Theory of GNSS Radio Occultation - GNSS Remote Sensing: Theory, Methods and Applications - page 105

The variation of n along a limb path in the Earth's atmosphere is dominated by
the vertical density gradient so that, to the first order, we can assume the gradient of
n is directed radially and the local refractive index field is spherically symmetrical,
i.e., n D n ( r ). Combining Eq. ( 5.9 ) with the Bouguer's rule in Eq. ( 5.7 ), the bending
angle increment along the ray path can be simplified as

d ln.n/
dr

a
p n 2 r 2 d˛ D

a 2 dr:

(5.10)

Since the refractivity generally decreases at higher altitudes, to allow the bending
angle to be positive values, a negative sign is added. The total bending angle thus
becomes

˛.a/ D 2 Z 1
r t

d˛ D 2a Z 1
r t

d ln.n/
dr

dr
p n 2 r 2 (5.11)

a 2

where r is distance from the center of curvature of a ray path and the integral is over
the portion of the atmosphere above r t , the radius of the tangent point (i.e., the point
on the ray path that is closest to the Earth's center). By introducing the integration
variable x D nr ,Eq.( 5.11 ) can be modified as
˛.a/ D 2a Z 1
a

d ln.n/
dx

dx
p x 2 (5.12)

a 2

Equation ( 5.12 ) provides the forward calculation of bending angle ˛ given
the refractive index profile n ( r ). By inverting the equation through the Abelian
transformation, the n ( r ) can be expressed as a function of ˛ and a (Fjeldbo et al.
1971 ):

n.r/ D exp 1

Z 1 :

˛ .x/dx
p x 2 (5.13)

a 2

a

Given impact parameter a and the refractive index n , the radius r at each tangent
point can be derived according to Bouguer's formula:

a
n.r/ :

r D (5.14)

Note that Eq. ( 5.13 ) embeds the assumption of local spherically symmetric
atmosphere, i.e., the refractive index only varies along radius direction. However,
the ellipsoidal shape of the Earth (with an equatorial radius roughly 20 km larger
than its polar radius) and horizontal gradients in atmospheric structure produce non-
spherical symmetry in the refractive index field. Moreover, the ray paths for a given
occultation do not necessarily scan the atmosphere vertically nor are they coplanar.
Therefore, measurement of ˛( a ) will be affected by the tangential refractivity
gradients and occultation geometry, and Eq. ( 5.13 ) could introduce systematic errors