Usual method of tectonic maps construction is the gravity eld analysis. However, this method is signi cantly limited because the gravity eld includes integral information about density features in lithosphere. This makes impossible to split selected tectonic blocks by depth. We suggest a new technique, which is based on lithostatic pressure calculation. Its idea can be applied to both two- and three-dimensional cases. In the 2D case we show the way to split the mantle to blocks with vertical boundaries. If lithostatic compensation hypothesis is adopted, the method also allows one to calculate density value for each block. Such separation of the mantle can help to diminish discrepancy between model and observed elds. In 3D case we suggest a method, which can be used to construct tectonic structure maps with information about approximate depth and height of each tectonic block.
The Earth crust and upper mantle are often predicted to have block structure. However, a common way to select these blocks is the analysis of the gravity eld or its derivative. But the gravity eld contains integral information about all lithosphere features in the whole depths interval from surface to the mantle. Thus, blocks selected by the gravity eld could not be split by depth. Moreover, these blocks are usually invisible in horizontal maps of density distribution. So, even having the gravity eld inverted, an attempt to separate lithosphere blocks by depth will most likely be unsuccessful.
We propose a di erent approach. Assume that we have a density model,
which can be obtained as a result of the gravity inversion or seismic velocity
model conversion. Such a model is the input data for calculation of lithostatic
pressure distribution. We calculate lithostatic pressure in a point by a mass
of vertical rock column, which top is on the Earth surface and the lower end
contains the point of calculation. Then, the mass is converted to the weight. But
the lithostatic pressure itself is not representative parameter because pressure
deviations inducted by density variations have much smaller value than pressure
associated with absolute density values. Instead, we analyze lithostatic pressure
anomaly, which is calculated as di erence between the actual pressure value
at the point and the mean pressure (hydrostatic) on this depth level. Similar
approach was used by Jimnez-Munt et al. in [
In this paper, we describe our method for both two- and three-dimensional
cases. In the two-dimensional case, we also show how adoption of an idea of
isostatic compensation helps one to reduce the error of density modeling (i.e.,
to minimize di erence between the observed gravity eld and one of the model).
Isostatic compensation is the hypothesis that, starting from some depth level,
there are no more lateral changes in pressure. For out study region (Ural
mountains in Russia and neighboring areas), there is a theory of compensation on level
of 80 km [
We used gradient velocity cuts of the Timan-Pechora region as the input data.
Velocities were converted to density values using empirical formula for
TimanPechora area [
8>0:113V + 2:034; (V ) = <0:2V + 1:6; 5 >:0:25V + 1:3; Then we performed averaging ltration of density values and, thus, the initial model was obtained. Figure 1 presents one of the model cuts. All the models are constructed down to 80 km only, and we accepted the hypothesis of the existence of isostatic compensation on this level.
Since the model is obtained as the result of seismic data interpretation, its calculated gravity eld (Fig. 1a, red curve) has signi cant discrepancy comparing with the observed one (Fig. 1a, purple curve). Usage of isostasy hypothesis helps to reduce this di erence.
The lithostatic anomaly P (x; z) is de ned as di erence between the lithostatic pressure P (x; z) on given level h and hydrostatic pressure calculated as mean pressure along the pro le on the same depth
P (x; z) = ga P (x; h) = P (x; h)
Here, ga = 9.80665 m/s2 is the average value of gravity acceleration, (x; z) is the density value at the point (x; z) of the cut, (z) is the mean value of density on a depth z
Isostatically compensated model with the compensation level hi should have no lateral pressure variations In our case, hi={80 km.
The lithostatic model for our density cut is presented in Fig. 2. As it can be clearly seen, there is no constant value on hi={80 km. (3) (4) (5) (6)
To construct such a compensated model, we introduced compensation function (x). This compensator shows what density value should be subtracted from the mantle (in our case, this is the layer between the Mohorovicic discontinuity and the hi level) to make condition (6) satis ed.
Let Phom and hom be the deviations of pressure and density from their mean values on given depth for the model with the homogeneous mantle. Lithostatic anomaly after addition of (x) is
P (x; hi) =
Phom(x; hi) ga[hM(x) hi] (x) Here z = hM(x) is Mohorovicic discontinuity position.
From condition (6), we have (x) =
Phom(x; hi) ga[hM(x) hi] =
1 hM(x) hi
The compensator function for study cut is presented in Fig. 3. As it could be easily predicted, it qualitatively repeats form of the Mohorovicic discontinuity. Zeros of compensator function were taken as boundaries of the mantle blocks. After, we distributed these excess densities in the upper mantle and performed density averaging inside blocks, and by this we obtained resulting density model (Fig. 4).
As it can be seen from Fig. 4a, the model eld (red curve) now have a good match with the observed one (purple curve). Figure 5 presents the lithostatic model of resulting density distribution. There is isostatic compensation on hi={ 80 km now and the lithostatic anomaly on hi is zero for almost all pro le length.
It is interesting to note that the same block boundaries could be selected without performing model compensation at all. Positions of blocks are clearly seen on the initial lithostatic model (Fig. 2). Thus, for the case of relatively at Mohorovicic discontinuity (when denominator of Eq. 6 is close to constant) (7) (8) blocks selection can be performed even for isostatically non-compensated model. We will use this approach in the three-dimensional case in the next section. 3
Input data for the three-dimensional case are a tectonic structures map of the
study region (Fig. 6) and a set of two-dimensional pro les (constructed as
described in Section 2. The tectonic map composed of di erent previously published
maps of tectonic structures [
The lithostatic anomaly in the three-dimensional case is de ned similarly to
But for the blocks selection, we used a di erent technique than for the 2D
case. Firstly, we equalized the model eld with the observed one. This was done
by density inversion using local corrections method. We omit description of this
procedure. It was presented in [
Then we calculated anomaly lithostatic pressure distribution for the resulting density model. Its horizontal cut is presented in Fig. 8b. Although we used isostatically compensated cuts, neither the initial nor the resulting models are isostatically compensated. This is related to the interpolation. Since we have no information of 3D positions of blocks, which were selected earlier on 2D cuts, we cannot perform correct continuation.
However, we are not obligated to compensate the model to detect blocks. As it was seen from the two-dimensional case, positions of block boundaries could be selected using the initial lithostatic model. We can perform matching of map of tectonic structures with horizontal cuts of the lithostatic model (Fig. 8b). This operation was performed for four depths: 10 km, 20 km, 40 km, and 60 km. Result is presented in Fig. 9. 4
The main problem in lithosphere blocks selection is impossibility to separate blocks by depth. Maps of tectonic structure are usually created by gravity eld, which contains only integral information about all features on all depths. In this paper, we suggested a new technique, which is based on lithostatic pressure calculation. Although the method is rather simple, it produces interesting results.
In Figure 9 four cuts of density and lithostatic models are presented. It can be seen that di erent structures are traced on di erent depths of lithostatic pressure cuts. For example, structure I (which is the Timan ridge) can be identi ed on 10 km and 20 km cuts, but disappears deeper. On the other hand, structure II (which is the East Urals De ection) cannot be traced in upper half of lithosphere, but is clearly seen on 40 km and deeper. And these structures are not visible on density cuts on corresponding depths.
For the two-dimensional case we suggested the technique of mantle splitting into blocks, which can be used to improve the model. Gravity eld of the updated model has noticeably lower discrepancy relative to the observed eld than that of the initial model. Homogeneous mantle with constant density 3.36 g/cm3 (Fig. 1) was split into 9 blocks with densities in range [3.25-3.4] g/cm3. This was performed under the hypothesis of isostatic compensation. But it was shown that blocks can be selected by the lithostatic model itself without need to equalize lithostatic pressure on some depth. However, hypothesis of local isostasy gives a simple method for calculation of resulting densities of selected blocks.