RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 9, ES1001, doi:10.2205/2007ES000217, 2007
[9] (1) As noted in Introduction, geoinformatics does not exist presently as an independent science because, although it has its own subject matter (The essence of this subject matter is systematization and accumulation of information that is related to the Earth and various geophysical fields and can be claimed in scientific research.), it does not have its own method as yet. As before, graphic (primarily cartographic) means of the geoinformation representation prevail.
[10] (2) The aforesaid immediately raises the question of what is the essence of the method inherent in the geoinformation science. This question can be answered very briefly: the geoinformatics method consists in an analytical (naturally approximate) description of geoinformation.
[11] However, such a brief answer will be unintelligible to the vast majority of readers. For this reason, the meaning of the answer is explained in detail below.
[12] First of all, I present a rather evident classification of geoinformation. This classification includes the following four types of geoinformation:
[13] (I) geoinformation defined on the set of land surface points;
[14] (II) information related to the Earth's interiors;
[15] (III) information related to water media (oceans and seas); and
[16] (IV) information related to the Earth's atmosphere.
[17] It is clear that the first two types of geoinformation are most significant. The following examples of these types can be presented.
[18] Example 1. Information on the land topography of the Earth.
[19] Example 2. Information on the Earth's gravity field (the complete field and its normal and anomalous components).
[20] Example 3. Information on the magnetic field of the Earth (the complete field and its normal and anomalous components).
[21] It is clear that these examples by no means exhaust the types of surface geoinformation. As regards examples of the second types of geoinformation, they are as follows.
[22] Example 1. Information on the occurrence depths of interfaces between layers in the crust and primarily in the sedimentary cover of platform regions.
[23] Example 2. Information on the spatial position of crustal faults.
[24] (3) The essence of linear analytical approximations of surface geoinformation is explained fairly comprehensively in this and subsequent paragraphs of the section.
[25] To begin with, note that geoinformation can be of three types:
[26] (a) local (the sphericity of the Earth can be neglected);
[27] (b) regional (geoinformation is specified on a fairly large area and the sphericity of the Earth should be taken into account);
[28] (c) global (geoinformation is specified on the entire surface of the Earth).
[29] Below I consider only the cases of surface geoinformation of the first type, implying that the Earth can be treated as a half-space bounded by a relatively small part of the Earth's surface. I address here the analytical description of the Earth's surface topography in a local variant. Let an orthogonal coordinate system
![]() | (1.1) |
![]() |
Figure 1 |
be introduced, with the 0 x3 axis directed upward (Figure 1) and let
![]() | (1.2) |
be the elevations of the Earth's surface above the plane
![]() | (1.3) |
which is regarded as the surface of a normal Earth.
[30] It is clearly seen that the function h(x) can be treated as limiting values of a function harmonic in the half-space x3>0. It is in terms of this treatment that the sought linear analytical approximations can be constructed from given values:
![]() | (1.4) |
where
![]() | (1.5) |
and dhi are uncertainties in the given values hi.
[31] There exist two main constructions, each involving a distribution of formal point sources on the plane
![]() | (1.6) |
[32] First construction consists in an integral representation of the function h(x):
![]() | (1.7) |
where
![]() | (1.8) |
Below, this function is denoted as
![]() | (1.9) |
[33] It is clear that the sought function in (1.7) is
r(x1, x2) (it should be found from the given values
hi,d,
1 i
N, in accordance with (1.4) and (1.5)).
[34] The function r(x1, x2) can be determined in terms of the constrained variational problem
![]() | (1.10) |
Problem (1.10) is evidently solved by the method of Lagrangian multipliers, reducing it to a family of unconstrained variational problems of the type
![]() | (1.11) |
where all values
li,
1 i
N, are
Lagrangian multipliers accounting for the equality conditions in
problem (1.10).
[35] Using the necessary (and, in this case, sufficient) criterion of an extremum and writing out the corresponding Euler equation [Buslaev, 1980; Kosha, 1979; Lavrentyev and Lyusternik, 1950], we obtain the following expression for the function r(x1, x2):
![]() | (1.12) |
As can be shown (see Section 2 of this work), the vector
l with the components
li,
1 i
N,
satisfies the system of linear algebraic equations (SLAE)
![]() | (1.13) |
where the
N N matrix
A possesses the property
![]() | (1.14) |
and its elements are
![]() | (1.15) |
It is easy to see that
![]() | (1.16) |
The integrals on the right-hand sides of relations (1.15) have fairly complex analytical expressions and this is a significant drawback of the method described (which is a variant of the method of linear integral representations, described in a more general form in the next section of the paper).
[36] We emphasize that the integrals determining the elements aij of the matrix A can be calculated by cubature formulas with a relatively small number of nodes (30 to 40); the integral
![]() | (1.17) |
should be preliminarily transformed into the integral of the type
![]() | (1.18) |
It is natural that we should actually find a stable approximate solution l of the SLAE
![]() | (1.19) |
this solution should be consistent with a fairly large
volume of a priori information on the vectors
h (with the
components
hi,
1 i
N ) and
dh (with the
components
dhi,
1
i
N ).
[37] The main aspects of a new theory developed by the author for the determination of stable approximate solutions to SLAEs of form (1.19) are described in Section 2.
[38] (4) Now, we characterize, in the local variant, the second (in a sense, more effective) method of constructing linear analytical approximations of the Earth's surface topography.
[39] The function h(x) = h (x1, x2, 0) in this method is represented as
![]() | (1.20) |
where we set
![]() | (1.21) |
The points
![]() | (1.22) |
coincide in coordinates (x(j)1, x(j)2) with the coordinates (x<undef>(j)1, x2<undef>(j)) of the points specifying the function h(x) = h(x1, x2, 0). In other words, a point source is located above each point (x1(j), x2(j), 0) at a depth H, and a set of such sources approximates the function h(x) = h(x1, x2, 0) treated in terms of limiting values of function harmonic in the exterior of the upper (relative to the normal surface of the Earth) half-space.
[40] It is clear that analytical expression (1.20) leads to a SLAE for
the determination of the coefficients
c(j),
1 j
N:
![]() | (1.23) |
hd is the
N -vector with the components
hi, d = hi + dhi,
c is the
N -vector of the sought
coefficients
c(j),
1 j
N, and
A is an
N
N matrix possessing the property
![]() | (1.24) |
and consisting of the elements
![]() | (1.25) |
Evidently, we have
![]() | (1.26) |
and the matrix A strongly gravitates toward the banded type; i.e. the value | A | E is determined by the elements of the main diagonal and a small number of diagonals parallel to the main one.
[41] Due to the above considerations, the determination of stable approximate solutions c to SLAEs of form (1.25) is not very difficult even in the case of large dimensions of the systems, namely,
![]() | (1.27) |
[42] The following method is quite rational (effective). SLAE (1.23) is rewritten in the form
![]() | (1.28) |
where A0 is a banded matrix with a band width 2m + 1,
![]() | (1.29) |
DA is the matrix complementing the matrix A, and a>0 is a small number,
![]() | (1.30) |
[43] The following iterative method is then used:
![]() | (1.31) |
and
![]() | (1.32) |
where we set
![]() | (1.33) |
The quantities gn in (1.32) are found from the conditions
![]() | (1.34) |
[44] Evidently, condition (1.32) can be written as
![]() | (1.35) |
where q(n-1) and p(n-1) are SLAE solutions.
![]() | (1.36) |
![]() | (1.37) |
[45] The criterion for stopping this iterative process is defined by a priori information available for the vectors h and dh and is discussed in greater detail in the next section of the paper.
[46] We believe that the readers are now convinced that the second method of constructing linear approximations of the Earth's surface topography (in a local variant) is much more effective than the first method for the following two reasons:
[47] (a) the calculation of elements of the matrix A is much simpler; and
[48] (b) the matrix A has specific properties facilitating the determination of the vector c.
[49] (5) Now, after the problem of constructing linear analytical approximations of the Earth's surface topography (in a local variant) is considered in rather great detail, we address the problem of constructing linear analytical approximations of other types of geoinformation specified at the physical surface of the Earth (pertinent examples are given above in paragraph 2 of this section). Evidently, we should first consider this problem in a local variant. The only distinction of problems of constructing geoinformation on a physical surface S from the above problem of the topography approximation is that functions defined at the Earth's surface S should be treated as the limits of functions harmonic in the exterior of S (whereas the topography heights h(x) = h(x1, x2, 0) are approximated by functions harmonic above the plane x3 = 0).
[50] Let
![]() | (1.38) |
be the coordinates of points above the plane x3 = 0 at which approximate values of the function
![]() | (1.39) |
are given; more specifically, these values can be written as
![]() | (1.40) |
(We mean that u(x) is not a component of the gravitational or magnetic field but describes other geoinformation, for example, the intensity of the surface heat flow of the Earth.)
[51] Then, the coordinate system ( 0x1 x2 x3 ) is defined in
such a way that the axis
0x3 is directed downward and all points
x(i),
1 i
N, at which the function
u(x) is
specified have vertical coordinates
x(i)3 < 0. In this case it
is possible (and appropriate) to use the following linear analytical
approximation of the function
u(x):
![]() | (1.41) |
![]() |
Figure 2 |
![]() | (1.42) |
are the same as in points x(i), while the coordinates x(j)3 are equal in absolute value but opposite in sign to the corresponding coordinates of observation points (Figure 2):
![]() | (1.43) |
Substituting the coordinates of observation points into
(1.41) and replacing
u(x(i)),
1 i
N, by the given
values
u<undef>i,d = u (x<undef>(i)) + dui, we obtain
![]() | (1.44) |
where
c is the
N -vector with the components
ci,
ud is the
N -vector with the components
ui,d,
and
A is the
N N matrix possessing the property
![]() | (1.45) |
and consisting of the elements
![]() | (1.46) |
Evidently, we have
![]() | (1.47) |
and hence
![]() | (1.48) |
[52] It is clear that SLAE (1.44)-(1.46) is computationally rather simple, and the determination of its stable approximate solution c should not encounter great difficulties even in the case of SLAEs of large dimensions.
![]() |
Figure 3 |
[54] Let we have n layers and, consequently, n+1 surfaces bounding these layers from above and from below. These boundaries can be described analytically if a sufficiently great number of points with known coordinates ( x1, x2, x3 ) in an a priori given coordinate system are present on each of the surfaces (Figure 3).
[55] Note. If the boundaries of the layers lie at depths of about 1 km or more and the number of boreholes crossing these boundaries is fairly small (no more than 5-7), a sufficiently great number of interface points with known, albeit approximate, coordinates can be found solely geophysical, primarily seismic and electric, methods. This fact is a very weighty argument indicating that the new (third) epoch in the development of Earth sciences is an epoch of geophysics and geoinformatics.
[56] An arbitrary boundary between layers G can be advantageously described in the following analytical form:
![]() | (1.49) |
where Pm(x1, x2) and Qm(x1, x2) are algebraic polynomials of the given degree m,
![]() | (1.50) |
![]() | (1.51) |
All coefficients kp, q and cp, q in (1.50) and (1.51) should be found from the given values
![]() | (1.52) |
The determination of the coefficients kp, q and cp, q evidently reduces to the solution of the SLAE derived from the relations
![]() | (1.53) |
It is clear that the degrees m of the polynomial Pm(x1, x2) and Qm(x1, x2) should be small. The most important is the case of
![]() | (1.54) |
In this case, the number of coefficients to be determined is 19; therefore, it is convenient to have the same (even greater) number of the points x(i) = (x(i)1, x2(i), x3(i)) with known coordinates on a boundary G between layers. It is clear that this condition is valid in the majority of cases.
[57] (7) Finally, we have to consider (as briefly as possible) the case when the surfaces to be analytically approximated are fault planes and boundaries of geological bodies.
[58] It is appropriate to specify analytical approximations of G in the following general form:
![]() | (1.55) |
where F(x1, x2, x3) is either an algebraic polynomial or any other function including linear coefficients to be determined from coordinates of the points given on the surface G. It is clear that these coefficients should be found from a system of linear equations of the form
![]() | (1.56) |
The system can be either normally determined (if the number of the known points on the surface G is the same as the number the coefficients to be found) or overdetermined (if the former is greater than the latter). The variant of an overdetermined SLAE is evidently preferable. However, the required number of points on G can actually be obtained from detailed geophysical investigations, preferably by a complex of methods.
[59] This is an additional argument indicating that the new (third) epoch in the development of Earth sciences is an epoch of geophysics and geoinformatics.
Citation: 2007), Change of epochs in Earth sciences, Russ. J. Earth Sci., 9, ES1001, doi:10.2205/2007ES000217.
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