the variance information.
It follows for the linearized model as introduced with Eq. (5.15), that the estimated unknowns are corrections for the B-spline coefficients collected in the vector x = ∆Dd and l is replaced by the reduced observation vector L. The consideration of different observation techniques in a combined equation system by means of normal equation stacking is thus defined with
* . ,
G
X
g=1
1
σg2ATgPgAg+ 1 σ2µPµ+
/
| {z }
-NEQ
∆dD=* . ,
G
X
g=1
1
σg2ATgPgLg+ 1 σ2µPµµ+
/
-. (5.51)
Substituting the prior information term σ12
µPµ by the extended weight matrixW with block diagonal structure
W = Σ−1µ =diag* ,
Pκ1 σ2κ1,Pκ2
σ2κ2, . . . , PκR σ2κH+
-(5.52) yields the solution for the B-spline coefficient corrections
∆dD= * . ,
G
X
g=1
1
σ2gATgPgAg+W+ /
-−1
* . ,
G
X
g=1
1
σ2gATgPgLg+W M+ /
-. (5.53)
In this notation, key parameter specific variance factors and weight matrices are contained inW. It follows from the linearization, that Eq. (5.53) is solved iteratively. Therefore, µ is replaced by the iteratively adapted vector M which stabilizes the system and counteracts a possible ill-conditioning of the normal equations in the presence of data gaps.
Due to the inhomogeneous data distribution provided by almost every observation technique, the pres-ence of large data gaps cannot be excluded. Or in other words, the selected uniform B-splines may not be able to represent the data distribution. As a consequence, the Ng×UJacobi matricesAgmay have a rank deficiency ofU − rg(Ng) with rg(Ng) < U and the corresponding sub normal equation ma-trixNEQg = σ12
g
ATgPgAg will be singular. However, under consideration that the weight matrices of the observationsPgare positive-definite, eachNEQg is symmetric and at least positive-semidefinite.
Thus, the eigenvalues ofNEQgare real and≥ 0 where it follows that the diagonal elements ofNEQg are non-negative. The positive-definite prior information matrixW is finally introduced to regularize the system leading to a stacked NEQ system that is of full rank, positive-definite and regular, i.e., invertible.
5.6 Adjustment sequence
The adjustment comprises a nested iteration sequence accounting for (1) the linearization of the model and (2) the estimation of variance factors for the observation techniques and the prior information. A schematic flowchart of the fundamental processing steps is provided by Fig. 5.5.
The data sources for the prior information and initial coefficients may coincide, i.e., both can also be obtained from the same background model. Based on these data, the model is linearized and the normal equations under consideration of weighted observation groups are established to solve for corrections of the B-spline series coefficients. Input data originate from the actual measurements, the background model (for the derivation of prior information) and optionally from additional external sources (for the provision of initial B-spline series coefficients).
94 Chapter 5. Modeling the ionosphere
Figure 5.5: Flowchart of the iterative solving procedure with nested variance component estimation.
As the observation models are based on the non-linear Chapman function, the linearization by means of a Taylor series approximation is performed. Estimations for the target parameters follow from an iterative computation where the initial coefficients are iteratively improved. According to Fig. 5.5, the VCE is embedded into the solving procedure as a second iteration that is carried out for each linearization step. The sequences of both the linearization and VCE iterations will be described in the following.
5.6.1 Outer iteration: Linearization
The outer iteration loop, accounting for the linearization, will henceforth be denoted by the subscript Lit whereas the inner loop with respect to the VCE will be identified by the subscript Vit.
Initial iteration
For the0thiteration, denoted as Lit=0, the prior information vector µis set to
µ =
d0,κ1 d0,κ2 ...
d0,κH
(5.54)
and
MLit=0= µ−dLit=0 (5.55)
wheredLit=0is a vector containing starting values for the coefficients. The closer the entries ofdLit=0 to the final values, the faster the convergence of the iterative solver. Such as for the prior information
5.6. Adjustment sequence 95 µ, initial coefficients can be derived from a given background model according to Eq. (5.46). In case thatµ contains the initial coefficientsdLit=0, it follows that
MLit=0= µ−dLit=0 =
dLit=0=µ 0. (5.56)
→Run VCE (Section 5.6.2)→The0th iterationyields the solution for∆DdLit=0. Iteration 1
The1st iterationstarts with the definition of new initial coefficients
dLit=1= dLit=0+∆DdLit=0 (5.57)
and
MLit=1= µ−
dLit=0+∆dDLit=0
dLit=0==µ −∆dDLit=0 (5.58)
where only those coefficients are updated, which fall into the support region of the given measure-ments. Coefficients which are located in data gaps remain untouched.
→Run VCE (Section 5.6.2)→The1st iterationyields the solution for∆DdLit=1. Iteration 2
In the2nd iteration, the prior information is derived as
dLit=2= dLit=0+∆DdLit=0+∆dDLit=1 =dLit=1+∆dDLit=1 (5.59) and
MLit=2= µ−
dLit=0+∆dDLit=0+∆dDLit=1
dLit==0=µ −
∆dDLit=0+∆dDLit=1
. (5.60)
→Run VCE (Section 5.6.2)→The2nd iterationyields the solution for∆dDLit=2. Iteration P-1
Assuming the convergence with a sufficient accuracy of the solution is reached after thePthiteration, the initial scaling coefficient vector for the antecedent(P−1)th iterationis obtained from
dLit=P−1 =dLit=0+
P−2X
p=0
∆DdLit=p (5.61)
with the adapted prior information vector MLit=P−1= µ−*
. ,
dLit=0+
P−2
X
p=0
∆dDLit=p+ /
-dLit=0==µ −* . ,
P−2
X
p=0
∆dDLit=p+ /
-. (5.62)
→Run VCE (Section 5.6.2)→TheP−1th iterationyields the solution for∆DdLit=P−1. Iteration P
The final solution for the B-spline series coefficients in thePth iterationis then obtained from dLit=P =dLit=0+
P−1
X
p=0
∆DdLit=p. (5.63)
96 Chapter 5. Modeling the ionosphere
5.6.2 Inner iteration: Variance component estimation
The determination of the variance components for observation groupsσg2and prior informationσ2κh is realized as an iterative estimation procedure starting with approximate values until a point of con-vergence is reached. The VCE procedure considered, e.g., by P-I: Limberger et al. (2013) and P-II:
Limberger et al. (2014), is based on the approach published by Koch and Kusche (2002). It is embed-ded into the iterations of the linearization as shown in Fig. 5.5. A similar realization can be found in Heiker (2012). In each VCE iteration step, variance components for the observation techniques and the prior information are calculated with
Dσ2g= DeTgPgDeg rg
and Dσ2κh = DeTκ
hPκhDeκh
rκh . (5.64)
The vectors of residualsDeg andDeκh are determined with
Deg = Ag∆dD−Lg and Deκh = ∆dDκh− Mκh (5.65) whereas the partial redundanciesrg,rκh can be obtained as
rg = Ng−Tr* . . . ,
1
σg2ATgPgAg(NEQ)−1
(U×U)
+ / / /
-and rκh =Uκh −Tr
* . . . . ,
1
σ2κhPκh(NEQκh)−1
(Uκh×Uκ
h)
+ / / / /
-. (5.66)
NEQis the matrix of normal equations as included in Eq. (5.51) and is defined with NEQ
(U×U) =
G
X
g=1
1
σg2ATgPgA+ 1
σ2µPµ. (5.67)
Furthermore, NEQκh indicates the key parameter specific sub-matrix on the diagonal ofNEQ. Ac-cording to Fig. 5.5, the VCE iteration yields the following steps where it is important to note, that the vectorsLgandM change only in the "Linearization"-step and not during the VCE iterations. The calculation sequence for the VCE consists of the following steps.
Initial iteration
1. Deployment of the system defined by Eq. (5.53) under consideration of a priori variance com-ponentsσ2g,Vit=0,σ2κ
h,Vit=0and initial coefficientsdLit=0 2. Estimation of corrections∆DdVit=0(Eq. (5.53))
3. Computation of residualsDeg,Deκh with∆DdVit=0(Eq. (5.65)) 4. Computation of partial redundanciesrg,rκh withσ2g,Vit=0σ2κ
h,Vit=0(Eq. (5.66)) 5. Calculation of updated variance componentsDσg,Vit=02 ,Dσ2κ
h,Vit=0(Eq. (5.64)) Iteration 1
1. Deployment of the system defined by Eq. (5.53) under consideration of the variance components σ2g,Vit=1 =Dσ2g,Vit=0,σ2κ
h,Vit=1 =Dσ2κ
h,Vit=0and initial coefficientsdLit=0 2. Estimation of corrections∆DdVit=1(Eq. (5.53))
3. Computation of residualsDeg,Deκh with∆DdVit=1(Eq. (5.65)) 4. Computation of partial redundanciesrg,rκh withσ2g,Vit=1σ2κ
h,Vit=1(Eq. (5.66)) 5. Calculation of updated variance componentsDσg,Vit2 =1,Dσ2κ
h,Vit=1(Eq. (5.64))
5.7. Parameter variances and correlations 97