18.2.2006
A METHOD FOR MEASURING ATMOSPHERIC WATER VAPOUR CONTENT USING
PHASE MEASUREMENTS FROM FIXED GNSS RECEIVERS
Antti Lange Ph.D.
Tilanhoitajankaari 18 C 39
FI-00790 Helsinki
Abstract
It is assumed that the satellite orbits of Global Navigation Satellite Systems (GNSS) are provided
by a global or local computing centre and a fast processor for computing Kalman Filters (FKF) is
locally available. The carrier-phase measurements of GNSS receiver(s) are used for estimating
water vapour content of the atmosphere.
Description of the Method
The Observation Equation is obtained from phase measurements φi,t of a receiver as follows:
yi,t = φi,t - ρi,t = γi,t + τ t + g i,t c t + e i,t
( i =1,2,…,n and t=1,…,T)
(1)
where
i = index of signals
t = index of epoch times
n = number of signals
T = number of epochs in a moving sample
φ = measured total phase of the reconstructed carrier signal i at epoch t
ρ = distance of the satellite for signal i from the receiver at epoch t
γ = clock error of the satellite for signal i at epoch t
g = vector of the propagation path for signal i at epoch t
c = water vapour content
e = random measurement error.
There are several System Equations as follows:
γ i,t = Ai γi,t-1 + η i,t
τ t = B τ t-1 + ζ t
c t = C c t-1 + ξ t
where the state transition matrices A, B and C and the random walk components are η, ζ and ξ,
respectively.
The contents of atmospheric water vapor for the sampled space and time volume can now be
estimated with an accuracy and resolution that depends on overdetermination of the Augmented
Model:
The semi-analytical FKF formula:
K
K
k 0
k 0
ĉ t =   Gt,k'Rt,kGt,k-1  Gt,k'Rt,kyt,k
where for t = 1, 2,… :
ĉ t = estimated information at time t
for k = 1, 2,…, K:
b̂ t,k = estimated information on the kth position of a balloon
Rt,k = I - Xt,k (Xt,k' Xt,k) -1Xt,k'
and, for k = 0:
Gt,0 = I
Rt,0 = Cov( ĉ t-1 - ct-1) + Cov(at,c)-1
yt,0 = ĉ t-1
ĉ 0 = estimated information on calibration at time 0
c0 = vector of initial calibration.
It is the index value k = 0 that makes Formula (12) the Fast Kalman Filtering (FKF)
formula for the statistical calibration of sounding systems. Matrices G t,0 and Rt,0
represent the vehicle that conveys the following calibration information: ct = ct-1 + at,c;
from a previous sounding t-1 to the Minimum Least Squares Estimation (MLSE) of
the present sounding t. For t = 1, the estimate of c0 is given by ĉ 0. This is a statistical
regularization of the Measurement Equation system (1) that may otherwise be
singular.