Geoscience Reference
In-Depth Information
Table 3 Polynomial function for surface reconstruction
No. of variables
Descriptive terms
Row
Formula
Z = A 0
+ A 1 X + A 2 Y
+ A 3 X 2 + A 4 Y 2 + A 5 XY
+ A 6 X 3 + A 7 Y 3 + A 8 X 2 Y + A 9 XY 2
+ A 10 X 4 + A 11 Y 4 + A 12 X 3 Y + A 13 X 2 Y 2 + A 14 XY 3
1
Flat
Zero
2
Linear
First
3
Quadratic
Second
4
Cubic
Third
5
Quartic
Fourth
7.1 Using Genetic Algorithms in Polynomial Optimization
Heights of polynomials can be useful for interpolation. The most common func-
tion to achieve this integration is the general polynomial function shown in Table 3
(Petrie and Kennie 1990 ).
It is clear that the single polynomial function has a special characteristic shape.
Using specific terms, unique surface features can be created.
For the actual surface production in a particular model, it is not necessary to
use the entire function. The operating system has the responsibility to determine
what is used. Only in a few cases it is possible for the user to select which parts of
the function to model the particular piece of land that is more relevant.
The first step is to determine the optimal use of polynomial functions in terms
of these functions. Shape optimisation of polynomials is related to the geometry
and topography of the region. GA is used to evaluate the effect of the presence or
absence of various terms where the polynomial functions are used to find the most
effective functions. For this purpose, a singular binary chromosome in the form of
a series of zeros and ones is used. The digit zero indicates non-interference and the
digit one indicates the interference. In the process of GA, optimal chromosomes
that show the best polynomial term obtained. Coefficients of the terms are deter-
mined by the least squares method during this process. In this study, quartic poly-
nomials are examined. For the GA optimisation, firstly the chromosomes must be
formed and an initial population created. Each chromosome is made up of vari-
ables that are essentially the polynomial coefficients, which is interpreted as gene.
Gene 1 represents in the desired term of polynomial and gene 0 represents the
interference term in the polynomial. The first algorithm optimisation process con-
sists of an initial population of chromosomes and the coefficients that can be cal-
culated by control points through the least squares method and using checkpoints
to determine the remaining residue. So by employing control points, checkpoints
and the dependent variable (RMSE), optimal chromosomes are formed. After fin-
ishing the optimal processes, the other checkpoints, which have no interference
in the process optimisation, the obtained chromosomes will be evaluated. In other
words, the process of determining proper coefficients for polynomials with genetic
algorithm as well as control points is used check points that consist of two parts.
One of these parts is for the optimisation of the process including control points to
Search WWH ::




Custom Search