CN107730586B - Method and system for modeling stratum - Google Patents
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Abstract
The invention discloses a method and a system for stratum modeling. The stratum modeling method comprises the following steps: acquiring formation data, wherein the formation data comprises a map, data or an exploration position; establishing an octree index according to the density of the stratum data; constructing geological data into a multi-layer hierarchical point set according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index; the formation is modeled by tight-support radial basis functions, where the tight-support radial basis functions are determined from the set of data points for each level. The invention solves the problems of uneven data distribution and large data scale difference in the prior art, improves the accuracy of stratum three-dimensional modeling and reduces the time of stratum modeling.
Description
Technical Field
The invention relates to the field of three-dimensional stratum modeling, in particular to a stratum modeling method and a stratum modeling system.
Background
At present, with the continuous improvement of the automation degree of mines, the precision requirement on the three-dimensional geological model is higher and higher. The mine three-dimensional stratum modeling is that a stratum body model of a target area is established by integrating multi-source and multi-scale data generated in each production stage, so that production personnel can carry out mining planning, safety early warning, achievement statistics and other work under the support of a visual three-dimensional model, production arrangement is facilitated, and working efficiency is improved. The research of the three-dimensional stratum surface modeling of the mine originated in the last 70 th century and is started on the basis of the increasingly urgent cognition on the underground structure of the mine. In the prior art, various problems exist when a complex geological condition is modeled, for example, when three-dimensional modeling is carried out on a three-dimensional mine stratum, the data source of the three-dimensional mine stratum is wide, the data scale is different, and the data volume is large, so that the defects exist in the process of establishing a geological surface model as follows:
(1) the wide data source causes the uneven data distribution, the large data scale difference and the difficult interpolation algorithm selection;
(2) the data volume is huge, so that a computer needs to spend a large amount of time on analyzing and establishing the layered geologic body;
(3) the difference between the established layered geologic body and the actual situation is larger, so that the precision of three-dimensional modeling is lower;
in view of the above problems in the prior art, no effective solution has been proposed.
Disclosure of Invention
The invention provides a method and a system for modeling a stratum, which aim to solve the problems of uneven data distribution, large data scale difference, time consumption for analysis and low modeling precision in the prior art.
According to an aspect of an embodiment of the present invention, there is provided a method of modeling a formation, including: acquiring formation data, wherein the formation data comprises a map, data or a survey location; establishing an octree index according to the density of the stratum data; constructing the geological data into a multi-layer hierarchical point set according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index; the formation is modeled by tight-support radial basis functions, wherein the tight-support radial basis functions are determined from the set of data points for each level.
Further, before establishing the octree index according to the density of the formation data, the method comprises the following steps: and preprocessing the formation data.
Further, pre-processing the formation data comprises: calculating a corresponding normal vector for each data point of the formation data; obtaining an orientation matrix through a normal vector of each data point; and obtaining the eigenvalue and the eigenvector of the orientation matrix through the orientation matrix.
Further, establishing an octree index according to the density of the formation data comprises: and layering the data points of the stratum data through recursive segmentation according to the density of the stratum data, averagely segmenting each segmentation layer into eight spaces, judging the number of the data points in each space, and stopping layering when the judgment result shows that the number of the data points in the space is less than 8.
Further, modeling the formation with tightly-supported radial basis functions includes: calculating an anisotropic distance of each hierarchical point set through the orientation matrix, the eigenvalue and an original support radius, wherein the original support radius is determined by a central point position of each data point of the octree and a normal vector of each data point; obtaining the tight support radial basis function according to the anisotropic distance; modeling the formation through the tight support radial basis functions.
Further, modeling the formation with the tight support radial basis functions includes: obtaining a tight support radial basis function of each level point set through the anisotropic distance; updating the interpolation function of each layer of point set according to the tight support radial basis function; and modeling the stratum according to the interpolation function.
Further, modeling the formation according to the interpolation function includes: updating an implicit function through the interpolation function, wherein the implicit function is a function enabling a last layer of interpolation function to be zero, and the last layer of interpolation function is a hierarchical point set with the highest density; and modeling the stratum according to the implicit function.
Further, constructing the geological data into a multi-level hierarchical point set according to the octree index comprises: and arranging the hierarchical point set according to the density of each layer of data points.
There is also provided, in accordance with another aspect of an embodiment of the present invention, a formation modeling system, including: the device comprises an acquisition unit, a processing unit and a processing unit, wherein the acquisition unit is used for acquiring formation data, and the formation data comprises a map, data or an exploration position; the index unit is used for establishing an octree index according to the stratum data; the construction unit is used for constructing a hierarchical point set of each layer of data according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index, and the hierarchical point sets are arranged according to the density of each layer of data points; a computing unit for modeling the formation by tightly-supported radial basis functions, wherein the tightly-supported radial basis functions are determined by the set of data points for each level.
According to the embodiment of the invention, the formation data is acquired, wherein the formation data comprises a graph, data or exploration position; establishing an octree index according to the density of the stratum data; constructing the geological data into a multi-layer hierarchical point set according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index; the formation is modeled by tight-support radial basis functions, wherein the tight-support radial basis functions are determined from the set of data points for each level. The invention solves the problems of uneven data distribution and large data scale difference in the prior art, improves the accuracy of stratum three-dimensional modeling and reduces the time of stratum modeling.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a method of formation modeling according to an embodiment of the present invention;
FIG. 2 is a detailed flow chart of a method for modeling a formation according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the vector results of the dotting algorithm for all samples of the formation S according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of establishing a hierarchy of point sets for data of a mine formation S according to an embodiment of the present invention;
FIG. 5 is a schematic representation of a surface model of all the strata of the mine created according to an embodiment of the invention;
fig. 6 is a block diagram of a formation modeling system according to an embodiment of the present invention.
Detailed Description
It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict. The present invention will be described in detail below with reference to the embodiments with reference to the attached drawings.
In order to make the technical solutions of the present invention better understood, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the terms "first," "second," and the like in the description and claims of the present invention and in the drawings described above are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used may be interchanged under appropriate circumstances in order to facilitate the description of the embodiments of the invention herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
The embodiment of the invention provides a stratum modeling method and a stratum modeling system. FIG. 1 is a flow chart of a method of formation modeling according to an embodiment of the present invention. As shown in fig. 1, the method comprises the steps of:
step S102, obtaining stratum data, wherein the stratum data comprises a graph, data or an exploration position;
step S104, establishing an octree index according to the stratum data;
step S106, constructing a hierarchical point set of each layer of data according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index, and the hierarchical point sets are arranged according to the density of each layer of data points; and layering the data according to the density of the data, so that the geological layer is modeled in multiple layers.
Step S108, modeling the stratum through a tight support radial basis function, wherein the tight support radial basis function is determined by the data point set of each layer,
the formation data in the above steps are from different sources, for example, the position of an exploration borehole or other working positions, the position data may be predetermined, and the formation data further includes all address-related maps and data such as geological sketch, profile map and the like, so that the data are non-uniform, the data have large scale difference, and the data amount is large, and the data complexity is large when the terrain is complex.
The tight support radial basis function is different from a common radial basis function in the prior art, and the modeling of the tight support radial basis function is modeling with a radius as a range instead of full-space search modeling, so the embodiment has certain limitation, and the modeling precision is higher.
In the steps, the octree is established for all the data, and the data are layered according to the density of the data, so that the multilayer modeling of the geological layer is realized; the modeling problem of large geological data volume, uneven distribution and large data scale difference in the prior art is solved. The stratum is modeled by tightly supporting the radial basis function, so that modeling errors are reduced, the accuracy of three-dimensional modeling of the stratum is improved, and the time of modeling the stratum is reduced.
Because formation data is multi-sourced, pre-processing of the formation data is required in building octree indexes from the formation data. In an alternative embodiment, the process of preprocessing the formation data is to first calculate a normal vector corresponding to each data point of the formation data; secondly, obtaining an orientation matrix through a normal vector of each data point; finally, the eigenvalue and the eigenvector are obtained through the orientation matrix.
Through the steps, all different source data are unified in a normal vector mode, and data are input, so that subsequent processing modeling is facilitated.
In the method, after geological data is preprocessed, octree indexes are built for the formation data, and when the octree indexes are built, in an optional embodiment, data points of the formation data can be layered through recursive segmentation according to the density of the formation data, each segmentation layer is averagely segmented into eight spaces, the number of the data points in each space is judged, and when the judgment result shows that the number of the data points in the space is less than 8, segmentation is stopped.
The octree index divides data into eight spaces according to the density of geological layer data, judges the number of data in each space, if the number of data in a certain space is more than or equal to 8, continues layering, namely, continues to divide the layer into 8, and if the number of data in a certain space is less than 8, stops layering.
The uneven stratum data are divided into different layers through the steps, for example, in a part of stratum data points are concentrated, the number of layers of the part divided through the octree index is large, and therefore the problem that modeling is not accurate due to uneven data distribution is solved. After layering, arranging the data of each layer from less to more according to the number (density) of the data, calculating the stratum data by layering, and arranging the layering according to the density solves the problem of difficulty in self-adaptive selection of the support radius in subsequent calculation.
In an alternative embodiment, modeling the formation by tight support radial basis functions, the anisotropic distance of each hierarchical point set can be calculated from the orientation matrix, eigenvalues, and original support radius, where the original support radius is determined by the location of the center point of each data point in the octree and the normal vector of each data point; obtaining a tight support radial basis function according to the anisotropic distance; the formation is modeled by tightly-supported radial basis functions.
The method calculates the anisotropy of the stratum data by the tightly supported radial basis function in the steps, and can better reflect the geometrical characteristics of the stratum and improve the modeling precision of the stratum surface model by adding the anisotropy influence in the calculation process of the radial basis function.
In an alternative embodiment, modeling the formation with the tight support radial basis functions comprises: obtaining a tight support radial basis function of each level point set through the anisotropic distance; updating the interpolation function of each layer of point set according to the tight support radial basis function; and modeling the stratum according to the interpolation function. For example, when the stratum carries out implicit surface interpolation through tightly-supported radial basis functions under the condition of anisotropy, the transformation formula of the radial distance can be passed
And radial support radius transformation formula
And (6) performing calculation.
The embodiment also considers a tight support radial distance transformation method and a radial support radius transformation method when the stratum interpolates the implicit curved surface through the tight support radial basis function under the anisotropic condition, thereby reducing the modeling error and the computation complexity and reducing the time of stratum modeling.
In an alternative embodiment, modeling the formation according to the interpolation function may update an implicit function by the interpolation function, where the implicit function is a function that makes a last layer interpolation function zero, and the last layer interpolation function is a set of hierarchical points with the highest density; and modeling the stratum according to the implicit function.
After geological data is layered through octree indexes according to data uniformity in the steps, the layer with the highest density in the layers can be taken as the last layer, the layer with the lowest density is the first layer, parameters of the tight support radial basis function are calculated by solving a linear equation system from the first layer, an updated interpolation function is calculated, the interpolation function until the last layer is calculated and is enabled to be equal to zero, then the function is the built target geological layer implicit function, the parameters of the tight support radial basis function are solved according to the function, and a model of the visual geological layer is built according to the target geological layer implicit function.
In an alternative embodiment, constructing the geological data into a multi-tiered hierarchical set of points according to an octree index comprises: the hierarchical point sets are arranged according to the density of each layer of data points, and the support radius of each layer of data is obtained according to the mean value of each layer of data set in the octree, so that the tight support radial basis function of each layer of hierarchical point set is obtained, the problem caused by uneven geological data is solved, and the modeling precision is improved.
The above steps are described below in connection with an alternative embodiment:
the embodiment provides a multi-level radial basis function modeling method of an anisotropic stratum surface model, which is based on multi-source and multi-scale data of a mine stratum, can fit the stratum surface model according to non-uniform and large amount of data, and improves modeling efficiency and model precision.
Step I, extracting stratum bottom plate discrete points rho ═ p from all types of datai1,2, N; discrete sample point p obtainediThe array is a numerical array of x, y and z coordinates in a local coordinate system obtained by translating a Xian 80 coordinate system, and each array is uniquely extracted aiming at the target stratum.
Step two, calculating a unit normal vector { n) for the data points of the target stratumi1,2, N; calculating the normal vector by adopting a mode of constructing a K-D tree in a Point Cloud Library (PCL, http:// points. org /), and finally obtaining the unit normal vectors of all the sample points.
Step three, calculating the orientation matrix of the stratum on the plane through the normal vector
And performing eigenvalue analysis on the orientation matrix to obtain an eigenvalue E1 < E2 and an eigenvector
I.e. the eigenvalues and eigenvectors of the two main directions of change on the plane are calculated.
Establishing a hierarchical point set { rho ] with density from low to high by establishing octree of the whole area for all sample point data1,ρ2,...,ρMρ }; the process can be as follows: first, an octree index is established for the whole space, and the number of sample data points in each leaf space is recursively divided to be less than 8. Then, for each node of the octree, the position of the center point and the unit normal vector are calculated according to the contained sample data points. Finally, each layer of the octree corresponds to one point set, and therefore the hierarchical point set is formed.
Fifthly, point set rho is arranged at each levelkIn the above, a new interpolation function f is defined by the error between the anisotropic tightly-supported radial basis function (interpolation basis function) and the interpolation function of the previous layerk(x) Solving the function and passing it to the next set of points, the process of which may beSo that:
I. setting an initial interpolation function to f0(x)＝-1；
According to the formula
raRepresenting anisotropic distance, for any layer p in the set of layer pointskThe radial basis function anisotropy distance is calculated, and the formula transforms the coordinate difference in the x and y directions into a coordinate difference by anisotropy transformation
The radius of the plane is also converted into the major and minor axes of the ellipse in different directions
Where σ is the original support radius, S is the scaling matrix,
p for any layer in the layer hierarchy point setkDefining an interpolation function fk(x)＝fk-1(x)+ok(x) 1,2, M, wherein,
is a set of points ρkUpper tightly supported radial basis function, λiFor the coefficients to be solved, by solving a system of linear equations fk-1(ρk)+ok(ρk) 0 to obtain λiThe solution of (1). Interpolation basis function phiσ(ra) Is the Wendland tightly-supported radial basis function, where raThe anisotropic distance calculated in the previous step.
In the above step, for each layer, all points ρ passing through the layerkConstructing radial basis functions ok(x) Interpolation error f of radial basis function and previous layerk-1(ρk) Adding to obtain the interpolation function f of the layerk(x) By constructing an implicit function fk(x) 0 and solve the system of linear equations to get ok(ρk) Each layer conveys fk-1(x) At a set of points ρkThe interpolation error of (3).
Step six, calculating the final layer of point set rho through tightly supporting radial basis functionMImplicit function of fM(x) Interpolating the target formation 0; the target formation passing through an implicit function f in the scalar fieldM(x) Is represented by 0, fM(x) That is, the last layer of point set rhoMThe interpolation function of (1). And fM(x) < 0 indicates on the formation side, fM(x) > 0 denotes the other side of the formation, x being the spatial arbitrary point coordinate within a scalar field containing x, y, z.
And step (c), realizing stratum model visualization, completing modeling, wherein the stratum visualization can be realized by using a Marching Cubes algorithm (MC) in a PCL library or a Blomenthal's method, and preferably by using the Blomenthal's method.
Some of the alternatives in the above embodiments are based on non-uniformities (10)-1To 104Rice), a large amount of (>103Data stratum data, an efficient surface modeling method is provided; meanwhile, the influence of anisotropy is eliminated and the surface model precision is improved by considering the geological anisotropy factor in the data and transforming the radial basis function.
The above process is described in an alternative embodiment as set forth below in connection with fig. 2:
as shown in FIG. 2, six stratum related data of a mine in Hebei C05, C07, C08, C09, C11 and C12-1 are collected. Taking C07 as an example, 36 drilling points, 57 underground stratum points and 2395 underground geological sketch points are collected;
as shown in fig. 3, normal vector calculation is performed by a PCL library, and unit normal vectors of all sample points of the formation S are calculated;
calculating characteristic values and characteristic vectors of the three most main change directions of the stratum S;
as shown in FIG. 4, for all sample point data of the formation S, a hierarchical point set { rho) with density from low to high is established by creating an octree of the whole area1,ρ2,...,ρM＝ρ}，M＝7，The built 7 hierarchical point sets are visualized in the graph, and the data distribution is from sparse to dense;
computing an interpolation function f by anisotropically transformed tightly-supported radial basis functions at each hierarchical point setk(x)＝fk-1(x)+ok(x) Is calculated by solving a linear equation set fk-1(x)+ok(x) 0 and the interpolation error fk(x) The attribute to be interpolated is transmitted as the next level;
by computing the implicit function f of the last layerM(x) Interpolating the formation S as 0;
and realizing visualization of the stratum model, and completing C07 stratum modeling. The effect of modeling all 6 formations is shown in fig. 5.
The above embodiment has the following advantages:
(1) the anisotropy of the stratum data is calculated, and the anisotropy influence is added in the radial basis function calculation process, so that the geometrical characteristics of the stratum can be better reflected, and the modeling precision of a stratum surface model is improved.
(2) The problem that self-adaptive selection of the support radius is difficult under non-uniform data is solved through a multi-level method; meanwhile, with the reduction of errors, the resolving matrix is gradually sparse, and the waiting time required by stratum establishment is reduced.
The embodiment of the invention also provides a stratum modeling system, which can realize the functions through the acquisition unit 62, the indexing unit 64 and the construction unit 66. It should be noted that the formation modeling system according to the embodiment of the present invention may be used to execute the formation modeling method according to the embodiment of the present invention, and the formation modeling method according to the embodiment of the present invention may also be executed by the formation modeling system according to the embodiment of the present invention. FIG. 6 is a schematic illustration of a formation modeling system, according to an embodiment of the present invention. As shown in fig. 6, fig. 6 is a structural diagram of a formation modeling system according to an embodiment of the present invention. A formation modeling system, comprising:
an acquisition unit 62 for acquiring formation data, wherein the formation data includes a map, data or a survey location;
an index unit 64, configured to establish an octree index according to the stratum data;
a constructing unit 66, configured to construct a hierarchical point set of each layer of data according to the octree index, where the hierarchical point set is a data point set corresponding to each layer of the octree index, and the hierarchical point sets are arranged according to the density of each layer of data points;
a calculation unit 68 for modeling the formation by tightly-supported radial basis functions, wherein the tightly-supported radial basis functions are determined by the set of data points for each level.
The embodiment of the formation modeling system corresponds to a formation modeling method, so that the beneficial effects are not described again.
It should be noted that, for simplicity of description, the above-mentioned method embodiments are described as a series of acts or combination of acts, but those skilled in the art will recognize that the present invention is not limited by the order of acts, as some steps may occur in other orders or concurrently in accordance with the invention. Further, those skilled in the art should also appreciate that the embodiments described in the specification are preferred embodiments and that the acts and modules referred to are not necessarily required by the invention.
In the foregoing embodiments, the descriptions of the respective embodiments have respective emphasis, and for parts that are not described in detail in a certain embodiment, reference may be made to related descriptions of other embodiments.
In the embodiments provided in the present application, it should be understood that the disclosed apparatus may be implemented in other manners. For example, the above-described embodiments of the apparatus are merely illustrative, and for example, the division of the units is only one type of division of logical functions, and there may be other divisions when actually implementing, for example, a plurality of units or components may be combined or may be integrated into another system, or some features may be omitted, or not implemented. In addition, the shown or discussed mutual coupling or direct coupling or communication connection may be an indirect coupling or communication connection of some interfaces, devices or units, and may be an electric or other form.
The units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of the embodiment.
In addition, functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may exist alone physically, or two or more units are integrated into one unit. The integrated unit can be realized in a form of hardware, and can also be realized in a form of a software functional unit.
The integrated unit, if implemented in the form of a software functional unit and sold or used as a stand-alone product, may be stored in a computer readable storage medium. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes several instructions for causing a computer device (which may be a personal computer, a mobile terminal, a server, or a network device) to execute all or part of the steps of the method according to the embodiments of the present invention. And the aforementioned storage medium includes: a U-disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a removable hard disk, a magnetic or optical disk, and other various media capable of storing program codes.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (5)
1. A method of formation modeling, comprising:
acquiring formation data, wherein the formation data comprises a map, data or a survey location;
pre-processing the formation data, comprising:
calculating a corresponding normal vector for each data point of the formation data;
obtaining an orientation matrix through a normal vector of each data point;
obtaining an eigenvalue and an eigenvector of an orientation matrix through the orientation matrix;
after preprocessing, establishing an octree index according to the density of the formation data;
constructing the stratum data into a multi-layer hierarchical point set according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index;
modeling the formation with a tight support radial basis function, wherein the tight support radial basis function is determined from the set of data points for each level;
modeling the formation with tightly-supported radial basis functions includes:
calculating an anisotropic distance of each hierarchical point set through the orientation matrix, the eigenvalue and an original support radius, wherein the original support radius is determined by a central point position of each data point of the octree and a normal vector of each data point;
obtaining the tight support radial basis function according to the anisotropic distance;
modeling the formation through the tight support radial basis functions;
modeling the formation with the tight support radial basis functions comprises:
obtaining a tight support radial basis function of each level point set through the anisotropic distance;
updating the interpolation function of each layer of point set according to the tight support radial basis function;
and modeling the stratum according to the interpolation function.
2. The method of modeling a formation of claim 1, wherein building an octree index based on the density of the formation data comprises:
and layering the data points of the stratum data through recursive segmentation according to the density of the stratum data, averagely segmenting each segmentation layer into eight spaces, judging the number of the data points in each space, and stopping layering when the judgment result shows that the number of the data points in the space is less than 8.
3. The method of modeling a formation of claim 1, wherein modeling the formation according to the interpolation function comprises:
updating an implicit function through the interpolation function, wherein the implicit function is a function enabling a last layer of interpolation function to be zero, and the last layer of interpolation function is a hierarchical point set with the highest density;
and modeling the stratum according to the implicit function.
4. The method of claim 1, wherein constructing the stratigraphic data into a multi-level hierarchical set of points according to the octree index comprises: and arranging the hierarchical point set according to the density of each layer of data points.
5. A formation modeling system that applies the formation modeling method of any of claims 1-4, comprising:
the device comprises an acquisition unit, a processing unit and a processing unit, wherein the acquisition unit is used for acquiring formation data, and the formation data comprises a map, data or an exploration position;
the index unit is used for establishing an octree index according to the stratum data;
the construction unit is used for constructing a hierarchical point set of each layer of data according to the octree index, wherein the hierarchical point set is a data point set corresponding to each layer of the octree index, and the hierarchical point sets are arranged according to the density of each layer of data points;
a computing unit for modeling the formation by tightly-supported radial basis functions, wherein the tightly-supported radial basis functions are determined by the set of data points for each level.
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