NOAA KLM User's Guide

Appendix I.2

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I.2 Calculating the Earth Coordinates of a Scan Spot


Consider an inertial coordinate system, I, whose origin is the center of the Earth (Figure I.2-1). The line joining the center of the earth to the Vernal Equinox constitutes the X-axis. The Z-axis is perpendicular to the equatorial plane and in the direction of the North Pole. The Y-axis is defined such that the vectors Vectors X. Y and Z define a right handed coordinate system constitute a right handed coordinate system. Let Position Vector P sub sat and Position Vector P sub spot be position vectors of the satellite and the scan spot, respectively. Then the position of the scan spot on the earth in the inertial coordinate system can be expressed in equation I-1.


P sub spot matrix equals the P sub sat matrix plus Range times the directional cosine matrix (I-1)

or

Vector P sub spot equals vector P sub sat plus Range times the dirctional cosine vector (I-2)

Figure I.2- 1.  The geometry of the satellite and its scan spot relative to the earth and
the earth-centered-inertial coordinate system.

where R is the range or distance from the satellite to the scan spot and (dx,dy,dz) are the direction cosines of the scan spot from the satellite. The subscript I designates the inertial coordinate system. See Figure I.2-1.

In order to solve for Position vector P sub spot is composed of the componants X sub spot, Y xub spot and Z sub spot , a new coordinate system centered at the spacecraft is established; call it the nominal scanning coordinate system. The scanner mounting frame is taken as the origin. The positive Xns-axis is in the direction of the satellite's subpoint (See I.3 "Defining the Satellite Subpoint"). The Zns-axis is along the nominal spin axis of the mirror, perpendicular to Xns and positive in the direction of the velocity vector. The Yns-axis completes a right handed system. If there are no misalignments, the instrument mirror will scan perpendicular to the Xns-Zns plane. See Figure I.2- 2.

Define

Vector P is defined with componants X sub p, Y sub p and Z sub p in an earth-centered inertial coordinate 
system(I-3)

where Xp, Yp and Zp are the direction cosines, in earth-centered-inertial coordinates, of the satellite subpoint from the satellite.


The speed of the satellite relative to the earth-centered-inertial frame is


The absolute value of the vector V sub sat equals the square root of X sub sat dot squared plus Y sub sat dot squared plus Z sub sat 
dot squared


The nominal scanning coordinate system



The scan angle within the nominal scanning coordinate sysem

Define


The velocity vector is defined as X sub sat dot divided by the absolute value of V sub sat, Y sub sat dot divided by the
absolute value of V sub sat and Z sub sat dot divided by the absolute value of V sub sat (I-4)

defined as vector componants X sub v, Y sub v and Z sub v in the earth-centered-inertial frame

where Xv, Yv and Zv are the direction cosines of the satellite’s velocity relative to the earth-centered-inertial frame and expressed as components in the earth-centered-inertial system.

The vector Q equals the cross product of the velocity vector v with the position vector P divided by the absolute value of the cross 
product and is can be resolved into the quantity of Y sub v times Z sub p minus Z sub v times Y sub p, Z sub v times X sub p minus X sub v times Z sub p and X sub v times Y sub p minus Y sub v times X sub p divided by the absolute value of the cross product of vectors V and P

= (Xq, Yq, Zq)I                                                                                                      (I-5)                                                                                     


where

The absolute value of the cross product of the velocity vector V and the postion vector P is the square root of the quantity of Y 
sub v times Z sub p minus Z sub v times Y sub p squared plus Z sub v times X sub p minus X sub v times Z sub p squared plus X sub v times Y sub p minus Y sub v times X sub p 
squared

and Xq, Yq and Zq are the direction cosines of Earth centered direction cosine Q in the earth-centered-inertial system. The spin vector

The Spin vector S is equal to the cross product of the vectors P and Q and is also equal to the cross procduct of P with the cross 
product of the vectors v and P all divided by the absolute value of the cross product of the vectors v and P

equal to the values of Y sub p times Z sub q minus Z sub p times Y sub q, Z sub p times X sub q minus X sub p times Z sub q and X 
sub p times Y sub q minus Y sub p times X sub q

equals the vector composed of X sub spin, Y sub spin and Z sub spin all in the earth-centered-inertial coordinate 
system(I-6)

where Xspin, Yspin, and Zspin are the direction cosines of Spin direction cosine 
S in the earth-centered-inertial system.

In Figure I.2-3, the scanner direction Scanning direction vector d is measured in the PQS system, that is, the nominal scanning system. The unit vector Scanning direction vector 
d is really Vector P rotated about the vector S rotated an angle sigma rotated about spin direction vector S by the right-handed scan angle σ as shown in Figure I.2-3. It should be noted that the scan direction of the AVHRR instrument is right-handed and opposite that of the TOVS instruments. Each of the TOVS instruments scan in the same direction, sun to anti-sun, and its scan is a left-handed rotation about the scan axis.

To transfer any vector, Vector A , from the PQS coordinate system to the inertial system,

Conversion of Vector A to the PQS coordinate system using the earth centered, scanning direction and spin vectors as matrix 
elements(I-7)

Case 1: It there are no instrument mounting errors, the scan is measured in the nominal scanning coordinate system. The unit vector Scanning direction vector d (along the direction toward the scanned spot) in the PQS system can be written as:


The vector d sub PQS equal the three by three matrix with elements cos sigma, -sin sigma, zero, sin sigma, cos sigma, zero, zero, 
zero and one multipled times the one by three martix zero, zero and one(I-8)

Using the coordinate transformation given by Eq. I-7 and employing Eqs I-3, I-5 and I-6, the vector can be written in the inertial system as:

d equal to the product of three matrices. The first three by three matix has the elements X sub p, X sub q and X sub spin, Y sub p, 
y sub q, Y sub spin, Z sub p, Z sub q and Z sub spin.  The second three by three matrix has the elements cos sigma, -sin sigma zero, sin sigma, cos sigma,zero, zero, zero 
and one. The third matris is a one by three with values of zero, zero and one.(I-9)

Case 2: If there are instrument mounting errors, expressed as nonzero right-handed roll(R), pitch(P) and yaw(Y), then the unit vector in the PQS system in equation I-8 becomes:

The vector d sub PQS is equal toproduct of  the yaw rotation matrix B times the pitch rotation matrix C and the roll rotation 
matrix E times the one by three matrix with elements values of one, zero and zero(I-10)

where yaw axis rotation matrix B is the rotation matrix about the yaw axis, nominally Xns, for an angle (-Y) (to undo the yaw)

The yaw axis rotation matrix B is defined as a three by three matrix with elements of one, zero,zero,zero,cos Y, minus sin Y, 
zero, sin Y and cos Y .gif(I-11)

Pitch rotation matrix C is the rotation matrix about the pitch axis, nominally Yns, for an angle (-P)(to undo the pitch)

The pitch axis rotation matrix C is defined as a three by three matrix with elements of cos P, zero, sinP, zero, one, zero, minus sin P, 
zero and cos P(I-12)

Roll rotation matrix D is the rotation matrix about the roll axis, nominally Zns, for an angle -(σ + R) ( to undo both the roll and the scan)

The roll axis rotation matrix D is defined as a three by three matrix with elements of cos of sigma plus R, minus sin of sigma plus R, 
zero, sin of sigma plus R, cos of sigma plus R, zero, zero, zero, one(I-13)

and

The vector d is defined as the three by three matrix with elements X sub p, X sub q, X sub spin,Y sub p, Y sub q, Y sub spin, Z sub p, Z 
sub q and Z sub spin multiplied by the D sub PQS vector(I-14)

Since The vector P sub spot is equal to the Vector P sub sat plus Range times the result of the rotation matrices and the point L lies on the surface of the Earth, it satisfies the equation:

X sub spot squared divided by r sub e squared plus Y sub spot squared divided by r sub e squared plus Z sub spot squared divided by r 
sub p squared equals one(I-15)

or

the quantity X sat plus R times d sub x squared divided by r sub e squared plus Y sat plus R times d sub y squared divided by r sub e 
squared plus Z sat plus R times d sub z squared divided by r sub p squared equals one(I-16)

where re is the equatorial radius and rp is the polar radius of the Earth (The WGS72 values,re= 6378.135 km and rp = 6356.75052 km, are used at NOAA for NOAA satellites). Expanding

X sub sat squared divided by r sub e squared plus the quantity 2 X sub sat 
times R time d sub x divided by r sub e squared plus R squared times d sub x squared divided by r sub e squared plus Y sub sat squared divided by r sub e squared plus the quantity 2 Y 
sub sat times R time d sub y divided by r sub e squared plus R squared times d sub y squared divided by r sub e squared plus Z sub sat squared divided by r sub p squared plus the 
quantity 2 Z sub sat times R time d sub z divided by r sub p squared plus R squared times d sub z squared divided by r sub p squared equal one(I-17)

or, on simplification,

AR2 + BR + C = 0                                                                                                      (I-18)                                                                                     

where

A equals d sub x squared divided by r sub e squared plus d sub y squared divided by r sub e squared plus d sub z squared divided by r 
sub p squared.(I-19) B is equal to two times the quanity X sub sat times d sub x divided by R sub e squared plus Y sub sat times d sub y divided by r 
sub  e squared plus Z sub sat times D sub z divided by r sub p squared.(I-20) C is equal to X sub sat squared divided by R sub e squared plus Y sub sat squared divided by r sub e squared plus Z sub sat squared 
divided by r sub p squared minus 1.(I-21)

Solving for R in equation I-18,

R is equal to minus B plus or minus the square root of B squared minus four times A times C divided by two A (I-22)

Equation I-18 is a quadratic in R and, if the scan ray does in fact intersect the surface of the Earth (i.e., real and positive roots), B above should always be negative. Since A is always positive and C is positive (whenever the satellite is above the surface of the Earth), then the radical in Eq I-22 must be less than -B. Therefore, in the case of two different real positive solutions for R, the smaller one, closer to the satellite, is visible to the satellite and the larger one, the point away from the satellite, is on the opposite side of the Earth. As such, the smallest of the two solutions should be taken as the distance of the satellite from the scan spot.

In Equation I-1,

The marix composed of the spot elements is equal to the matrix composed of the sat elements plus the range times the distance matrix (I-23)

The earth-centered-inertial and the earth-centered-fixed coordinate systems.

everything on the right hand side is known, so the coordinates of the scan point on the Earth in the inertial coordinate system can be calculated. All that remains is to rotate the Earth using time t obtained from the data sample to calculate a point fixed to the spinning Earth.

The Earth Centered Fixed coordinate system (ECF) rotates with the Earth. It has its center at the center of mass of the Earth with the following defined axes: (See Figure I.2-4)

xECF = the axis from the center of the Earth through Greenwich meridian at the equator

yECF = toward 90 degrees East longitude

zECF = points north along the spin axis of the Earth

Next, we calculate the rotation of the ECF system (rotating with the Earth) with respect to the inertially fixed (I) system. The angle G(t), is the rotation of the Greenwich meridian relative to the inertial x-axis. As a function of time,

G of t equals G sub zero plus the rate G sub one increases times the day of year plus G sub two 
( the rotation rate of the Earth) times the fraction of the day of interest(I-24)

where

G(t) = Hour angle of Greenwich (that is, the eastward angle from the direction of the vernal equinox to the direction of (0 degrees N, 0 degrees E) measured at the earth's center) at time t (radians)

G0 = Hour angle of Greenwich at the beginning of the year of interest (radians)

Increase in the hour angle of Greenwich per day = Increase in the hour angle of Greenwich per day ( =0.0172027912 radians/day)

t1 = Day of year for time of interest, t

Rotational rate of the Earth = Rotational rate of the Earth ( =6.300388098 radians/day)

t2 = Fraction of a day for time of interest, t


Values of G0 can be computed or found in the American Ephemeris and Nautical Almanac for the current year. (These should be updated as appropriate to account for leap seconds.)

In order to transform inertial coordinates to geocentric Earth Centered Fixed coordinates, the following equations are used:

XECF = XIcos(G(t)) + Yisin(G(t))                                                                    (I-25)                                                                                                      

YECF = YI cos(G(t)) - Xisin(G(t))                                                                    (I-26)                                                                                                      

          ZECF = ZI                                                                                                      (I-27)                                                                                                     


When the "rotating" coordinates are found, the geocentric latitude, φgc, and longitude, θ, can then be calculated according to equations I-28 and I-29.

Phi sub gc equals arctan of Z sub ECF divided by the square root of X sub ECF squared plus Y sub ECF squared(I-28)
theta equals the arctan of Y sub ECF divided by X sub ECF(I-29)


Geocentric latitude, φ is the angle between the equatorial plane and a line joining the point, L (the scan spot), on the Earth's surface to the center of mass of the Earth. This is in contrast to the geodetic latitude, φgd, which is the angle between the normal at L and the plane of the equator. The longitude, θ, is the angle between two meridian planes both containing the earth's axis of rotation; one of the planes contains L, and the other contains the Greenwich meridian.


Values of latitude given on standard maps are usually ‘geodetic’ latitude. The geodetic latitude, φgd, of a point on the earth ellipsoid (that is, at Mean Sea Level) has the following value:

Phi sub gd equals arctan of the quantity r sub e squared times Z sub ECF divided by r sub p squared times the square root of X sub ECF squared plus Y sub ECF squared (I-30)


(See I.4 "Conversion between Geodetic and Geocentric Latitude")

Amended May 8, 2006


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