# 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 constitute a right handed coordinate system. Let and 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.

(I-1)

or

(I-2)

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 , 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

(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

Define

(I-4)

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.

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

where

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

(I-6)

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

In Figure I.2-3, the scanner direction is measured in the PQS system, that is, the nominal scanning system. The unit vector is really rotated about 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, , from the PQS coordinate system to the inertial system,

(I-7)

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

(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:

(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:

(I-10)

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

(I-11)

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

(I-12)

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

(I-13)

and

(I-14)

Since and the point L lies on the surface of the Earth, it satisfies the equation:

(I-15)

or

(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

(I-17)

or, on simplification,

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

where

(I-19) (I-20) (I-21)

Solving for R in equation I-18,

(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,

(I-23)

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,

(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 ( =0.0172027912 radians/day)

t1 = Day of year for time of interest, t

= 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.

(I-28)
(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:

(I-30)

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

Amended May 8, 2006