Article in Journal of Applied Geophysics · January 2017 doi: 10. 1016/j jappgeo
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| r, θ, φ)
relative to the point. An unknown boundary of the body was determined by the equation
, where
r(θ, φ) is a single-valued function. The unknown function r(θ, φ) was found by solving the following nonlinear integral equation
∬
x,y,z , (1) Fig. 3. Regional field model and approximation with line segments. Top: zoomed gravity anomaly, initial data (left), model of the regional field – 2D harmonic function (right). Bottom: residual anomalies after subtracting the regional field model (left), their approximation with two 3D line segments, RMS = 0.41mGal (right)
where G is the gravitational constant, Δσ
is the mass density contrast, U(x,y,z) is the given field. For a fixed value of
, the solution of the inverse problem is unique according to the Novikov’s (1938) theorem. We employed our new integral equations for the gravity and magnetic inverse problems (Prutkin, 2008). Integrands of the integral equations are algebraic relative to the function sought and do not contain its derivatives. In case of the function
in Eq. (1), we used a special combination of the potential and its derivatives. Based on the approximation of the observed field by the field of 3D line segments, this combination could readily be calculated. For the parametrization of the contact surface, we applied the Cartesian coordinates. The unknown 3D topography was then defined by z=z(x, y) . For this unknown function
z=z(x, y), there is a nonlinear integral equation similar to Eq. (1), see (Prutkin and Saleh, 2009). For the contact surface, we could use the gravity or magnetic data, which are given directly on the physical surface. For the restricted body, we calculated the mentioned combination of the potential and its derivatives on the closed surface
r=ρ(θ, φ) , which allegedly contains the unknown body in its interior. Integral equations were solved by the method of local corrections for both, the contact surface (Prutkin and Saleh, 2009) as well as for the restricted body (Prutkin et al., 2011). This method is belong to iteration methods such as the method of Cordell and Henderson (1968). It is based on computing the integral while the same grid is used for evaluating the given field. In each iteration, an attempt is made to decrease differences between the given and approximate field values at a fixed node solely by means of modifying the value of the unknown function at the same node. This idea leads to a decomposition of the inverse problem and to reduction of time expenditures needed to solve it approximately by an order of magnitude. For the restricted object, we refined the obtained solution by applying Newton’s method. An initial approximation provided by the method of local corrections ensured its fast convergence. First, we took the medium wavelengths of observed gravity without negative anomalies (they were substituted for the three local areas by the regional field models) and inverted them for a 3D topography of density interface between the crystalline basement (Neoproterozoic gneiss, density 2750 kg/m 3 ) and the lower crustal layer with the density of 2850 kg/m 3 . The density values were assumed according to previous gravity modelling and deep seismic profiles. The density interface is located in depths between 6 and 13 km, the average depth is approximately 10 km. The obtained topography is shown in Fig. 4 (left).
The main feature of the topography is a mountain range of the southwest-northeast direction. In late Paleozoic, the area of investigation was affected by a subduction with the stress direction from the south-east (Paleo-Tethys Ocean) to the north-west (Euramerica Continent). The mountain range looks like a fold belt orthogonal to the direction of stress. For all three local areas, we transformed the sets of 3D line segments found by an approximation into the 3D restricted bodies with the same field. Each time, we introduced spherical coordinates with the origin located in the centre of mass and then solved Eq. (1). As the density value, we assumed 2600 kg/m 3
crystalline rocks intruded by Variscan granites. Since intrusions are contained in the denser ambient medium (crystalline), they cause the presence of large negative gravity anomalies. All three granitic intrusions are located above the density interface. The northern intrusion corresponds to the most intensive negative anomaly (see Figures 2 and 3). It is stretched up to about 41 km in the west-east direction and up to 23 km in the south-north direction. The intrusion is located at depths from 4 down to 9.5 km. This body is the closest one to the Earth's surface and this is why it generates the anomaly with the largest amplitude. Two southern intrusions
lay in valleys to the west and to the east from the above-mentioned mountain range. The southwest intrusion is spread over 44 km in the south-north direction and over 28 km from the west to the east. It is located at depths between 5 and 10 km, a bit deeper than the northern intrusion, so the corresponding anomaly has a smaller amplitude. The southeast intrusion is 30 km long in the south- north direction and 26 km wide in the east-west direction, with depths ranging from 6 to 10 km. It is the smallest and the deepest body, this is why it causes the smallest negative anomaly. The obtained 3D model for the medium wavelengths includes three low-density bodies, which we interpret as granitic intrusions, and the density interface below, see Fig. 4 (right). Yüklə 1,42 Mb. Dostları ilə paylaş:
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