Physics of the Earth and Planetary Interiors 114 Ž1999. 49–58
Moho discontinuity in central Balkan Peninsula in the light of the
geostatistical structural analysis
Antoaneta Boykova
)
Geological Institute, Bulgarian Academy of Science, Acad. G. BoncheÕ St., bl.24, 1113 Sofia, Bulgaria
Received 19 December 1997; accepted 3 November 1998
Abstract
The map of Moho discontinuity in the central part of Balkan Peninsula has been obtained applying the geostatistical
procedures Žvariogram analysis and kriging.. The information about the depth of Moho in 144 points, situated irregularly
was published during the last some 10 years period by different authors on the basis of deep seismic profiling and
seismological data. The methods of geostatistics was chosen to show their possibility for analysis of the character of
geological and geophysical structures. The mathematical structure and the geometric anisotropy of the data variability has
been discovered using the variogram analysis. The main axes of the anisotropy correspond to main direction of the
development of the geodynamic processes forming the contemporary structure of Moho discontinuity in the observed region.
New configuration of Moho discontinuity has been obtained with evaluation of standard deviation of the estimation in each
cell Ž20 = 20 km2 . of territory. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: Moho discontinuity; Geostatistics; Variogram analysis; Kriging
1. Introduction
The depth of Moho discontinuity is always a
pressing geological problem because it is the limit of
sharp change of surrounding physical parameters by
the transition from the earth crust to upper mantle.
Many authors published a data and drew maps of
Moho discontinuity for the territory of central part of
Balkan Peninsula on the basis of different types of
)
Fax: q359-2-72-46-38; e-mail: [email protected]
interpretations. In one part of them the interpretation
is according to one preliminary accepted geological
hypothesis, the others are too generalised. For interpolation of the isolines few methods are used but
more frequently it has been made by hand.
During the last 10 years some Bulgarian authors
ŽDachev and Volvovsky, 1985; Shanov and Kostadinov, 1990, 1992; Riazkov, 1992. tried to obtain a
revised map of Moho discontinuity.
The purpose of this paper is to show the possibility of the methods of geostatistics for analyses of the
character of geological and geophysical structures
and discontinuity and to research and discover the
0031-9201r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 1 - 9 2 0 1 Ž 9 9 . 0 0 0 4 5 - X
50
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
geometric andror zonal anisotropy using variogram-structural analysis.
Applying kriging to all published and new found
data for this part of Balkan Peninsula for its creation,
the new configuration of map of Moho discontinuity
pretends to be more precise and interesting than the
existing maps. The results are compared with the
investigations of some other authors.
2. Initial data
The information about the depth of Moho discontinuity is irregularly situated in 144 points on the
territory ŽFig. 1.. This information was presented
from different authors at different times on the basis
of deep seismic sounding, geophysical profiles made
in the frame of international projects and seismic
refraction profiles, and published by Bulgarian and
Greek authors—Dachev Ž1980., Panagiotopoulos
Ž1984., Dachev and Volvovsky Ž1985., Shanov and
Kostadinov Ž1990, 1992., and Riazkov Ž1992., which
synthesise the researches, made before. According to
Dachev Ž1980. Moho discontinuity is marked on
DSS-profiles with stable seismic velocities y8.0–8.2
kmrh.
The precision of determination of the depth of
Moho discontinuity from the different methods varies
from 0.6 to 2.1 km according to Panagiotopoulos
Ž1984. and is equal to "1 km according to Riazkov
Ž1992.. Dachev and Volvovsky Ž1985. determines
the precision of interpretation of the data to be 2.2 to
3.1 km. For a big part of the input data points the
precision of their determination is not given. Table 1
presents the input points with their geographical
Fig. 1. Scheme of territorial distribution of the initial data: Ž . . received from DSS; Ž(. seismic refraction; Ž=. deep seismic sounding
Žindustrial blasting..
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
51
Table 1 Žcontinued.
Table 1
Input data
Longitude Ž8.
Latitude Ž8.
Depth Žkm.
MethodU
27.18
27.05
27.89
27.70
27.26
28.14
28.03
28.18
23.385
22.908
22.401
23.103
21.983
22.893
21.771
22.490
23.982
23.446
22.106
23.276
22.008
23.354
23.592
22.965
20.411
20.243
19.893
19.867
20.455
20.799
21.44
22.57
25.583
25.35
25.172
23.334
27.757
29.059
25.034
22.903
29.00
22.88
23.56
23.55
23.54
23.52
23.52
23.34
26.09
27.68
22.38
21.96
21.41
21.33
44.02
43.85
43.50
43.48
42.52
45.70
45.49
43.51
40.757
40.673
40.957
40.608
40.358
41.162
40.307
40.101
40.334
40.374
40.314
40.689
40.181
40.822
41.117
40.632
42.076
42.262
42.043
41.347
44.822
41.111
41.972
41.321
42.05
41.641
43.147
42.685
41.822
41.066
45.268
45.883
42.83
44.14
45.95
45.55
45.07
44.54
44.31
44.02
44.44
44.89
43.09
42.95
42.76
42.02
y31
y31
y31
y30
y29
y47
y47
y32
y32.9
y32.9
y31.4
y30.8
y31.5
y32.9
y32.4
y30.6
y29.6
y27.3
y34.2
y32.5
y32
y32
y29.8
y31.4
y42.6
y39.8
y38.4
y42.1
y29
y34.6
y31.3
y31.9
y29
y28.7
y30.2
y34.3
y24.9
y28.6
y31.4
y28.3
y25
y32.5
y42.5
y40
y34
y32
y29
y30
y32.5
y38
y30
y35
y37.5
y40
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
SR
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
U
Longitude Ž8.
Latitude Ž8.
Depth Žkm.
Method
23.68
21.91
21.19
21.44
22.56
22.19
23.65
23.53
23.57
23.43
23.51
23.48
23.63
23.64
23.57
23.63
23.77
23.63
24.96
24.57
24.40
24.26
25.33
25.62
25.98
25.28
25.02
25.04
26.68
26.84
26.39
26.15
26.33
26.54
26.93
26.17
26.74
26.49
26.50
26.21
26.18
26.58
27.12
27.29
27.42
28.00
27.94
27.83
27.73
27.52
27.44
27.41
27.26
39.927
44.16
43.73
43.87
44.33
44.55
43.63
42.77
42.69
42.61
42.3
42.23
42.21
42.00
41.85
41.83
41.77
41.45
43.56
43.50
43.47
43.12
43.44
43.40
43.35
42.97
42.25
42.09
45.80
45.75
44.66
43.86
43.83
43.82
43.68
43.44
43.43
43.38
41.80
42.60
42.10
41.78
45.68
45.64
45.59
45.44
45.35
45.15
45.07
44.57
44.46
44.38
44.14
y29.6
y30
y33
y31
y29
y29
y31
y40
y40
y35
y50
y46
y52
y47
y48
y50
y48
y45
y32
y32
y32
y38
y32
y32
y37
y39
y43
y39
y45
y47
y35
y31
y33
y32
y32
y34
y35
y35
y26
y25
y33
y35
y47
y41
y40
y47
y48
y42
y40
y37.5
y36
y32
y29
SR
DSS
DSS
DSS
DSS
DSS
DSS
DSS-IB
DSS-IB
DSS
SR
DSS-IB
SR
DSS
DSS-IB
DSS-IB
DSS-IB
SR
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
SR
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
SR
SR
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
52
Table 1 Žcontinued.
Longitude Ž8.
Latitude Ž8.
Depth Žkm.
MethodU
21.59
22.1
22.05
22.43
25.39
25.88
26.51
27.21
27.44
24.65
24.47
24.12
23.95
23.84
23.73
23.57
23.45
23.42
23.7
23.1
23.4
23.5
23.4
24.1
23.9
23.8
23.3
25.9
24.1
24.2
25.0
25.05
25.2
25.3
25.2
25.9
26.5
42.01
42.99
41.99
41.97
41.52
41.93
42.49
43.17
43.32
43.39
43.15
42.67
42.45
42.28
42.13
41.89
41.72
41.67
42.9
42.6
42.1
41.9
41.9
41.6
41.7
41.5
40.9
40.5
42.2
42.0
42.2
42.2
42.1
42.2
42.2
42.1
41.8
y37.5
y32.5
y35
y32.5
y37
y36
y35
y34
y34
y33
y35
y40
y43
y45
y46
y48
y47
y46
y40
y48
y55
y52.5
y51.5
y50.5
y46
y39
y37
y38
y36
y36
y39
y40
y40
y40.5
y41
y35
y35
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS
DSS-IB
DSS-IB
SR
SR
SR
SR
SR
SR
SR
SR
SR
DSS-IB
DSS-IB
DSS-IB
DSS-IB
DSS-IB
DSS-IB
SR
SR
In statistics it is common to assume that the
variable is stationary, that its distribution is invariant
under translation. In the same way, a stationary
random function is homogeneous, and self-repeating
in space. This makes statistical inference possible.
Since the information about the studied discontinuity is fragmentary, it needs a model to be able to
draw any conclusions about parts where it has no
any data. Matheron chose the term ‘regionalized
variables’ to emphasise the two apparently contradictory aspects of these types of variables:
– a random aspect, which accounts for local irregularities, and
– a structural aspect, which reflects the large scale
tendencies of the phenomenon.
The best way of representing the reality is to
introduce randomness in terms of fluctuations around
a fixed surface, which we shall call the ‘drift’ to
avoid any possible confusion with the term ‘trend’.
Fluctuations are not ‘errors’ but rather fully fledged
features of the phenomenon, with a structure of their
own. The first task in a geostatistical study is to
identify these structures, hence the name ‘structural
analysis’. After this the geostatistician can go on to
solve various types of problem such as estimation or
simulation.
The function-variogram is necessary to make this
structural analysis. It is defined by the equation:
g Ž h . s 0.5 Var Z Ž x y h . y Z Ž x .
The experimental variogram can be calculated
using the following formula:
U
DSS—deep seismic sounding.
DSS-IB—deep seismic sounding Žindustrial blasting..
SR—seismic refraction.
coordinates, depth and method of receiving of information.
3. Methods of geostatistical analysis
Geostatistics created by George Matheron Ž1962,
1970. is based on the theory of random functions. A
random function is characterised by its finite dimensional distribution.
g U Ž h. s
1
2 N Ž h.
N Ž h.
Ý
z Ž x i . y z Ž x i q h.
2
is1
where z Ž x i . are the data values, x i are the locations
of the samples, N Ž h. is the number of pairs of points
Ž x i , x iqh . —that is the number of pairs separated by
a distance h, i.e., those which will be actually taken
into account.
When the data are two-dimensional, the variograms should be calculated in at least four directions to check for anisotropy. The anisotropy, in the
sense of the geostatistics ŽArmstrong and Carignan,
1997, p. 24. could be of two types: geometric and
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
53
estimation ŽMatheron, 1970; Journel, 1977; Deutsch
and Journel, 1992.. For this, the solutions of the next
system of N q 1 linear equations should be found,
which are written in terms of the variogram model:
N
Ý l jg Ž x i , x j . q m s g Ž x i , n .
js1
N
Ý li s 1
i
s k2 s Ý l i g Ž x i , n . y g Ž n , n .
where l i is the weighting factor, m the Lagrange
multiplier, and s k2 the kriging variance.
Fig. 2. Histogram of the initial data.
zonal. For a fixed angle, the variogram indicates how
different the values become as the distance increases.
When the angle is changed, the variograms disclose
directional features of the phenomenon such as its
anisotropy. In this study directional variograms were
calculated in four directions—08, 458, 908 and 1358.
The ranges of variogram Žthe distance where the
variogram reaches its variance. in different directions draw an ellipse or ellipsoid of anisotropy. Then
the received experimental variograms must be compared with the different mathematical models upon
receiving the maximal coincidence Žfitting a variogram model..
The properties of the function-variogram, the theoretical mathematical models and the principles of
the structural analysis of variograms were published
in many monographs and textbooks ŽMatheron, 1970;
Journel, 1977; Englund and Sparks, 1991; Deutsch
and Journel, 1992..
After that to make the most precise interpolation
between the points with the initial data the geostatistics uses the estimation method called ‘kriging’,
which is a way of finding the best Žin the sense of
least variance. linear unbiased estimator. The purpose is to minimise the standard deviation of the
Fig. 3. Directional variograms in azimuth 438N and 1338N: the
values of the experimental variogram are marked by points and
the fitted variogram model is drawn by solid line.
54
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
The calculations are very difficult and it is impossible to do them without suitable software. Here the
software GEO-EAS ŽEnglund and Sparks, 1991. for
two-dimensional case was applied.
Table 2
Parameters for the chosen model
Mathematical model
Azimuth 438
Azimuth 1338
Sill
Range
Sill
Range
3
43
10
160
500
3
43
10
250
500
4. Data processing and results
Nugget effect
Spherical-1
Spherical-2
In Fig. 2 is shown the histogram of the initial
data. The values of the depths of Moho vary from 25
to 55 km. The mean is 36.5 km and the median is 35
km. Seventy-six percent of the values are disposed
between 27.5 and 42.5 km.
The variogram analysis shows complicated dependence on the data and existence of geometric anisotropy. The ellipse of anisotropy ŽFig. 3. is drawn
using the values of variogram ranges in directions 08,
458, 908 and 1358. The large axes of anisotropy has
an azimuth of 1338 and the short axes shows the
direction of 438.
The experimental variogram was fitted by double
spherical model ŽFig. 4. combined with a nugget
effect:
The nugget effect constitutes only 5% of the
variability of the data. It could be caused by the
precision of the data and its irregularly distribution.
The small value of the nugget effect and the large
ranges of mathematical models indicate that the data
are good structured.
The combined double spherical model presents
two levels of variation. The two models correspond
to different character of structure of Moho. At the
nearest zone the structure of Moho discontinuity is
anisotropic and it could be considered as a result of
geodynamic processes which formed its contemporary character in central part of Balkan Peninsula.
g438 s 3 q 43 sph 1 Ž 160 . q 10 sph 2 Ž 500 .
g 1338 s 3 q 43 sph 1 Ž 250 . q 10 sph 2 Ž 500 .
The parameters of the model are written in Table
2.
Fig. 4. Ellipse of geometric anisotropy for the first spherical
model and a circle for the second spherical model.
Fig. 5. Histogram of the estimation: H U —estimated values of the
depth; H —the initial values of the data for the depth; St. Dev.—
standard deviation.
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
The correlation to the direction NW–SE Žup to 250
km. is stronger than to the direction NE–SW, where
the correlation distance of 160 km shows the faster
changing of the relief of Moho discontinuity.
The second spherical model is isotropic with a
range of 500 km and it reflects the general continental tendencies in the development of the earth’s crust.
The testing of the variogram model was realised
by the application of the procedure ‘cross validation’.
This procedure permits to estimate every point from
the initial data using the points around it. After the
application of the cross validation the histogram of
55
the ratio between estimated and initial values reported to standard deviation is obtained ŽFig. 5.. It
has near normal distribution with maximum around
zero, which shows that the variogram model is good
enough.
The kriging procedure is performed for elementary estimation cells Ž20 = 20 km2 .. This size of the
grid has been chosen after testing different cells
Ž30 = 30, 40 = 40, 50 = 50 km2 , etc.. because it
gives the minimum standard deviation of the estimation error. As a result of this estimation the map of
Moho discontinuity was obtained ŽFigs. 6 and 7..
Fig. 6. Map of the Moho discontinuity in Central part of Balkan Peninsula. The isolines are in km from the sea-level: Ž1. major isolines; Ž2.
secondary isolines.
56
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
Fig. 7. Map of Moho discontinuity in three-dimensional projection.
The major isolines are drawn through the value of 3
km as a value of standard deviation of estimated
data. They are in km from the sea-level. The secondary isolines help to see better the character of
Moho discontinuity.
5. Discussion
The configuration of the presented map of Moho
ŽFigs. 6 and 7. is different from the published maps.
The negative structure at the southwest of Bulgaria,
where the greatest depth of 51 km is, is the most
interesting. The structure is oriented to the azimuth
1338, which corresponds to geometric anisotropy of
the data, received by variogram-structural analysis.
This negative structure continues to the southeast in
Greece. It is limited by steep gradients parallel to the
front of the movement of the zone of paleosubduction beneath the Rhodopes, dipping to the northeast
as supposed by Boccaletti et al. Ž1974., Hsu et al.
Ž1977., Spakman Ž1986., Botev Ž1987., Shanov and
Kostadinov Ž1990, 1992..
The other interesting structure is the structure in
the region of East Rhodopes, where the Moho dis-
continuity lies at the depth of 30 km. The last data,
not used in this study, according to the interpretation
of seismic profiles in the East Rhodopes in Zdraveva
et al. Ž1996. confirm the depth of 30–32 km. This
coincidence serves as a control of the efficiency of
the geostatistical processing of the data presented in
this study.
The least values of the depth Žup to 30 km. are at
the territory of northeast Bulgaria, at the aquatory of
the Black sea and at the territory southward of
Thessaloniki.
If the depths of Moho discontinuity in general
coincide with the published interpretation, the configuration of Moho discontinuity is different and it
includes some new structures. The geometric anisotropy correlates well with regional velocity anisotropy Žaccording to Botev, 1987. of the mean with an
azimuth of 1208 at the depth of about 250 km.
And finally the map of the error of estimation
ŽFig. 8. shows that only at the periphery of the
investigated territory the error is greater than standard deviation Žup to 7 km. and it is a result of a
border effect. The mean value of the error at the
whole territory is 2–3 km.
The calculation of the error of estimation is very
important because it gives the limits of the probable
A. BoykoÕar Physics of the Earth and Planetary Interiors 114 (1999) 49–58
57
Fig. 8. Map of the estimation error of the depths in km.
values of the depth in every point. This error depends on the space distribution of the initial data.
6. Conclusions
The presented map of Moho discontinuity for the
central part of Balkan Peninsula is drawn on the
basis of all published data for its depth. The geostatistics is used for the first time for analysis and
mapping. This permits to investigate the existence of
geometric anisotropy between the data and to use the
most precise interpolator for irregularly disposed data
set. The parallel presentation of the map of estima-
tion error is one advantage for correct utilisation of
the depth of Moho discontinuity.
This map of Moho discontinuity could serve as a
basis for future adjustment of different new geological and geophysical data at this part of Balkan
Peninsula.
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