1.

Majdański et al. (2009). An example velocity fields sampled from profiles P4 and CEL02 are presented in Fig. 3. The next step of the analysis was a se-lection of velocities verified with refracted waves. Comparing with rays cov-erage for each individual profile, the velocity fields were truncated at the edges and at the depth where the maximum depth of bending refracted waves was observed. Truncated models are also presented in Fig. 3. As de-scribed in several papers (e.g., Grad et al. 2008) the uncertainties in veloci-ties in those models are estimated as 0.1-0.2 km/s and for a depth of deep boundaries as 1-2 km. We need to remember that sparse sampling every 10 km along profiles can incorporate an additional uncertainty that we esti-mate to be lower than 0.1 km/s.

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Fig. 3. Two examples (P4 and CEL02 profiles) of selecting the part of the input models with verified velocity field. Verification was based on published ray cover-age of refraction rays in each profile separately.

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3. LINEAR INTERPOLATION The gathered set of 71 038 velocity data containing latitude, longitude, depth and P-wave velocity value were interpolated using linear interpolation. Interpolation grid was set as 0.1 degree for both longitude and latitude and 1 km with depth. Two algorithms were used: interpolation of 2D layer for data at the same depth calculated for each depth separately, and single 3D interpolation of all data at once. The results were identical for the major part of the volume. The only differences were found at the edges for deep layers (about 45 km). For further use, the 3D result was selected. Slices through the resulting model are presented in Fig. 4.

At the 10 km depth slice we recognize a zone of low velocities stretching from NW to SE that corresponds to deep sediments in the TESZ area. On both sides we observe crustal velocities of about 6.3 km/s for EEC and about 6.1 km/s for the Palaeozoic complex. The velocities lower than 6 km/s in the

Fig. 4. Horizontal slices through interpolated (linear interpolation) velocity field at 10, 20, 30, and 40 km. Colour scales represent P-wave velocity. Black lines mark positions of the profiles used in interpolation that is why on deeper slices some lines are missing. At shallow slices we see low velocities corresponding to deep sedi-ments in Polish Basin. Clear edge of the EEC is visible at 40 km.

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Carpathians area correspond to complicated mixture of rock types. At 20 km we still see a similar suture zone structure but all velocities suggest crystal-line composition. Significantly smaller velocities, 5.9 km/s, are observed be-low the Carpathians (21.5°E, 49.5°N). At 30 km depth the situation changes and we see crystalline rock’s velocities (about 6.8 km/s) for both TESZ and EEC but higher lower-crustal and upper-mantle velocities to the south of TESZ. In the lower left corner between profiles ALP01 and C10, low veloci-ties of 6.4 km/s are visible, marking the southern edge of the Bohemian Massif. At 40 km we have got clear division for crustal velocities in EEC area and upper mantle velocities for both TESZ and Palaeozoic complex. In this figure we also see that the area of estimated velocities is getting smaller with depth. This is a result of a smaller number of data points at larger depths, because of truncation of initial models. Please notice that at 40 km slice some profiles, like S02, S03, and S06, are completely missing, because reliable information on them was shallower than 40 km. However, we can easily observe several artificially looking sharp edges in all slices. Those are artifacts generated by linear interpolation. To overcome those effects we need a more advanced method of interpolation.

4. KRIGING INTERPOLATION The Kriging method is a linear estimation algorithm based on observed val-ues of surrounding data points, weighted according to spatial covariance val-ues. Comparing to simpler methods (nearest neighbor or linear interpolation) Kriging is the best linear unbiased predictor in the mean squared error sense. In analysis presented in this paper we are using the Ordinary Kriging that assumes stationarity of the data. The estimated value V in a given point is a weighted sum of values (j index) in the surrounding points,

j jV Vω=∑ , (1)

where ω are weights calculated according to the equation

1 1

2 21

( )( )

1

i i

i i

hh

ω γω γ

λ

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

C (2)

by inversion of covariance matrix C defined as in Eq. (3) below. The i index corresponds to unknown estimation point and hi1, ωi1 are, respectively, the distance and weight between the first data point and calculation point. The λ parameter in Eq. (2) is the so-called Lagrange multiplier that is helping to overcome deviation in the data from the assumed stationarity.

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11 12 13

21 22 23

31 32 33

( ) ( ) ( ) 1( ) ( ) ( ) 1( ) ( ) ( ) 1

1 1 1 0

h h hh h hh h h

γ γ γγ γ γγ γ γ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

C

…

. (3)

Matrix C is derived from all pair combinations in our input data and ex-presses variability in analyzed data. Elements of matrix C are semivariances calculated for each combination of pairs of data points according to the semivariance used. The estimation variance is called Kriging standard devia-tion (KSD) (see the following Eq. (4)) and includes the sum of weighted variances and Lagrange multiplier. It is not strictly a standard deviation, be-cause it includes deviations from assumed stationarity and isotropy in the in-put data. It has to be interpreted with care

2 ( )i i ihσ γ ω λ= +∑ . (4)

For the analysis presented in this paper we are using the most common spherical divergence form (Eq. (5)) to describe semivariance in the input data. This curve fitted to input data was used as an estimator for our predic-tion (see Fig. 5). Because of the assumed stationarity, the semivariance depends only on the distance between points, not the direction.

Fig. 5. Schematic example of semivariogram. Empirical semivariance values calcu-lated from input data are represented by circles. Fitted spherical divergence curve plotted as a line. Range a marks the saturation point where theoretical semivario-gram riches constant value of the sill (C0 + C1). The nugget value C0 is a value of theoretical semivariance at zero distance.

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3

0 1

0 1

1.5 0.5 , if | |( )

, if | |

h hC C h ah a a

C C h a

γ

⎡ ⎛ ⎞⎛ ⎞+ − ≤⎢ ⎜ ⎟⎜ ⎟⎜ ⎟= ⎝ ⎠⎢ ⎝ ⎠⎢ + >⎣

(5)

C0 (nugget), C1, and range a are parameters fitted to the data presented as an empirical semivariance

[ ]2

,

1ˆ( ) ( ) ( )2 i j

h V i V jN

γ = −∑ . (6)

For a large number of points, the empirical semivariogram is computed by binning the differences and averaging results around specified offset ranges. Schematic fit and shape of spherical divergence function with description of its parameters is presented in Fig. 5. Using this typical approach, a single theoretical semivariogram with defined saturation range a should be fitted to the empirical semivariogram representing complete data set.

Although it is possible to use 3D data for kriging interpolation, it is slow, because a large number of points have to be included. In this paper each layer was interpreted separately from data points at the same depth, which is typical for Kriging. The Kriging estimation assumes that the estimated sur-face is changing slowly, but in case of complicated tectonic structure this as-sumption might not be fulfilled. Fortunately, the study area contains neither major faults nor subduction zones, thus it is suitable for this method. The Ordinary Kriging used in this analysis assumes that observed values oscillate around a common mean value, so no general trends are observed. Such a trend can be observed in large scale tectonic structures. To overcome this it is possible to use only limited data around calculation point with smaller var-iations in observed values, assuming that the number of data points is large enough for statistical analysis. The spherical semivariogram form for data points lying closer than defined maximum range a is simplified to the single equation

3

0 1( ) 1.5 0.5 , if | |h hh C C h aa a

γ⎛ ⎞⎛ ⎞= + − ≤⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (7)

Because for each point there was a different number of input data, semi-variogram parameters have to be fitted separately for each data set. A simple rule was used in the algorithm: C0 was the minimum and C1 was the maxi-mum value from semivariogram. Range a was constant and, based on semi-variogram fit to complete data set, was estimated as 2.5 degree. The nugget parameter C0 was not set to zero allowing the interpolated surface to differ from values in data points. This helps to obtain a better fit to our data that contains uncertainty in values and possible mismatches on crossing profiles.

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In this analysis we assume stationarity and isotropy of the input data. To test this assumption, surface variograms that depend on two parameters (longi-tude and latitude distance) were prepared for selected depths. In Figure 6 we see semivariograms calculated in bins selected every 1° degree in Easting and Northing directions. All slices show angle dependence that is changing with depth. For shallow surface variograms (10 and 20 km), the maximum observed values are small comparing to uncertainty in the input data, thus using rough assumption of stationarity seems to be correct. For deeper sur-face variograms we see larger values, but they are observed for large dis-tances from the calculation point (center of each semivariogram), thus using limited range (a = 2.5°) approach to Kriging should give correct results. For defined limited range we can claim that the input data are isotropic. The Krig-

Fig. 6. Semivariogram surfaces at selected depths of 10, 20, 30, and 40 km. Two dimensional semivariogram that depends on distance separately in Easting and Northing directions. Empirical semivariogram values increase with distance from the origin. Visible symmetries changes with depth. For short distances variations (< 2.5°) are isotropic, thus one dimensional parameterization based on total distance is used for analysis.

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Fig. 7. Comparison of two Kriging strategies: using all data with specified range of variogram saturation (a, d) or using points in limited range (b, e) presented by inter-polation result (top) and its uncertainties (bottom), and also differences between them (c, f). The results are almost identical in the central area and differences are visible at the edges where no points were present. A similar situation is observed for differences in the uncertainty.

ing estimations were calculated in a regular grid of 0.5 degree for longitude, and 0.3 degree for latitude, which correspond to about 33 km at the surface. Test results of two versions of Kriging estimation are presented in Fig. 7, where we see a comparison of Kriging for all points using Eq. (5), and

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Kriging for limited range points using Eq. (7). The results are identical in the central part and the only differences are observed at the edges and in the cor-ners, where the distance to the nearest data point was large. Still, differences are relatively small. This is another proof that using Ordinary Kriging was reasonable. Comparing the uncertainties for both approaches we see even smaller relative differences which prove that the used automatic semi-variogram fit works well comparing to the manual fit. The results for limited range Kriging are presented in Fig. 8 as a series of horizontal slices at 10, 20, and 40 km through velocity field, uncertainty estimation and number of points used for estimation. Comparing the results with Fig. 4 we see similar structures, but we got values estimated in the whole area. Additionally, the

Fig. 8. Example slices through interpolated model obtained by limited range Kriging at 10, 20, and 40 km depths (top) with corresponding uncertainties (middle), and number of points included in the interpolation (bottom). The results are similar to those in Fig. 4 but exist in the whole space. Uncertainty changes slightly with depth, but the most certain results are around coordinates (21°E, 52°N) where we have a lot of profiles from the CELEBRATION experiment.

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obtained velocity field is smoother and no sharp artifacts are present. At 10 km we see a clear sedimentary basin in TESZ area. At 20 km without sharp artifacts from linear interpolation we see southern edge of the Bohe-mian Massif with its lower velocities. At 40 km we see a clear edge of EEC with large contrast of velocities. As an additional result we got estimation of Kriging standard deviation for each slice. It is not the classical standard deviation, because besides of uncertainty of the results it includes also uncer-tainty of method and all assumptions that may not be true, like stationarity and isotropy. It is rather a relative index of the reliability of estimation in different regions. The global characteristic of these uncertainties is similar for all depths, showing the most reliable results around coordinates (21°E, 52°N), but we can observe changes with depth. Surprisingly, the number of points taken for estimation does not always correspond with the uncertainty values. This might be an effect of discrepancies in velocities at crossing pro-files, resulting from different quality of seismic data or anisotropy.

5. CONCLUSIONS The velocity field interpolation is one of the ways to solve the difficult prob-lem in crustal structure modelling – creating 3D model from 2D models with a different number of layers. The Ordinary Kriging estimation presented here gives similar results to a standard linear interpolation, but unlike the latter method it can extrapolate data and estimate the uncertainty of result. The method is easy to implement and produces no sharp artifacts. The obtained result, based on 33 high-quality 2D raytracing profiles, correctly describes crustal structures of the discussed area. With known uncertainty, it is a good reference model for other analyses, like teleseismic tomography, receiver function, potential fields modelling or even full waveform inversion. The Kriging method is not perfect. It tends to produce smooth images of the real-ity and cannot reproduce small scale variability. However, it seems to be precise enough for tectonic scale models in the study area.

As a simple use for the resulting model we calculated an average P-wave velocity in the crystalline crust, realized in each column as a mean velocity for values higher than 5.8 km/s (to exclude sedimentary layers) and lower than 7.8 km/s (mantle material). The results estimated using linear interpola-tion and Kriging are presented in Fig. 9. Both results show similar structures, like the localized higher velocity anomalies (17°E, 54°N) or (22°E, 53°N). We can recognize the low velocity crust (6.3 km/s) of the Bohemian Massif, and similar low velocity crust beneath the Carpathians (22°E, 48°N). An interesting result at (20°E, 48°N) shows different velocities at crossing pro-files. The uncertainty for this point is very small and this discrepancy can be interpreted as local anisotropy.

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Fig. 9. Average P-wave velocity in the crystalline crust for 5.8 < Vp < 7.8 km/s using linear interpolation (a) compared to Kriging result (b) with its uncertainty es-timation (c). The results are similar. We can distinguish significantly different areas, like high velocity crust at (16°E, 54°N) or (22°E, 52°N). An interesting result at (20°E, 48°N) with small uncertainty shows different velocity for crossing profiles: higher velocity for S04 profile, and lower velocity for CEL05 profile marking poten-tial anisotropic area.

Acknowledgemen t s . This work was financed by MNiSW grant Iuventus Plus No. IP2010 023370. I would like to thank Mateusz Moskalik for useful discussions and also thank to my colleagues for providing digital raytracing models, especially Dr. Tomasz Janik for unpublished results. The map was prepared using GMT tools (Wessel and Smith 1998). I thank two anonymous reviewers whose comments significantly improved this paper.

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Received 11 April 2011 Received in revised form 12 August 2011

Accepted 6 September 2011

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