Bernese GPS Software Version 5.0 - Bernese GNSS Software

11. Troposphere Modeling and Estimation
β
Z'
β
Z
z
∆rt (α,z)
Figure 11.3: Tilting of the tropospheric zenith by the angle β.
One way to represent azimuthal asymmetries is a tilting of the zenith the mapping function
is referred to (see Figure 11.3). The troposphere gradient parameters then comply with
the fact that the direction to the so-called tropospheric zenith (i.e., the direction with
minimal tropospheric delay) and the corresponding tropospheric zenith distance ˜z might
not be identical to the geometrical (or ellipsoidal) zenith distance z . Having introduced
the tropospheric zenith angle as a parameter of the mapping functions, the tropospheric
delay from Eqn. (11.16) would be given by
∆̺ i k(t,z) = ∆̺apr,k(˜z i k) + ∆̺k(t)f(˜z i k) . (11.17)
However, due to the fact that we usually do not have any a priori information on the
tropospheric zenith, the geometrical zenith is used in the a priori part of the Equation
(11.17):
∆̺ i k (t,z) = ∆̺apr,k(z i k ) + ∆̺k(t)f(˜z i k ) . (11.18)
Assuming a small angle β between the tropospheric and geometrical zenith, the two zenith
angles are related to each other by the equation
˜z i k = z i k + β = z i k + xk cos(A i k) + yk sin(A i k) ,
where A i k is the azimuth of the direction station–satellite and xk,yk are two stationdependent
parameters. Using this equation and approximating linearly, the time dependent
part of Eqn. (11.18) may be written as
∆̺k(t,A,z) f(˜z i k) = ∆̺k(t) f(z i k + xk cos(A i k) + yk sin(A i k))
Introducing the notation
= ∆̺k(t) f(z i ∂f
k ) + ∆̺k(t)
∂z xk cos(A i ∂f
k ) + ∆̺k(t)
∂z yk sin(A i k ) .
∆ h ̺k(t) = ∆̺k(t) zenith delay parameter,
∆ n ̺k(t) = ∆̺k(t) xk gradient parameter in north-south direction, and
∆ e ̺k(t) = ∆̺k(t) yk gradient parameter in east-west direction,
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