）

：

B(h)



(h

2



R

2

)ln(h

2



R

2

)

张力样条函数（

spline with tension

）

：

B(h)



ln(

规则样条函 数（

completely regularized spline

）

：

< br>R



h

)



K

0

(R



h)

2



C

E

2

(-1 )

n



r

2n

R



h

2

R



h

2

B(h)









ln(

)



E

1

(

)

C

E

n!n

2

2

n



1



高次曲面样条函数（< /p>

multiquadric spline

）

：

B(h)



h

< br>2



R

2

反高次曲面样条函数（

inverse multiquadric spline

）

：

B(h)



1

h



R

2

2

< br>

各式子中

h

为表示由点

(x

，

y)

到第

i

个数据点的距离，

R

参数是用户指定的

平滑因子，

K

0

为修正贝塞尔函数，

E

1

为指数积分函数，

C

E< /p>

为

Euler

常数，其值

约为

0.577215

。< /p>

Radial Basis Function

interpolation is a diverse group of data

interpolation methods. In terms of the ability to fit your data and to

produce a smooth surface, the

Multiquadric

method is considered by many

to be the best. All of the

Radial Basis Function

methods are exact

interpolators, so they attempt to honor your data. You can introduce a

smoothing factor to all the methods in an attempt to produce a smoother

surface.

Function Types

The basis kernel functions are analogous to variograms in

Kriging

. The

basis kernel functions

define

the optimal

set

of weights to

apply to the

data points when interpolating a grid node. The available basis kernel

functions are listed in the

Type

drop-down list in the

Radial Basis

Function Options

dialog.

Inverse Multiquadric

Multilog

Multiquadratic

Natural Cubic Spline

Thin Plate Spline

where:

h is the anisotropically rescaled, relative distance from the point to

the node

R

2

is the smoothing factor specified by the user

Default R

2

Value

The default value for R

2

in the Radial Basis Function gridding algorithm

is calculated as follows:

(length of diagonal of the data extent)

2

/ (25 * number of data points)

Specifying Radial Basis Function Advanced Options

1. Click on

Grid | Data

.

2. In

the

Open

dialog,

select

a

data

file

and

then

click

the

Open

button.

3. In

the

Grid

Data

dialog,

choose

Radial

Basis

Function

in

the

Gridding

Method

group.

4. Click

the

Advanced

Options

button

to

display

the

Radial

Basis

Advanced

Options

dialog.

5. In

the

General

page,

you

can

specify

the

function

parameters

for

the

gridding operation.



The

Basis Function

list specifies the basis kernel function to use during gridding. This

defines

the

optimal

weights

applied

to

the

data

points

during

the

interpolation.

The

Basis

Function

is analogous to the variogram in

Kriging

. Experience indicates that the

Multiquadric

basis

function

works

quite

well

in

most

cases.

Successful

use

of

the

Thin

Plate

Spline

basis function is also reported regularly in the technical literature.



The

R

2

Parameter

is

a

shaping

or

smoothing

factor.

The

larger

the

R

2

Parameter

shaping

factor,

the rounder the mountain tops and the smoother the contour lines. There is no universally

accepted method for computing an optimal value for this factor. A reasonable trial value

for

R

2

Parameter

is between the average sample spacing and one-half the average sample

spacing.

Triangulation with Linear Interpolation

The

Triangulation with Linear Interpolation

method in

Surfer

uses the

optimal Delaunay triangulation. The algorithm creates triangles by

drawing lines between data points. The original points are connected in

such

a

way

that

no

triangle

edges

are

intersected

by

other

triangles.

The

result is a patchwork of triangular faces over the extent of the grid.

This method is an exact interpolator.

Each

triangle

defines

a

plane

over

the

grid

nodes

lying

within

the

triangle,

with the tilt and elevation of the triangle determined by the three

original

data

points

defining

the

triangle.

All

grid

nodes

within

a

given

triangle

are

defined

by

the

triangular

surface.

Because

the

original

data

are used to define the triangles, the data are honored very closely.

Triangulation with Linear Interpolation

works best when your data are

evenly

distributed

over

the

grid

area.

Data

sets

that

contain

sparse

areas

result in distinct triangular facets on the map.

2.5

最小曲率法

Minimum Curvature

is widely used in the earth sciences. The interpolated surface

generated by

Minimum Curvature

is analogous to a thin, linearly elastic plate

passing through each of the data values with a minimum amount of bending.

The Minimum Curvature gridding algorithm is solves the specified partial

differential equation using a successive over-relaxation algorithm. The

interior is updated using a

(1988, p. 868). The only difference is that the biharmonic equation must have

nine different

Minimum Curvature

generates the smoothest possible surface while attempting to

honor your data as closely as possible.

Minimum Curvature

is not an exact

interpolator, however. This means that your data are not always honored

exactly.

Minimum Curvature

produces a grid by repeatedly applying an

equation

over the

grid in an attempt to smooth the grid. Each pass over the grid is counted as one

iteration. The grid node values are recalculated until successive changes in the

values are less than the

Maximum Residuals

value, or the maximum number of

iterations is reached (

Maximum Iteration

field).

The

Maximum Residual

parameter

has the same units as the data, and an

appropriate value is approximately 10% of the data precision. If data values are

measured to the nearest 1.0 units, the

Maximum Residual

value should be set at

0.1. The iterations continue until the maximum grid node correction for the

entire iteration is less than the

Maximum Residual

value. The default

Maximum

Residual

value is given by:

Default Max Residual = 0.001 (Z

max

- Z

min

)

The

Maximum Iteration

parameter

should be set at one to two times the

number of grid nodes generated in the grid file. For example, when

generating a 50 by 50 grid using

Minimum Curvature

, the

Maximum Iteration

value should be set between 2,500 and 5,000.

The

Internal Tension

and

Boundary Tension

,Qualitatively, the

Minimum

Curvature

gridding algorithm is attempting to fit a piece of sheet metal

through all of the observations without putting any creases or kinks in the

surface. Between the fixed observation points, the sheet bows a bit. The

Internal Tension

is used to control the amount of this bowing on the interior:

the higher the tension, the less the bowing. For example, a high tension makes

areas between observations look like facets of a gemstone. The

Boundary

Tension

controls the amount of bowing on the edges. The range of values for

Internal Tension

and

Boundary Tension

are 0 to 1. By default, the

Internal

Tension

and the

Boundary Tension

are set to 0.

the

Relaxation Factor

,

The Relaxation Factor is as described in Press et al. (1988). In general, the

Relaxation Factor should not be altered. The default value (1.0) is a good

generic value. Roughly, the higher the Relaxation Factor (closer to two) the

faster the Minimum Curvature algorithm converges, but the more likely it

will not converge at all. The lower the Relaxation Factor (closer to zero) the

more likely the Minimum Curvature algorithm will converge, but the

algorithm is slower. The optimal Relaxation Factor is derived through trial

and error.

2.6

近邻法

The Natural Neighbor gridding method is quite popular in some fields. What is

Natural Neighbor interpolation?

Consider a set of Thiessen polygons (the dual of a Delaunay triangulation). If a

new point (target) were added to the data set, these Thiessen polygons would be

modified. In fact, some of the polygons would shrink in size, while none would

increase in size. The area associated with the target's Thiessen polygon that was

taken from an existing polygon is called the

The

Natural

Neighbor

interpolation

algorithm

uses

a

weighted

average

of

the

neighboring observations, where the weights are proportional to the

area.

2.7

最近邻法

The Nearest Neighbor gridding method assigns the value of the nearest point to

each grid node. This method is useful when data are already evenly spaced, but

need to be converted to a Surfer grid file. Alternatively, in cases where the data

are nearly on a grid with only a few missing values, this method is effective for

filling in the holes in the data.

2.8

多项式回归方法

You can select the type of polynomial regression to apply to your data from the

Surface

Definition

group.

As

you

select

the

different

types

of

polynomials,

a

generic polynomial form of the equation is presented in the dialog, and the values

in

the

Parameters

group

change

to

reflect

the

selection.

The

available

choices

are:



Simple planar surfaceBi- linear saddleQuadratic surfaceCubic surfaceUser