Theory of GNSS Radio Occultation - GNSS Remote Sensing: Theory, Methods and Applications - page 104

Fig. 5.3 Araypath( thick
line ) in a medium with
spherical symmetry that
satisfies the Bouguer's
formula

where is the angle between the position vector r and the tangent of the ray path
(see Fig. 5.3 ), and the constant a in a spherically symmetric atmosphere is called
impact parameter and is also known as Bouguer's rule, which represents Snell's law
in a spherically symmetric medium.

5.2.4 Bending Angle and Refractive Index

The accumulated change in the ray path direction along a ray path is defined as
the bending angle. According to Eq. ( 5.3 ), the rate of change in ray path tangential
direction is given as

r n s

d s
ds

r ?

s n ;

1
n

dn
ds

1
n

D D (5.8)

Thus, the bending is only due to the refractive index gradient that is orthogonal
to the ray path tangent direction s , i.e., the projection of r n into the plane
perpendicular to the ray direction s (i.e., r ? s n). Now we can define a local
coordinate system where x is orthogonal to r and lies in the plane defined by s
and r , and y is in the direction orthogonal to the r - x plane (Kursinski et al. 2000 ).
The bending angle increment along the ray path can be written as

ˇ ˇ ˇ

d s ˇ ˇ ˇ
ˇ ˇ ˇ s ˇ ˇ ˇ

" @n
@r

2 # 1=2

cos 2

@n
@y

ds
n

@n
@x

d˛ D D sin C C : (5.9)

The largest gradients of refractivity are generally found in the lower level of the
atmosphere near the tangent point ( 90 ı , cos 0) along a ray path. Since the
magnitude of horizontal gradient is generally much smaller than those of vertical
gradients in the Earth's atmosphere, the bending of a ray path is largely caused by
the refractivity gradient in radial direction. But it is worth noting that the greatest
horizontal gradient contribution will come from the gradient in y direction (i.e.,
perpendicular to the ray tangent direction).