CN113312695A - Ancient landform restoration method based on trend surface analysis - Google Patents

Ancient landform restoration method based on trend surface analysis Download PDF

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CN113312695A
CN113312695A CN202110575415.1A CN202110575415A CN113312695A CN 113312695 A CN113312695 A CN 113312695A CN 202110575415 A CN202110575415 A CN 202110575415A CN 113312695 A CN113312695 A CN 113312695A
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trend surface
trend
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马宁
丁晓琪
祁壮壮
朱颖
刘鑫
陈乙莱
刘璇
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Chengdu Univeristy of Technology
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Abstract

The invention discloses an ancient landform restoration method based on trend surface analysis, which collects enough single-well target stratum thickness data h for stratum ancient landforms needing to be restored1Single well target overburden thickness data h2And single well geographic coordinates x, y; and then using the calculated superposed stratum thickness data z-h1+h2Constructing a trend surface equation by using a least square method principle, programming a calculation program by using MATLAB software to obtain different-order trend surface model equations, and substituting geographic coordinates x and y into the equations to obtain a series of trend surface values
Figure DDA0003084176830000011
Thereafter using R2Carrying out moderate inspection on the model by three different methods of inspection, F inspection and moderate successive inspection of the trend surface to obtain the trend surface of the optimal model, and finally carrying out the obtained trend surface and the target formation thickness h1And overlapping to obtain the final restored ancient landform. Compared with the traditional residual thickness method, the method combines the residual thickness method and the trend surface method, the technical means are properly fused, and the reliability of the comprehensive calculation result is high.

Description

Ancient landform restoration method based on trend surface analysis
Technical Field
The invention relates to the field of ancient landform restoration in mineral exploration technology, in particular to a method for restoring weathering crust type ancient landform.
Background
Ancient landforms are units of landforms whose basic form is completely decoupled from the modern landform shaping process and which are not compatible with modern tectonic conditions. The ancient landscape is often not easily left intact, especially in denuded zones where elevations are constructed. The ancient landform can be exposed on the earth surface in a lifting area, such as a quasi-plain of a mountain top, an ancient valley of a wind gap and the like; in subsidence areas, they are buried under the surface of the existing landform, such as ancient river channels, ancient buried mountains, ancient sand dunes, etc. The ancient landform is an important part of ancient geographic environment research and has important significance for finding oil gas.
The general idea of recovering the ancient landform is to find out the ancient structural surface of the underlying stratum by various methods and then accumulate the ancient structural surface and the thickness of the current stratum to obtain the ancient landform. The method mainly used for recovering the ancient landform at present is a residual thickness method, and the principle of the method is that stratum deposition is a filling and filling process, the deposition thickness at the high places of the terrain is small, the deposition thickness at the low places of the terrain is large, therefore, the top interface of the stratum is leveled, and then the ancient structural surface of the underlying stratum can be obtained according to the thickness of the stratum, as shown in figure 1.
However, this method has its limitations, and works best when the formation is well preserved, and once the formation has degraded, it can be less accurate. In particular, when ancient landforms are restored in a weathering crust type of ground layer, the residual thickness method is not applicable at all. The reason is that the weathering crust is strongly degraded, and although the degradation intensity is influenced by the terrain, the influence factors of the degradation intensity on the top boundary of the stratum are various, so that the weathering crust often shows non-uniformity at the same height. The unconsolidated formation thus recovered has more localized undulations, which is not consistent with the actual situation. In fact, the paleo-tectonic surface should be a relatively smooth surface reflecting the spatially varying tendency of the formation, which may fluctuate over a large range, but is usually locally stable, without significant fluctuations. As shown in fig. 2, for the weathering crust type formation, the ancient structural surface recovered from the direct flat top interface is greatly different from the actual structural surface, and therefore, the ancient geomorphology recovery of the weathering crust formation needs to be performed by improving the original method or using a new method.
Disclosure of Invention
In order to solve the problems in the prior art, the invention aims to provide an improved method for recovering ancient landforms based on a residual thickness method.
The specific technical scheme is as follows:
an ancient landform restoration method based on trend surface analysis comprises the following steps:
step 1, collecting thickness h of weathering crust of target stratum1Thickness h of overburden2And geographic coordinates x and y, calculating the thickness z of the superposed stratum according to the obtained data, constructing a trend surface equation by using a least square method principle, and programming a calculation program by using MATLAB software to obtain a primary simple trend surface equation z11Once bilinear trend surface z12Equation z of the quadratic trend surface model2；
Step 2, substituting the independent variables x and y into the three trend surface model formulas to obtain a series of trend surface values
Figure BDA0003084176810000021
Step 3, the result is processedLine R2Checking; will zi、
Figure BDA0003084176810000022
And
Figure BDA0003084176810000023
substituting into correlation formula to obtain SSD、SSRAnd SSTThen, R is found2Comparison of
Figure BDA0003084176810000024
The higher the numerical value is, the higher the fitting degree is;
step 4, carrying out F inspection on the result; substituting the correlation data into a formula to obtain an F value of a corresponding model, and looking up a table to obtain the F of the single linear trend surface when the confidence level alpha is 0.050.05(p11，n-p11-1), a one-time bilinear trend surface F0.05(p12，n-p12-1), a secondary trend surface F0.05(p11，n-p11-1); comparison F0.05And F, if F is larger, the trend surface equation is obvious, otherwise, the trend surface equation is not obvious;
step 5, carrying out successive testing on the trend surface of the result, and respectively comparing the primary single-linear trend surface with the primary bilinear trend surface, the primary single-linear trend surface with the secondary trend surface and the primary bilinear trend surface with the secondary trend surface;
step 6, integrating the three model test results of the step 3-5, and selecting an optimal model;
step 7, utilizing a trend surface model formula z2A series of results
Figure BDA00030841768100000214
Value and weathering crust thickness h1And (5) overlapping, and importing related mapping software to finally obtain the final ancient landform recovery map.
The trend surface equation in the step (1) can also use a higher-order trend surface model equation of a cubic trend surface and a quartic trend surface according to the complexity of the ancient landform to be restored.
Z in the above steps (2) and (3)i、
Figure BDA0003084176810000025
And
Figure BDA0003084176810000026
respectively indicating the stratum stacking thickness, the trend face value and the stratum stacking thickness average value,
Figure BDA0003084176810000027
in the form of a remaining sum of squares,
Figure BDA0003084176810000028
to return the sum of squares, SST＝SSD+SSRIs the sum of the squares of the total deviations.
The formula in the step (4) is
Figure BDA0003084176810000029
p is the number of coefficients except for a constant term in the trend surface equation; n is the number of observation points.
The method for successive detection in the step (5) comprises the following steps: first, the regression sum of squares of the higher order polynomial equation is found
Figure BDA00030841768100000210
Regression sum of squares with lower degree polynomial equation
Figure BDA00030841768100000211
The difference between the two; this difference is then divided by the difference in degrees of freedom p-q of the sum of squares of the regression to yield the regression mean square error due to the increase in the polynomial degree
Figure BDA00030841768100000212
Finally, the mean square error is divided by the residual mean square error of the higher-order polynomial
Figure BDA00030841768100000213
Obtaining the comparative appropriateness check value F of the two successive order trend surface modelsK+1→K(ii) a Wherein, MSR＝SSR/p，MSD＝SSD/(n-p-1)。
The method provided by the invention is based on the residual thickness method, combines the overburden stratum thickness and trend surface analysis method, performs more refined recovery on the ancient landform, can perform relatively accurate recovery under the condition that the stratum is degraded, and has wider application range compared with the residual thickness method.
Drawings
FIG. 1 is a diagram of a recovery pattern of ancient landform by a residual thickness method in the prior art;
FIG. 2 is a schematic diagram of the restoration of the weathering crust ancient landform by using the residual thickness method according to the embodiment;
FIG. 3 is a diagram of an example weathered shell and overlying strata;
FIG. 4 is a graph illustrating differential compaction of an overburden of an example weathering crust;
FIG. 5 is a stratigraphic diagram after an example leveling of the overburden top boundary;
FIG. 6 is a graph of paleotopographic structures calculated by the example trend surface analysis method;
FIG. 7 is a schematic diagram of the final restoration of ancient geomorphic geometry of the embodiment;
FIG. 8 is a diagram of the final ancient landform restoration results in the example.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the following embodiments and accompanying drawings. The exemplary embodiments and descriptions of the present invention are provided to explain the present invention, but not to limit the present invention.
The method is mainly used for improving the residual thickness method from two aspects, so that the method is suitable for restoring the weathering crust type ancient landform. First, the target formation thickness plus the overburden thickness is used as a recovery basis to reduce the effects of degradation without weathering degradation of the overburden, as shown in fig. 3, where the top interface is stable and flat compared to the weathering crust top interface and is suitable for leveling. Second, although the overlying strata is stable and flat relative to the weathering crust, due to differential compaction, local undulations may still be present at the same terrain elevation, as shown in FIG. 4, and the topographical surface obtained after flattening it along the top interface may still be inaccurate, as shown in FIG. 5. Therefore, further correction was performed using trend surface analysis.
The trend surface analysis is a mathematical method for simulating the distribution and variation trend of geographic system elements in space by using a mathematical curved surface. The method is characterized in that a two-dimensional nonlinear function is fitted by using a least square method through a regression analysis principle, the distribution rule of the geographic elements on the space is simulated, and the change trend of the geographic elements on the region space is displayed. The ancient earth structural surface is exactly the change trend of the stratum in space, so that more accurate ancient structural surface can be obtained by using trend surface analysis.
The observation surface of the trend surface analysis is composed of a trend surface part and a residual part. The trend surface part reflects the change condition in a regional large range, the residual error part is the difference between the measured value and the corresponding value of the trend function, the local change condition is reflected, and the combination of the two parts is helpful for deep analysis. One basic requirement of trend surface analysis is that the selected trend surface model should have the smallest residual value and the largest trend value, so that fitting accuracy can be sufficient. The spatial trend surface analysis is to analyze trend values and residual values from actual data of geographic element distribution, so as to reveal the trend and law of the geographic element spatial distribution. For data with mild changes, analysis can be performed by using a low-frequency trend surface; for complex and fluctuating data, a polynomial order higher trend surface can be used.
First, a model of the trend surface is built. Thickness z of superposed stratai(xi，yi) (i ═ 1, 2.. times.n) as a dependent variable, geographic position coordinates (x) are determinedi，yi) As an independent variable, the trend fit value is
Figure BDA0003084176810000041
εiAs residual, then:
Figure BDA0003084176810000042
the core of the trend surface analysis method is to calculate the trend surface from the actual observed value, and a regression analysis method is generally adopted, so that the sum of squares of residual errors Q tends to be minimum, namely:
Figure BDA0003084176810000043
this is a trend surface fit in the least squares sense.
Fitting values to trends
Figure BDA0003084176810000044
Formulas of different orders can be set according to needs, and generally polynomial functions of different orders are used for calculation, such as:
Figure BDA0003084176810000045
Figure BDA0003084176810000046
Figure BDA0003084176810000047
Figure BDA0003084176810000048
in the above formulas, formula (3) corresponds to a first simple linear trend surface, formula (4) corresponds to a first bilinear trend surface, formula (5) corresponds to a second trend surface, formula (6) corresponds to a third trend surface, and a higher-order standard preset formula is analogized, and the formula can also be customized.
After the models are established by using different formulas, the fitting degree of the models needs to be checked, and the models with the highest fitting degree are selected as the basis for the final ancient landform restoration. The model test mainly consists of the following three parts.
1)R2Examination of
Fitting degree coefficient R of trend surface and practical surface2Is an important index for determining the goodness of fit of the regression model. The goodness of fit of the regression model is generally expressed in terms of the proportion of the regression sum of squares in the total dispersion sum of the variable z.
The total sum of squared deviations equals the sum of the regression sum of squares and the remaining sum of squares, i.e.:
Figure BDA0003084176810000049
wherein the content of the first and second substances,
Figure BDA00030841768100000410
is the residual sum of squares, which represents the effect of random factors on dispersion,
Figure BDA00030841768100000411
it is the regression sum of squares, which represents the total effect of the dispersion of the independent variables on the dependent variables.
SSRThe larger (or SS)D) Smaller values indicate closer relationship between the dependent variable and the independent variable, and stronger regression regularity and better effect are achieved. This gives equation (8):
Figure BDA00030841768100000412
R2the larger the fit of the trend surface.
2) Significance F test
The trend surface moderated F test is a significance test for the whole trend surface regression model. The specific method is to determine whether the regression relationship between the variable z and the independent variables x and y is significant or not by utilizing the ratio of the residual square sum to the regression square sum in the total deviation square sum of the variable z. As shown in formula (9):
Figure BDA0003084176810000051
in the formula, p is the number of coefficients except for a constant term in the trend surface; n is the number of observation points.
At significance level α, look up F of the F distribution Tableα(p, n-p-1) if the calculated F value is greater than the threshold value FαThen, the trend surface equation is considered to be significant; otherwise, it is not significant.
3) Successive examination
First, the regression sum of squares of the higher order polynomial equation is found
Figure BDA0003084176810000052
Regression sum of squares with lower degree polynomial equation
Figure BDA0003084176810000053
The difference between the two; this difference is then divided by the difference in degrees of freedom p-q of the sum of squares of the regression to yield the regression mean square error due to the increase in the polynomial degree
Figure BDA0003084176810000054
Finally, the mean square error is divided by the residual mean square error of the higher-order polynomial
Figure BDA0003084176810000055
Obtaining the comparative appropriateness check value F of the two successive order trend surface modelsK+1→K. The specific formula can be seen in table 1.
TABLE 1 regression significance test with K-fold increase to (K + 1-fold) fold polynomial trend surface
Figure BDA0003084176810000056
Figure BDA0003084176810000061
Note: in the table, p and q are the number of coefficients except for constant terms in the trend surface equation; n is the number of observation points.
Looking up the table to obtain Fα(p-q, n-p-1) and F obtainedK+1→KIn comparison, if the resulting F value is significant, the higher order polynomial contributes to the regression, and if the F value is not significant, the higher order polynomial does not contribute new to the regression.
After the trend surface model is tested by the 3 methods, a proper model is selected, and then the earth structure as shown by a red dotted line in fig. 6 can be obtained. And then, the weathering crust stratum is superposed on the earth structure, and the ancient landform restoration map (shown in the attached figure 7) can be obtained.
The method according to the invention will be described by taking the example of a weathering crust in a certain area. As shown in Table 2, the data of the geographic coordinate positions of 100 wells and the thickness of the earth strata are combined, the distribution situation of the stratum thickness on the geographic space position is used as a trend surface model to be constructed for restoring the ancient landform, wherein the thickness h of the weathering crust1And thickness h of overburden2The thickness z of the stack of (a) as a dependent variable and the geographic coordinates as independent variables x, y.
TABLE 2 thickness and geographical coordinates of the strata in a certain area
Figure BDA0003084176810000062
Figure BDA0003084176810000071
Figure BDA0003084176810000081
Figure BDA0003084176810000091
Figure BDA0003084176810000101
Step 1, constructing a trend surface equation by using a least square method principle according to data in table 2, and obtaining a primary simple trend surface, a primary bilinear trend surface and a secondary trend surface model equation by using an MATLAB software to compile a calculation program as follows:
z11＝-5111+0.0004049x-0.0006157y
z12＝0.5797+0.0001457x-0.001306y+(3.464e-11)xy
z2＝(-8.264e+06)+0.6645x+0.8433y+(-1.44e-08)x2+(-2.451e-08)xy+(-4.272e-08)y2
step 2, substituting the independent variables x and y into the three trend surface model formulas (9), (10) and (11) to obtain a series of trend surface values
Figure BDA0003084176810000102
Step 3, performing R on the result2And (6) checking. Will zi、
Figure BDA0003084176810000103
And
Figure BDA0003084176810000104
substituting into correlation formula to obtain SSD、SSRAnd SSTThen, R is found2The correlation results are shown in table 3.
TABLE 3 analysis of the trend surface of the formation in a certain area R2Test sum F test results
Figure BDA0003084176810000105
Figure BDA0003084176810000111
Watch with watch3 according to R2As a result of the examination, the determination coefficients of the primary linear trend surface, the primary bilinear trend surface, and the secondary trend surface are 0.3047, 0.3760, and 0.5852, respectively, and the fitness of the secondary trend surface is higher.
And 4, carrying out F test on the result. The correlation data was substituted into equation (9) to find F values of the three models. In the case of confidence level α being 0.05, F of the one-time single-linear trend surface is obtained by table lookup0.05(2，97)＝3.09<Calculated value F1121.25; one time bilinear trend surface F0.05(3，96)＝2.70<Calculated value F1219.28; second trend surface F0.05(9，90)＝1.98<Calculated value F214.11. Therefore, the three equations are all significant from the F test, and the fitting result is reasonable.
And 5, carrying out successive testing on the trend surface of the result, and respectively comparing the primary single-linear trend surface with the primary bilinear trend surface, the primary single-linear trend surface with the secondary trend surface and the primary bilinear trend surface with the secondary trend surface.
The results of the successive-test analysis are shown in table 4 for the one-time single-linear trend surface and the one-time bilinear trend surface.
TABLE 4 successive testing analysis Table of Once-Mono-Linear and Once-bilinear trend surface regression model
Figure BDA0003084176810000112
Increase from a one-time single-linear trend surface to a one-time double-linear trend surface, F12→1110.97. In the case of the confidence level alpha being 0.05, F is obtained by looking up the F distribution table0.05(1，96)＝3.96<F12→1110.97. Therefore, the increase of the primary linear trend surface to the primary bilinear trend surface makes a new contribution to the regression equation, namely, the primary bilinear trend surface has higher fitting degree than the primary single linear trend surface.
The results of the sequential test analysis for the primary single linear trend surface and the secondary trend surface are shown in table 5.
TABLE 5 successive testing analysis of the Primary Mono-Linear and Secondary Trend surface regression models
Figure BDA0003084176810000113
Figure BDA0003084176810000121
Increase from a primary single linear trend surface to a secondary trend surface, F2→118.70. In the case of the confidence level alpha being 0.05, F is obtained by looking up the F distribution table0.05(7，90)＝2.11<F2→118.70. The rising of the primary, single-linear trend surface to the secondary trend surface makes a new contribution to the regression equation, i.e., the secondary trend surface has a higher fitness than the primary, single-linear trend surface.
The results of the sequential testing analysis for the primary bilinear trend surface and the secondary trend surface are shown in table 6.
TABLE 6 successive testing analysis of the Primary bilinear and Secondary Trend surface regression models
Figure BDA0003084176810000122
Increase from a primary single linear trend surface to a secondary trend surface, F2→127.57. In the case of the confidence level alpha being 0.05, F is obtained by looking up the F distribution table0.05(6，90)＝2.2<F2→127.57. Therefore, the rising of the primary bilinear trend surface to the secondary trend surface makes a new contribution to the regression equation, i.e., the secondary trend surface has a higher fitness than the primary bilinear trend surface.
By combining the three analyses, the model with the highest fitness of the secondary trend surface can be obtained.
Step 6, synthesizing the three model test results of steps 3-5, and obtaining the result model of the secondary trend surface with the highest fitting degree, so that the trend surface model formula is used:
z2＝(-8.264e+06)+0.6645x+0.8433y+(-1.44e-08)x2+(-2.451e-08)xy+(-4.272e-08)y2
step 7, utilizing a trend surface model formula z2A series of results
Figure BDA0003084176810000123
Value and weathering crust thickness h1And (4) superposing, and importing related mapping software to finally obtain the final ancient landform restoration map (figure 8).
Note: in this example, all the thickness values used for convenience in the calculation part are absolute values, and actually, if the ground is taken as a zero interface and the upward direction is positive according to a general situation, the trend face value is
Figure BDA0003084176810000124
Should be negative, so proceed
Figure BDA0003084176810000125
And h1Coping before superposition
Figure BDA0003084176810000126
And taking the negative value.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that are not thought of through the inventive work should be included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope defined by the claims.

Claims (5)

1. An ancient landform restoration method based on trend surface analysis is characterized by comprising the following steps:
step 1, collecting thickness h of weathering crust of target stratum1Thickness h of overburden2And geographic coordinates x and y, calculating the thickness z of the superposed stratum according to the obtained data, constructing a trend surface equation by using a least square method principle, and programming a calculation program by using MATLAB software to obtain a primary simple trend surface equation z11Once bilinear trend surface z12Equation z of the quadratic trend surface model2；
Step 2Substituting the independent variables x and y into the three trend surface model formulas to obtain a series of trend surface values
Figure FDA0003084176800000011
Step 3, performing R on the result2Checking; will zi、
Figure FDA0003084176800000012
And
Figure FDA0003084176800000013
substituting into correlation formula to obtain SSD、SSRAnd SSTThen, R is found2Comparison of
Figure FDA0003084176800000014
The higher the numerical value is, the higher the fitting degree is;
step 4, carrying out F inspection on the result; substituting the correlation data into a formula to obtain an F value of a corresponding model, and looking up a table to obtain the F of the single linear trend surface when the confidence level alpha is 0.050.05(p11，n-p11-1), a one-time bilinear trend surface F0.05(p12，n-p12-1), a secondary trend surface F0.05(p11，n-p11-1); comparison F0.05And F, if F is larger, the trend surface equation is obvious, otherwise, the trend surface equation is not obvious;
step 5, carrying out successive testing on the trend surface of the result, and respectively comparing the primary single-linear trend surface with the primary bilinear trend surface, the primary single-linear trend surface with the secondary trend surface and the primary bilinear trend surface with the secondary trend surface;
step 6, integrating the three model test results of the step 3-5, and selecting an optimal model;
step 7, utilizing a trend surface model formula z2A series of results
Figure FDA0003084176800000015
Value and weathering crustThickness h1And (5) overlapping, and importing related mapping software to finally obtain the final ancient landform recovery map.
2. The method according to claim 1, wherein the method comprises: the trend surface equation in the step (1) can also use a higher-order trend surface model equation of a cubic trend surface and a quartic trend surface according to the complexity of the ancient landform to be restored.
3. The method according to claim 1, wherein the method comprises: z in the above steps (2) and (3)i、
Figure FDA0003084176800000016
And
Figure FDA0003084176800000017
respectively indicating the stratum stacking thickness, the trend face value and the stratum stacking thickness average value,
Figure FDA0003084176800000018
in the form of a remaining sum of squares,
Figure FDA0003084176800000019
to return the sum of squares, SST＝SSD+SSRIs the sum of the squares of the total deviations.
4. The method according to claim 3, wherein the method comprises: the formula in the step (4) is
Figure FDA00030841768000000110
p is the number of coefficients except for a constant term in the trend surface equation; n is the number of observation points.
5. The method according to claim 1, wherein the ancient landform restoration method is based on trend surface analysisThe method is characterized in that: the method for successive detection in the step (5) comprises the following steps: first, the regression sum of squares of the higher order polynomial equation is found
Figure FDA00030841768000000111
Regression sum of squares with lower degree polynomial equation
Figure FDA00030841768000000112
The difference between the two; this difference is then divided by the difference in degrees of freedom p-q of the sum of squares of the regression to yield the regression mean square error due to the increase in the polynomial degree
Figure FDA0003084176800000021
Finally, the mean square error is divided by the residual mean square error of the higher-order polynomial
Figure FDA0003084176800000022
Obtaining the comparative appropriateness check value F of the two successive order trend surface modelsK+1→K(ii) a Wherein, MSR＝SSR/p，MSD＝SSD/(n-p-1)。
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