CN103544679A - Method for correcting distortion of foundation cloud pictures of all-sky imager - Google Patents

Abstract

The invention provides a method for correcting distortion of foundation cloud pictures of an all-sky imager, and belongs to the technical field of photovoltaic power generation and distribution. The method includes steps of creating a conversion function among spherical distortion zenith angles in the foundation cloud pictures and actual zenith angles; converting the foundation cloud pictures into cloud pictures in a spherical coordinate system; converting the cloud pictures in the spherical coordinate system into actual cloud pictures in a rectangular plane coordinate system. The method has the advantages that the quantity of required parameters is low, and only a few parameters such as the distance from a lens of the all-sky imager to a spherical lens and the radius of the spherical lens are required; input data are simple, the distortion can be corrected only by means of inputting original data of the foundation cloud pictures, and loss of information of the pictures in conversion procedures is prevented.

Description

A kind of total sky imager ground cloud atlas distortion correction method

Technical field

The invention belongs to photovoltaic and be transported to electro-technical field, be specifically related to a kind of total sky imager ground cloud atlas distortion correction method.

Background technology

In recent years, sun power science and technology and application development are swift and violent.By the end of 2010 Nian， world photovoltaic accumulative total installed capacitys, approached 40GW, nearly 10 annual years increase by 45%, become one of industry with the fastest developing speed.But because solar electrical energy generation has randomness and intermittent feature, the grid-connected of large solar generating brings charge and challenge by the safe and stable operation to electrical network.Therefore, region solar energy resources is carried out to monitoring and prediction timely and effectively and become the key factor that restricts photovoltaic power generation grid-connecting.

At present, carry out ground cloud amount, cloud form automatically the effective means of monitoring be the observation of ground cloud atlas.The observation of ground cloud atlas is by surface-based total sky imager, the atmospheric exploration mode that local sky is observed.Its advantage, for having high-spatial and temporal resolution, can obtain sky total amount of cloud, be less than the cloud form detailed information of 1km and the observing frequency of minute level.Chinese scholars has been carried out certain research at aspects such as the cloud identification based on ground cloud atlas, radiation predictions.Calb ó etc. are by having realized the identification of varieties of clouds types to the feature extraction of ground all-sky cloud atlas.The people such as Chow use TSI440A total sky imager to observe U.S. SanDiego area, and set up based on ground cloud atlas hour in cloud Forecasting Methodology.The employing WAVELET PACKET DECOMPOSITION such as domestic scholars Zhang Yonghong and morphological method have been carried out edge extracting to ground cloud atlas respectively, and utilize image fusion technology to obtain optimal edge image.But the distortion correction to ground cloud atlas, and the reduction of the actual cloud atlas of surface level, yet there are no correlative study achievement.

Utilize ground cloud atlas to carry out cloud track and radiation prediction, its key is direction of motion and the speed of prediction cloud.But because the imaging mode of total sky imager is the problems such as convex mirror imaging, and imaging visual angle, its image of taking the photograph exists certain distortion, this has all brought difficulty to cloud amount calculating, cloud trajectory predictions.This invention, by original ground cloud atlas being implemented to spherical mirror reflection distortion correction and spherical co-ordinate to rectangular coordinates transformation, has realized the distortion correction of ground cloud atlas, for cloud amount calculates, region sun power monitoring and prediction provides true and reliable Data Source.

Summary of the invention

In order to overcome above-mentioned the deficiencies in the prior art, the invention provides a kind of total sky imager ground cloud atlas distortion correction method, desired parameters is few, only needs total sky imager camera lens to a small number of parameters such as the distance of spherical mirror and spherical mirror radiuses; Input data are simple, and only need to input original ground cloud atlas data and can complete correction, and image information free of losses in transfer process.

In order to realize foregoing invention object, the present invention takes following technical scheme:

A ground cloud atlas distortion correction method, said method comprising the steps of:

Step 1: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle;

Step 2: ground cloud atlas is converted to spheric coordinate system cloud atlas;

Step 3: described spheric coordinate system cloud atlas is converted to the actual cloud atlas of plane right-angle coordinate.

Described step 1 comprises the following steps:

Step 1-1: set up described total sky imager image deformation model;

Step 1-2: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle according to described image deformation model.

In described image deformation model, camera lens is counted h to the distance on spherical mirror summit ₁, the minute surface radius of described spherical mirror is counted R, and sky point is counted a to the incident ray of spherical mirror at the angle of incidence point place and vertical direction ₂, the reflection ray that reflexes to described camera lens through spherical mirror is counted a at place, described optical center with the angle of vertical direction ₁;

Take the minute surface center of spherical mirror is true origin O, take horizontal direction as x axle, take vertical direction as y axle, and minute surface equation is:

x ²+y ²＝R ² （1）

From the equations of light ray of reflection ray, be:

y-(h ₁+R)＝-ctana ₁·x （2）

By equation (1) and equation (2), can obtain incidence point (x ₀, y ₀) lateral coordinates and along slope coordinate be respectively:

$x_0=\frac{1}{2(1+{c\tan }^2{a}_1)}(2\left(h_1+R\right)c\tan a_1-\sqrt{{\left(2(h_1+R)c\tan a_1\right)}^2-4({(h_1+R)}^2-R^2)(1+{c\tan }^2{a}_1)})---\left(3\right)$

$y_0=\sqrt{R^2-{x_0}^2}---\left(4\right)$

If incidence point (x ₀, y ₀) to the line at minute surface center, at the angle of minute surface center and vertical direction, be a ₃, a ₃be expressed as:

$a_3=\arcsin \frac{x_0}{R}---\left(5\right)$

Can obtain a ₁with a ₂between pass be:

$a_2=2\·a_3+a_1=2\·\arcsin \frac{x_0}{R}+a_1---\left(6\right)$

Simultaneous equations (6) and equation (3) are the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle; By transfer function, can be obtained the transfer function curve of sphere distortion zenith angle.

In described step 2, by transfer function, the sphere distortion zenith angle of every in described ground cloud atlas being proofreaied and correct is the actual zenith angle of this point, and it is constant to maintain position angle, can realize ground cloud atlas to the conversion of spheric coordinate system cloud atlas.

In described step 2, establish the image magnification ratio M of spheric coordinate system cloud atlas center ₀be 1, M ₀be expressed as:

${\mathrm{<figure-callout\; id="0"\; label="M"\; filenames="HDA00001863800300011.png,HDA00001863800300012.png"\; state="\{\{state\}\}">M</figure-callout>}}_0={\left(\frac{{\mathrm{\&Delta;<figure-callout\; id="2"\; label="a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">a</figure-callout>}}_2}{{\mathrm{\&Delta;a}}_1}\right)}_0\frac{p_2}{p_1}=1---\left(7\right)$

Wherein,

for the derivative of the actual zenith angle in spheric coordinate system cloud atlas center to sphere distortion zenith angle, p ₂for the size of unit zenith angle correspondence image in spheric coordinate system cloud atlas, p ₁size for unit zenith angle correspondence image in total sky imager image;

The image magnification ratio M that obtains any point in spheric coordinate system cloud atlas by equation (7), M is expressed as:

$M=\left(\frac{{\mathrm{\&Delta;<figure-callout\; id="2"\; label="a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">a</figure-callout>}}_2}{{\mathrm{\&Delta;a}}_1}\right)\frac{p_2}{p_1}=\left(\frac{{\mathrm{\&Delta;<figure-callout\; id="2"\; label="a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">a</figure-callout>}}_2}{{\mathrm{\&Delta;a}}_1}\right)\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_1}{{\mathrm{\&Delta;a}}_2}\right)}_0---\left(8\right)$

Wherein,

for the derivative of this actual zenith angle in some place to sphere distortion zenith angle,

for the derivative of spheric coordinate system cloud atlas center sphere distortion zenith angle to actual zenith angle;

The ratio R of corresponding point radius in the radius at maximum zenith angle place and total sky imager image in spheric coordinate system cloud atlas _maxbe expressed as:

$R_{\max }=\frac{a_{2\max }}{a_{1\max }}\frac{p_2}{p_1}=\frac{a_{2\max }}{a_{1\max }}\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_1}{{\mathrm{\&Delta;a}}_2}\right)}_0---\left(9\right)$

Wherein, a _2maxfor the actual zenith angle of maximum, a _1maxfor biggest ball area distortion zenith angle.

In described step 3: establish day aerial cloud in sustained height H, and the thickness of cloud can ignore, on cloud, any point is α with respect to the zenith angle of total sky imager ₂, the known distance that this is zero place to zenith angle on equal height is L, L is expressed as:

L＝Htana ₂ （10）

That is: in spheric coordinate system cloud atlas, any point is that the distance L at the zero place zenith angle by this point is multiplied by this point and obtains apart from the height on ground apart from zenith angle on equal height under plane right-angle coordinate.

In described step 3, establish the image magnification ratio M of the actual cloud atlas of plane right-angle coordinate center _l0be 1, M _l0be expressed as:

${\mathrm{<figure-callout\; id="0"\; label="M\; L"\; filenames="HDA00001863800300011.png,HDA00001863800300012.png"\; state="\{\{state\}\}">M</figure-callout>}}_L={\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)}_{{\mathrm{\&alpha;}}_2=0}\frac{P_{2\mathrm{<figure-callout\; id="2"\; label="L\; P"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">L</figure-callout>}}}{P_{}}=1---\left(11\right)$

Wherein,

for the derivative of the actual cloud atlas of plane right-angle coordinate center L to actual zenith angle, p _2Lfor the size of unit distance correspondence image in the actual cloud atlas of plane right-angle coordinate, p ₁size for unit zenith angle correspondence image in spheric coordinate system cloud atlas;

Obtain the image magnification ratio M of the actual cloud atlas any point of plane right-angle coordinate _l, M _lbe expressed as:

$M_L=\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)\frac{P_{2\mathrm{<figure-callout\; id="2"\; label="L\; P"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">L</figure-callout>}}}{P_{}}=\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)\&CenterDot;{\left(\frac{{\mathrm{\&Delta;<figure-callout\; id="2"\; label="a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">a</figure-callout>}}_2}{\mathrm{\&Delta;L}}\right)}_0---\left(12\right)$

Wherein,

for the derivative of this some L of place to actual zenith angle,

for the derivative of the actual zenith angle in the actual cloud atlas of plane right-angle coordinate center to L;

The ratio R of corresponding point radius in maximum radius and spheric coordinate system cloud atlas in the actual cloud atlas of plane right-angle coordinate _lmaxbe expressed as:

$R_{L\max }=\frac{L_{\mathrm{<figure-callout\; id="2"\; label="max\; a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">max</figure-callout>}}}{a_{}}\frac{P_{2\mathrm{<figure-callout\; id="2"\; label="L\; P"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">L</figure-callout>}}}{P_{}}=\frac{L_{\mathrm{<figure-callout\; id="2"\; label="max\; a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">max</figure-callout>}}}{a_{}}\&CenterDot;{\left(\frac{{\mathrm{\&Delta;<figure-callout\; id="2"\; label="a"\; filenames="HDA00001863800300021.png"\; state="\{\{state\}\}">a</figure-callout>}}_2}{\mathrm{\&Delta;L}}\right)}_0---\left(13\right)$

Wherein, L _maxfor this point in the actual cloud atlas of plane right-angle coordinate is apart from the ultimate range of the actual cloud atlas of plane right-angle coordinate center.

Compared with prior art, beneficial effect of the present invention is:

1) this method desired parameters is few, only needs total sky imager camera lens to a small number of parameters such as the distance of spherical mirror and spherical mirror radiuses;

2) input data are simple, only need to input original ground cloud atlas data and can complete correction, and correction accuracy is high;

3) image information free of losses in transfer process;

4) the method is simple and reliable, easily carries out.

Accompanying drawing explanation

Fig. 1 is total sky imager image deformation illustraton of model in the embodiment of the present invention;

Fig. 2 is total sky imager image deformation coordinate diagram in the embodiment of the present invention;

Fig. 3 is the conversion curve figure of actual zenith angle and sphere distortion zenith angle in the embodiment of the present invention;

Fig. 4 is that in the embodiment of the present invention, spheric coordinate system cloud atlas converts schematic diagram to the actual cloud atlas of plane right-angle coordinate.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

As Fig. 1-Fig. 4, a kind of total sky imager ground cloud atlas distortion correction method, said method comprising the steps of:

Step 1: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle;

Step 2: ground cloud atlas is converted to spheric coordinate system cloud atlas;

Step 3: described spheric coordinate system cloud atlas is converted to the actual cloud atlas of plane right-angle coordinate.

Described step 1 comprises the following steps:

Step 1-1: set up described total sky imager image deformation model;

Step 1-2: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle according to described image deformation model.

In described image deformation model, camera lens is counted h to the distance on spherical mirror summit ₁, the minute surface radius of described spherical mirror is counted R, and sky point is counted a to the incident ray of spherical mirror at the angle of incidence point place and vertical direction ₂, the reflection ray that reflexes to described camera lens through spherical mirror is counted a at place, described optical center with the angle of vertical direction ₁;

Take the minute surface center of spherical mirror is true origin O, take horizontal direction as x axle, take vertical direction as y axle, and minute surface equation is:

x ²+y ²＝R ² （1）

From the equations of light ray of reflection ray, be:

y-(h ₁+R)＝-ctana ₁·x （2）

By equation (1) and equation (2), can obtain incidence point (x ₀, y ₀) lateral coordinates and along slope coordinate be respectively:

$x_0=\frac{1}{2(1+{c\tan }^2{a}_1)}(2\left(h_1+R\right)c\tan a_1-\sqrt{{\left(2(h_1+R)c\tan a_1\right)}^2-4({(h_1+R)}^2-R^2)(1+{c\tan }^2{a}_1)})---\left(3\right)$

${\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HDA00001863800300011.png,HDA00001863800300012.png"\; state="\{\{state\}\}">y</figure-callout>}}_0=\sqrt{R^2-{x_0}^2}---\left(4\right)$

If incidence point (x ₀, y ₀) to the line at minute surface center, at the angle of minute surface center and vertical direction, be a ₃, a ₃be expressed as:

$a_3=\arcsin \frac{x_0}{R}---\left(5\right)$

Can obtain a ₁with a ₂between pass be:

$a_2=2\&CenterDot;a_3+a_1=2\&CenterDot;\arcsin \frac{x_0}{R}+a_1---\left(6\right)$

Simultaneous equations (6) and equation (3) are the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle; By transfer function, can be obtained the transfer function curve of sphere distortion zenith angle.

In described step 2, by transfer function, the sphere distortion zenith angle of every in described ground cloud atlas being proofreaied and correct is the actual zenith angle of this point, and it is constant to maintain position angle, can realize ground cloud atlas to the conversion of spheric coordinate system cloud atlas.

Described step 2 comprises the following steps:

Step 2-1: obtain the ground cloud atlas image that total sky imager is taken, deposit in array Array1;

Step 2-2: remove sky edge and above ground portion in image, retain cloud atlas central authorities zenith angle and be less than or equal to the region of 70 °;

Step 2-3: calculate spherical co-ordinate image size according to ground cloud atlas size and equation (9), the array variable Array2 of the corresponding size of definition is as the space that stores spherical co-ordinate image;

Step 2-4: to each pixel in Array1, calculate its sphere distortion zenith angle and position angle, obtain this according to equation (6) and put actual zenith angle, and utilize equation (8) to calculate the size of this pixel in spherical co-ordinate image; Position angle remains unchanged in conversion;

Step 2-5: according to actual zenith angle and the position angle of each pixel, the coordinate of calculating pixel in spherical co-ordinate image, the R of each pixel, G, each passage color value of B are write in the picture dot of respective coordinates in Array2, Color Channel and corresponding size, obtain spherical co-ordinate cloud atlas.

Described step 3 comprises the following steps:

Step 3-1: calculate the actual cloud atlas size of rectangular coordinate system according to spherical co-ordinate cloud atlas size and equation (13), the array variable Array3 of the corresponding size of definition is as the space that stores actual cloud atlas image;

Step 3-2: to each pixel in Array2, obtain this point to central point distance according to equation (10), calculate its coordinate in rectangular coordinate system image, and utilize equation (12) to calculate the size of this pixel in spherical co-ordinate image; Position angle remains unchanged in conversion;

Step3: the R of each pixel, G, each passage color value of B are write in the picture dot of respective coordinates in Array3, Color Channel and corresponding size, obtain the actual cloud atlas image of rectangular coordinate system.

Finally should be noted that: above embodiment is only in order to illustrate that technical scheme of the present invention is not intended to limit, although the present invention is had been described in detail with reference to above-described embodiment, those of ordinary skill in the field are to be understood that: still can modify or be equal to replacement the specific embodiment of the present invention, and do not depart from any modification of spirit and scope of the invention or be equal to replacement, it all should be encompassed in the middle of claim scope of the present invention.

Claims (7)

1. a total sky imager ground cloud atlas distortion correction method, is characterized in that: said method comprising the steps of:

Step 1: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle;

Step 2: ground cloud atlas is converted to spheric coordinate system cloud atlas;

Step 3: described spheric coordinate system cloud atlas is converted to the actual cloud atlas of plane right-angle coordinate.

2. total sky imager ground cloud atlas distortion correction method according to claim 1, is characterized in that: described step 1 comprises the following steps:

Step 1-1: set up described total sky imager image deformation model;

Step 1-2: set up the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle according to described image deformation model.

3. total sky imager ground cloud atlas distortion correction method according to claim 2, is characterized in that: in described image deformation model, camera lens is counted h to the distance on spherical mirror summit ₁, the minute surface radius of described spherical mirror is counted R, and sky point is counted a to the incident ray of spherical mirror at the angle of incidence point place and vertical direction ₂, the reflection ray that reflexes to described camera lens through spherical mirror is counted a at place, described optical center with the angle of vertical direction ₁;

Take the minute surface center of spherical mirror is true origin O, take horizontal direction as x axle, take vertical direction as y axle, and minute surface equation is:

x ²+y ²＝R ² （1）

From the equations of light ray of reflection ray, be:

y-(h ₁+R)＝-ctana ₁·x （2）

By equation (1) and equation (2), can obtain incidence point (x ₀, y ₀) lateral coordinates and along slope coordinate be respectively:

$x_0=\frac{1}{2(1+{c\tan }^2{a}_1)}(2\left(h_1+R\right)c\tan a_1-\sqrt{{\left(2(h_1+R)c\tan a_1\right)}^2-4({(h_1+R)}^2-R^2)(1+{c\tan }^2{a}_1)})---\left(3\right)$

$y_0=\sqrt{R^2-{x_0}^2}---\left(4\right)$

If incidence point (x ₀, y ₀) to the line at minute surface center, at the angle of minute surface center and vertical direction, be a ₃, a ₃be expressed as:

$a_3=\arcsin \frac{x_0}{R}---\left(5\right)$

Can obtain a ₁with a ₂between pass be:

$a_2=2\&CenterDot;a_3+a_1=2\&CenterDot;\arcsin \frac{x_0}{R}+a_1---\left(6\right)$

Simultaneous equations (6) and equation (3) are the transfer function that in ground cloud atlas, sphere distorts between zenith angle and actual zenith angle; By transfer function, can be obtained the transfer function curve of sphere distortion zenith angle.

4. total sky imager ground cloud atlas distortion correction method according to claim 1, it is characterized in that: in described step 2, by transfer function, the sphere distortion zenith angle of every in described ground cloud atlas being proofreaied and correct is the actual zenith angle of this point, and it is constant to maintain position angle, can realize ground cloud atlas to the conversion of spheric coordinate system cloud atlas.

5. total sky imager ground cloud atlas distortion correction method according to claim 4, is characterized in that: in described step 2, establish the image magnification ratio M of spheric coordinate system cloud atlas center ₀be 1, M ₀be expressed as:

$M_0={\left(\frac{{\mathrm{\&Delta;a}}_2}{{\mathrm{\&Delta;a}}_1}\right)}_0\frac{p_2}{p_1}=1---\left(7\right)$

Wherein,

for the derivative of the actual zenith angle in spheric coordinate system cloud atlas center to sphere distortion zenith angle, p ₂for the size of unit zenith angle correspondence image in spheric coordinate system cloud atlas, p ₁size for unit zenith angle correspondence image in total sky imager image;

The image magnification ratio M that obtains any point in spheric coordinate system cloud atlas by equation (7), M is expressed as:

$M=\left(\frac{{\mathrm{\&Delta;a}}_2}{{\mathrm{\&Delta;a}}_1}\right)\frac{p_2}{p_1}=\left(\frac{{\mathrm{\&Delta;a}}_2}{{\mathrm{\&Delta;a}}_1}\right)\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_1}{{\mathrm{\&Delta;a}}_2}\right)}_0---\left(8\right)$

Wherein, for the derivative of this actual zenith angle in some place to sphere distortion zenith angle, for the derivative of spheric coordinate system cloud atlas center sphere distortion zenith angle to actual zenith angle;

The ratio R of corresponding point radius in the radius at maximum zenith angle place and total sky imager image in spheric coordinate system cloud atlas _maxbe expressed as:

$R_{\max }=\frac{a_{2\max }}{a_{1\max }}\frac{p_2}{p_1}=\frac{a_{2\max }}{a_{1\max }}\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_1}{{\mathrm{\&Delta;a}}_2}\right)}_0---\left(9\right)$

Wherein, a _2maxfor the actual zenith angle of maximum, a _1maxfor biggest ball area distortion zenith angle.

6. total sky imager ground cloud atlas distortion correction method according to claim 1, is characterized in that: in described step 3:

If it aerial cloud is in sustained height H, and the thickness of cloud can ignore, and on cloud, any point is a with respect to the zenith angle of total sky imager ₂, the known distance that this is zero place to zenith angle on equal height is L, L is expressed as:

L＝Htana ₂ （10）

That is: in spheric coordinate system cloud atlas, any point is that the distance L at the zero place zenith angle by this point is multiplied by this point and obtains apart from the height on ground apart from zenith angle on equal height under plane right-angle coordinate.

7. total sky imager ground cloud atlas distortion correction method according to claim 6, is characterized in that: in described step 3, establish the image magnification ratio M of the actual cloud atlas of plane right-angle coordinate center _l0be 1, M _l0be expressed as:

$M_{L0}={\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)}_{{\mathrm{\&alpha;}}_2=0}\frac{P_{2L}}{P_2}=1---\left(11\right)$

Wherein,

for the derivative of the actual cloud atlas of plane right-angle coordinate center L to actual zenith angle, p _2Lfor the size of unit distance correspondence image in the actual cloud atlas of plane right-angle coordinate, p ₁size for unit zenith angle correspondence image in spheric coordinate system cloud atlas;

Obtain the image magnification ratio M of the actual cloud atlas any point of plane right-angle coordinate _l, M _lbe expressed as:

$M_L=\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)\frac{P_{2L}}{P_2}=\left(\frac{\mathrm{\&Delta;L}}{{\mathrm{\&Delta;a}}_2}\right)\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_2}{\mathrm{\&Delta;L}}\right)}_0---\left(12\right)$

Wherein,

for the derivative of this some L of place to actual zenith angle,

for the derivative of the actual zenith angle in the actual cloud atlas of plane right-angle coordinate center to L;

The ratio R of corresponding point radius in maximum radius and spheric coordinate system cloud atlas in the actual cloud atlas of plane right-angle coordinate _lmaxbe expressed as:

$R_{L\max }=\frac{L_{\max }}{a_{2\max }}\frac{P_{2L}}{P_2}=\frac{L_{\max }}{a_{2\max }}\&CenterDot;{\left(\frac{{\mathrm{\&Delta;a}}_2}{\mathrm{\&Delta;L}}\right)}_0---\left(13\right)$

Wherein, L _maxfor this point in the actual cloud atlas of plane right-angle coordinate is apart from the ultimate range of the actual cloud atlas of plane right-angle coordinate center.

Images