FITTING AN ELLIPSE TO THE SET OF INTENSITY DATA POINTS OF VRANCEA EARTHQUAKES

Fitting an ellipse to the set of intensity data points of earthquakes occurred on 11.10.1940, 07.04.1977, 31.08.1986, 30.05.1990 and 31.05.1990 is performed. Test criteria indicate that the ellipse smoothes the observed line of macroseismic field. For all earthquakes, focal axes of 5, 6 and 7 EMS-98 intensity zones are directed along the y-axis. For other zones are oriented along the parallels. This fact is one more acknowledgement of the anisotropy of a geophysical medium. Approximation of the set of points with an ellipse is performed using the method proposed by Fitzgibbon.


INTRODUCTION
It is known that the intensity attenuation of seismic shakes with the distance determined geometrical disperse of the wave front, energy dissipation on inhomogeneities and absorption [1,2].Features seismic impact of each earthquake is determined by such characteristics as tectonics, focal depth, mechanism, direction and development of the rupture of rocks [3].
Attenuation equation assumes an isotropic medium and uniformity of distribution of seismic energy [4,5].In this case, the theoretical isoseists have a circular shape.However, in reality, from the epicenter of the earthquake isoseists disperse as ovals or curved form lines.
The shape of these isolines is influenced by factors: such as common -geometry of the source, velocity of detrusion waves, and regional -a material composition, a thickness of a soil stratum, level of ground waters, distinction of velocity in base breeds and in overlying stratum, physical properties of medium, type of soils, features of a geological structure of medium etc [6,7].
Usually observed isoseists, when analyzing seismic hazard, approximate an ellipse.In the near zone: r ~ h, (repicenter distance, h -depth earthquake) geometry of the earthquake has a decisive influence on the configuration of the macroseismic field energy [8].Approximation by an ellipse of zones identical IDP is the simplified model of a real image, nevertheless, can be considered comprehensible for probability analysis of seismic hazard.
The purpose of the given work is the study of the possibility to fit an ellipse to the set of intensity data points of Vrancea earthquakes.

FITTING AN ELLIPSE TO A SET OF DATA POINTS OF MACROSEISMIC FIELD
For this analysis, the macroseismic data set of the earthquakes which have occurred 10.11.1940, 07.04.1977, 31.08.1986, 30.05.1990, 31.05.1990 are used [9].Large number of macroseismic observations in all azimuthal directions on earthquakes 1977 and 1986, 4088 and 2119 IDP (intensity data points) accordingly suffices, allows carrying out the detailed analysis of a macroseismic field.The macroseismic field is characterized by zones of identical intensity; the lines between zones of changing intensity in EMS-98 scale are very twisting.The breadth of zones of identical intensity is incremented in the process of removal from epicenter of earthquakes.For example, the breadth of 3 rd and 4 th of EMS-98 intensity zones reaches several hundred kilometers.The analysis is carrying out for approximation with ellipse of zones of identical EMS-98 intensity.
Besides, it is supposed that, the lines dividing areas of change of intensity are curves of the second order.In the near zone, they have the form close to ellipse that is defined basically by source geometry, for example, a horizontal expansion of the seismic source.In the process of increase epicenter distance, the shape of isolines changes according to singularities of the geological structure of medium.Analytically, these curves can be described by an implicit second order polynomial: Thus, the database is in the form of points in the xy-plane, the point set of macroseismic observations (x n ,y n ,I k ), n=1,…, N; k=1,…, 12, where x n and y n -are geographical coordinates of IDP points.It is required to fit an ellipse to these points, i.e. to discover coordinates of its focal points and an ellipse axis a, b.Clearly, that approximation should be done separately for each of k=3, 4, 5,…, I max grade of EMS-98 intensity zone of macroseismic field.In certain cases, the ellipse center is combined with arithmetic mean of latitude and longitude of IDP points accordingly.In our case, the center of an ellipse and an origin of coordinates are combined with epicenter of earthquakes.Let's write the canonical equation of an ellipse [10][11][12]  If points lie on an ellipse like in Figure 1, their coordinates satisfy the canonical equation (2).The values of semi-major axis (a), semi-minor axis (b), assuming b<a, are defined by a minimum of the sum of quadrates of deviations [13,14]: Let's introduce variables: (4) Further we will have: From the linear equation system solution: using expressions from equation (4), will define values of lengths of semiaxes of the ellipse approximating a line, connecting points (x n , y n ) of equal intensity.The canonical equation of an ellipse assumes that the ellipse center is the origin of coordinates, and ellipse axes lie on axes of Cartesian coordinates.
Points (x n ,y n ) for n=1,…, N, satisfying equation ( 2) should the equation of an ellipse also satisfy to an ellipse condition: where value c is half of focal length.For investigated macroseismic fields, the condition equation ( 7) is fulfilled.The focal semiaxis is oriented depending on intensity.The interesting fact, in the canonical equation of an ellipse for 5 th , 6 th and 7 th of EMS-98 intensity zones, all investigated earthquakes, the focal semiaxis is directed on an axis of ordinates grade 5, 6 and 7 of EMS-98 intensity (Table 1).It means that 5 th , 6 th and 7 th grade of EMS-98 intensity zones are oriented in a direction the north-south, i.e. meridians are prolated lengthways, and in this direction energy of seismic attenuated more slowly.The focal axis of ellipses, approximating remaining zones of equal intensity, is on an abscissa axis and is directed in a parallel direction.This fact is one more acknowledgment of anisotropy of a medium.

THE GENERAL EQUATION OF AN ELLIPSE
The canonical equation of an ellipse was applied to the detection of the anisotropy of a medium.However, definition of parameters of an ellipse (the center, lengths of semi-axes and their orientation concerning the coordinate system) is carried out from the general equation of curves of the second order: with an ellipse-specific constraint [15][16][17][18]: where a, b, c, d, e, f are coefficients of the ellipse and z=(x,y) are coordinates of IDP lie on it.The polynomial


is called the algebraic distance of the point z i =(x i , y i ) to the given conic.By introducing vectors: The equation can be rewritten in the vector form: In the equation ( 8) coefficients a, b, c, d, e enter linearly, hence, they can be calculated using the method of least squares [19,20].This problem can be considered also fitting an ellipse to a set of data points [11].For resolving this problem, we follow an approach suggested by Fitzgibbon [16,17,21], a numerically stable non-iterative algorithm for fitting an ellipse to a set of data points.The approach is based on a least squares minimization, and it guarantees an ellipse-specific solution even for scattered or noisy data [13,22].Let's designate through N i number of points in zone i th grade of EMS-98 intensity.The fitting of a general conic to a set of points (x n ,y n ) i , i = 1,…,N i , may be approached by minimizing the sum of squared algebraic distances of the points to the conic which is represented by coefficients a: The least squares method is centered on finding the set of parameters that minimize the squares sum of an error of fit between the data points and the ellipse.To ensure an ellipse-specificity of the solution, the appropriate constraint inequality, equation ( 9), has to be considered.Under a proper scaling, the inequality equation ( 9) can be changed into an equality constraint [14]: The ellipse problem becomes: where C and D are: The constraint equation ( 15) makes sure the resultant curve is an ellipse.Matrix C is called the constraint matrix, and matrix D, of the size (N i x6), defined with coordinates (x n ,y n ): The minimization problem equation ( 14) is ready to be solved by a quadratically constrained least squares minimization.First, by applying the Lagrange multipliers we get the following conditions for the optimal solution γ [14]: The elements of matrix elements are calculated under the formula [21]: Next, the system, equation (18), is solved by using generalized eigenvectors.There exist up to six real solutions (λ j , a j ), but because: We are looking for the eigenvector a k corresponding to the minimal positive eigenvalue λ k .Finally, after a proper scaling ensuring: We get a solution of the minimization problem, equation (14), which represents the best-fit ellipse for the given set of points.For solving the problem Fitzgibbons method is used [16,17]: The design matrix D equation ( 17) decomposed into its quadratic and linear parts [14]: where: Next, the scatter matrix S, equation (19), can be split as follows: The constraint matrix C, equation ( 16) can be expressed as: Further, we split the vector of coefficients, equation (10), into two vectors: Based on these decompositions, the rest condition, equation ( 18) can be rewritten as: Which is equivalent to the following two equations [14]: is exactly a scatter matrix of the task of a fitting a line through a set of data points IDP.This matrix is singular only if all the points lie on a line.In such situations, there is no real solution of the task of a fitting an ellipse through these points.In all other cases, the matrix S 3 is regular [19].Regarding that, a 2 can be expressed from equation (29) as: Including equation (31) into equation (29) we receive: The determinant of matrix C 1 is not equal to zero (the matrix is regular).Thus, the equation (32) can be rewritten as: Due to the special shape of matrix C equation (26) we get: Finally the conditions equation ( 18) can be expressed as the following set of equations: where M is the reduced scatter matrix of the size [17]: The computing program of realization of the described algorithm on «FORTRAN» demonstrate, that the condition of fitting with ellipse the set of zones identical IPP's is satisfied (Table 1).
Except for some zones equal grade of NSK intensity macroseismic fields of earthquakes 30.05.1990 and 31.05.1990 of coefficients of the general equation ( 8) satisfy to a condition of an ellipticity of a curve, equation (9).In the equation of the ellipse the coefficient b≠0, that is in the equation contains a member with product x and y.The Rotation of axes of coordinates on some angle α it is possible to reach that the member with product of these coordinates will disappear.For a solution is made matrixes where elements of matrix G 0 are connected with coefficients of the equation of ellipse [23]: where det (G 0 ) is determinant of matrix G 0 .To evaluation of lengths of axes of the ellipse and focal length are applied formulas: Matrix eigenvalues are ordered according to an inequality [19,20]: The angle α between the major axis of ellipse and axis X, of Cartesian coordinates, oriented against a direction of movement of an hour hand will be defined by the formula ellipse [23]: Then, the tangent of the angle will be defined by expression: Coordinates of focal points depend on the orientation of axes of the ellipse.If the big axis is directed along axis Y, a declination tangent of the angle accepts value [22]: Coordinates of focuses of the ellipse are defined by the formula:  Fx 1 =F 0 -p, Fy 1 =Λ 0 -kp  Fx 2 =F 0 +p, Fy 2 =Λ 0 +kp The angle is equal: The center of the ellipse, defined using formulas equation (39), slightly deviate coordinates of epicenters of investigated earthquakes. :