International Journal of Geosciences, 2012, 3, 14-20
http://dx.doi.org/10.4236/ijg.2012.31002 Published Online February 2012 (http://www.SciRP.org/journal/ijg)
A New Regularized Solution to Ill-Posed Problem in
Coordinate Tr ansformation
Xuming Ge1, Jicang Wu1,2
1Department of Surveying and Geo-Informatics, Tongji University, Shanghai, China
2Key Laboratory of Advanced Surveying Engineering of State Bureau of Surveying and Mapping, Shanghai, China
Received July 23, 2011; revised September 21, 2011; accepted November 15, 2011
ABSTRACT
Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with
the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very
poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition
number of the coefficient matrix A exists in the linear system of equations
x
b

A. Regularization is required to reduce
the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution.
Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transforma-
tion problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship
among different regularization approaches. The results indicate that the proposed new method can obtain a more rea-
sonable resolution with higher precision and/or robustness.
Keywords: Bursa; Coordinate Transformation; Ill-Posed; Methodology
1. Introduction
Coordinate transformation plays a very importance role
in the numerical treatment of global positioning system.
For transforming GPS coordinates from WGS-84 coordi-
nate system to a local coordinate system, the Bursa mo-
del is generally used to solve transformation parameters,
including three translation parameters, three rotation pa-
rameters and one scale parameter. From a theoretical point
of view, a great number of algorithms have been devel-
oped to solve these problems. Early publications such as
Grafarend et al. [1], Vanicek and Steeves [2], Vanicek et
al. [3], Yang [4], and Grafarend and Awange [5] have
given some detail algorithms of coordinate transformation.
In general, their algorithms to solve transformation pa-
rameters are all based on the classical least squares crite-
rion. Recently, better methods are available in literature,
e.g. hard or soft fixing of certain transformation parame-
ters (e.g. rotation around some coordinate axes are strongly
correlated with translations), reduction of coordinates to
the centre of “gravity” etc., however, ill-posed problems
were rarely considered in those methods. Actually, ill-
posed problems often impact this kind of data processing.
In the coordinate transformation, the Bursa model is
more suitable for global data, so global distributed data is
normally required to solve coordinate transformation pa-
rameters. However, an engineering GPS control network
covers only hundreds of square kilometers, or even
smaller. In this case, the translation and rotation parame-
ters are highly correlated and the mathematical model is
thus ill-posed. Generally, parameters obtained by tradi-
tional algorithms from the ill-posed model will have poor
precision and robustness.
As is well known, regularization is a powerful tool to
solve ill-posed problems, which have been widely ap-
plied to solve inverse ill-posed geodetic problems and sig-
nal processing. Tikhonov proposed a well-known regu-
larization method to ill-posed models [6,7]. Golub [8] pro-
posed a singular value decomposition (SVD) method to
least squares solutions and Hansen analyzed the truncated
SVD as a method for regularization [9] in mathematics.
Xu and Rummel [10] advanced a multiple parameters
regularization method to solve ill-posed problems in ge-
odesy [11]. These investigations are almost based on pure
mathematics and not considered the characteristics of
practical surveying applications.
In this paper, we will propose a new approach to solve
ill-posed problems in the coordinate transformation. The
process starts with discussion of early methods for solv-
ing ill-posed problems in mathematics, and analysis of
the characteristics of ill-posed coordinate transformation
problems. Taking account of the characteristics of coor-
dinate transformation in GPS applications, a new algo-
C
opyright © 2012 SciRes. IJG

X. M. GE ET AL. 15
rithm is proposed. On other hand, based on the new algo-
rithm we formulate a new approach to choose regulariza-
tion parameters. In the new algorithm, information borne
by small and large singular values has been kept by dif-
ferent methods.
2. Main Equations and Notations
We introduce main equations and notations used through-
out this paper. Matrices are represented by the uppercase
English alphabet and I denotes identity matrix. Scalars
are represented by lowercase Greek letters or English
alphabet with double subscripts. Superscript T denotes
the transpose of a matrix. Let 2
A
denotes the 2-norm
of matrix A. Let

A
 denotes the range of the matrix
A. In this paper, we deal with the linear finite-dimension
equations.
,,
mn md
,,
x
b b

 AA mnmd 
i
(1)
When the rotations are small, the mathematical model
of the Bursa model can recast as
000 1
000 1
00 0 1
y
x
T
y
z
x
R
z
z
yx
xx
yy
zz





 

 
 
 
 
  
 










 

T


(2)
where xyz

and

000
,
T
Txyz R

denote the translate parameters, rotation parameters and
scale parameter, respectively.
We also denote the SVD of A by
1
n
TT
iii
i
uv 



UVA (3)
with


12
,,Uuu, ,
ni
uum 
T
n
UU I ,


12
,,Vvv, ,
ni
vv n 
T
n
VV I

,nn
n



0
n
, ,
and
12
diag, ,   ,
12

 
i
 .
The numbers

are called the singular values of A,
while the vectors i and i are the left and right sin-
gular vectors of A, respectively.
u v
exact noise
,,,
mn md
3. Methodologies for Ill-Posed Problem
The ill-posed problems are with a very large condition
number of the coefficient matrix A in a linear system as
Equation (1), and most of discussion about solutions of
ill-condition-ed matrices require knowledge of the SVD
of the matrix A [12]. In particular, the condition number
of A is defined as the ratio between the largest and the
smallest singular values of A.
The numerical treatment of equations with an ill-condi-
tioned coefficient matrix depends on the type of ill-con-
dition of A. There are two classes of ill-posed problems.
The first is rank-deficient problems in which the matrix
A has a cluster of small singular values, and there is a
well-determined gap between large and small singular
values. The second is the discrete ill-posed problem that
all of the singular values of A, on the average, decay
gradually to zero, that means there is no gap in the sin-
gular value spectrum. For some more details on ill-posed
problems, the reader may refer to Hansen [13] and Tar-
antola [14].
Considering an ill-posed linear system as Equation (1),
with nonsingular matrix A and the right-hand side b pol-
luted by white noise, thus Equation (1) can be rewritten
in the form
x
bb

 AbA bb (4)
where
exact noise
22
bb
1
exact exact
(5)
We wish to approximate
x
Ab


bb

1
TT
LS
(6)
However, exact cannot be directly divided from a
observation vector , we get the direct least squares
solution
AA

Ab
x
(7)
It is well known that the direct solution is completely
dominated by noise, when the coefficient matrix A is
ill-posed. Based on the SVD of A, the direct solution can
recast
naive
1
T
n
i
i
ii
uv


b
x
(8)
exact noise
11
TT
nn
ii
ii
ii
ii
ub ub
vv







(9)
Considering Equation (8), to get an applicable directly
solution, it is necessary to make sure that the following
assumption which is called the Discrete Picard Condition
is satisfied: The exact SVD coefficient T ub
i decay fas-
ter than the i

. More details and analysis can also be
fund in Hansen [13,15]. When the coefficient matrix A is
ill-posed, the solution naive
x
is with poor robustness and
not applicable.
Next considering Equation (9), the first sum in Equa-
tion (9) must converge to exact
x
. It means the numerators
in the first sum in Equation (9) decay faster than (or, at

X. M. GE ET AL. 16
least, as fast as) the singular values in denominators, with
growing i. On other hand, noise represents white noise
in second sum in Equation (9). It means noise Fourier
coefficients
b
noise cannot decay faster than (or, at least,
after some point) the singular values, with growing i in
ill-posed problems. That means noise will dominate the
solution naive
T
i
ub
b
x
after some point. Consequently, ill-posed
problems with perturbations in observation vector may
magnify the noise information by the corresponding sin-
gular value which is very small. The magnified noise
completely cover the useful information,
1
naive
1
exact , and
noise
AbAb 
x
thus does not ap-
proximate to exact
x
.
For solving the first class ill-posed problems, for which
there is a well-determined gap between the large and
small singular value of coefficient matrix A, Hansen pro-
posed a well-known approach [9,13,16,]—truncated SVD.
The key idea in this approach is to obtain the truncated
point k, then Equation (8) can recast as
1
T
k
i
ki
ii
ub
x
v



k
(10)
The solution
x
is referred to as the truncated SVD
solution.
For solving the second class ill-posed problems in
which the singular values of matrix A decay gradually to
zero, the above TSVD method cannot applied for solving
this problem. Tikhonov regularization method is a well-
known method for solving this kind of ill-posed problem.
The key idea in Tikhonov’s method is to incorporate a
priori assumptions about the size and smoothness of the
desired solution. Tikhonov regularization in general form
leads to the minimization problem

22
2
22
min
A
xb Lx   (11)
where the regularization parameter λ controls the weight
given to minimization of the regularization term, relative
to the minimization of the residual norm, and L repre-
sents regularization matrix. In this paper, we only con-
cern L = I. Then the Tikhonov solution
x

is given by

1
2TT
x
AA

I Ab 

 (12)
Tikhonov method can be rewritten in SVD form with
filter factor defined as
2
22
i
i
i
f



 (13)
and the regularized solution is
rank( )
1
T
A
i
ii
ii
ub
reg
x
In Equation (14), if the filter factor defined as
fv



1
0
ii
i
ii
f
 (14)








x
(15)
Then the regularized solution is turn to the truncated
SVD solution. Clearly, the key idea in Equation (14) is to
modify the small singular values of matrix A, and Equa-
tion (15) is use to decide from where the singular values
should be modified. Here we propose a new method for
modifying the singular values in a specific method, and
its solution is noted as regk which is called as the
modified singular value (MSV) solution.
For solving ill-posed problems in GPS coordinate trans-
formation by the Buras model, the characteristics of in-
dividual real model should be considered. That is, after
the ith, the true Fourier coefficients exacti cannot
decay faster than singular values, however, there is still
some useful information exist in
T
ub
exacti, one of real
example is presented in Figure 1. In Figure 1, singular
values i
T
ub

is connected with circle dash line, true Fou-
rier coefficients exacti is connected with cross dash
line, and total Fourier coefficients
T
ub
T
ub
i is connected with
red circle solid line. We can see from i = 5, error Fourier
coefficients noise are greater than corresponding sin-
gular values i
T
i
ub

, but the true Fourier coefficients
exacti are still significant. If we use TSVD method,
the true Fourier coefficients from i = 5 to i = 7 will be
discarded. Our new MSV method will try to use these
discarded items for improving the solution.
T
ub
As above, Tikhonov regularization is to modify those
small singular values, so as to absorb discarded items by
TSVD. However, this method also modifies large singu-
lar values, and thus influences the precision of coordinate
transformation.
Our proposed MSV method is similar with Tikhonov
regularization method, but only items with small singular
values in Equation (14) will be modified and items with
large singular values will be kept unchanged, that means
f = 1 in Equation (14).
4. Estimation of Regularization Parameter
The key question in regularization method is how to
choose the regularization parameter, either the Tikhonov
method’s parameter

or the TSVD method’s parame-
ter k. Thus, estimation of regularization parameter plays
a very importance role in solving ill-posed problems.
Perhaps the most convenient graphical tool for deter-
mining of regularization parameter is so-called L-curve
which is a plot of regularization parameter solution (semi)
norm e.g., the 2-(semi)norm 2, for all possible regu-
larization parameters versus the corresponding residual
norm e.g.,
Lx
2
A
xb. The L-curve clearly displays the
compromise between minimization of these two quantities,

X. M. GE ET AL. 17
Figure 1. Compare singular values with Fourier coefficients.
Singular values (circle with dash line), true Fourier coeffi-
cients (asterisks with dash line), error Fourier coefficients
(crosses with dash line), and total Fourier coefficients (red
circle with solid line) for coefficient matrix A and observa-
tion vector . b
which is the main concerns of any regularization method.
For some more details and analysis on L-curve, the
reader may refer to Hansen [17], and also Hanke [18].
In geodetic problems, L-curve method has often in-
duced oversmoothed solutions [19]. Minimizing the trace
of mean squared error (MSE) method is also a powerful
approach to estimate regularization parameter

, Shen
and Li [20] presented this method in GPS numerical
treatment. In this method, minimizing the trace of mean
squared error is required, the criterion as
minTrace( )M (16)
where Trace(.) denotes the trace of the matrix, and M
denotes the mean squared error matrix. From Shen and
Li the second order derivative of Trace (M) satisfies


2
d T
2
race 0
d 

M
 

(17)
Thus the unique solution exists for Equation (16), and
it can be solved by letting the first-order derivative of
Trace(M) to zero.
Another approach to estimate regularization parameter
is through the condition number of coefficient matrix A
[21], in this method a relation between the regularization
parameter and the sensitivity of the regularized solution
is investigated. As in Regińska
max
min




A
A
A
(18)
where () 
A
shows the sensitivity of Equation (4) on
data perturbations. In this method, through analysis of the
optimal , the regularization parameter can be chosen
as


min max
  
A
A
mod ,1, 2, 3 ify iii
(19)
Many authors have discussed estimation of regularized
parameter, and there are also some other methods, like
Generalized Cross-Validation [22]. In fact, each method
for estimating regularization parameter has different ad-
vantages and disadvantages. There is no unique method
applicable to all ill-posed problems. Based on those early
methods, we propose a new method to estimate regulari-
zation parameter in coordinate transformation.
Considering Equations (14) and (15), and the charac-
teristics of coordinate transformation mentioned in the
above, we concern how to modify the small singular
values and from where they should be modified. The
form of coefficient matrix A in coordinate transformation
has been given by Equation (2), and the singular values,
the true Fourier coefficients and the error Fourier coeffi-
cients of an example of matrix A are presented in Figure
1. As in Figure 1, the first three singular values are sig-
nificant larger than zero and their corresponding total
Fourier coefficients, moreover, the true Fourier coeffi-
cients are larger than their corresponding error Fourier
coefficients, i.e., total Fourier coefficients are completely
dominated by true Fourier coefficients, so we decide to
keep these three singular values unmodified
(20)


 
The singular value decays at the fourth, and the last
three singular values are approximate to zero. The fourth
total Fourier coefficient is less than the fourth singular
value, thus ill-posed problems is not significant. How-
ever, Figure 1 shows the fourth error Fourier coefficient
is almost equal to the fourth true Fourier coefficient, thus
perturbations may influence the exact solution, so this
singular value is modified as
opt

mod ,4 ify iii





,
(21)
where


5cm
denote error level and error adjustment co-
efficient, respectively (In our simulation examples,

and choosing 4.53



mod ,max5, 6, 7 ify ii
).
The last three singular values are obviously smaller than
the corresponding total Fourier coefficients. Moreover, the
corresponding error Fourier coefficients are larger than
the corresponding true Fourier coefficients respectively.
Clearly, in this situation, the model is ill-posed, and the
useful information is completely covered by perturba-
tions. For keeping the useful information for coordinate
transformation, the singular values are modified as
(22)




So, we obtain “new” singular values using Equations
(20)-(22) so as to the solution of our MSV method, it can
be rewritten as

X. M. GE ET AL. 18

1mod ,
T
i
i
ify i
ub
rank
reg k
i
x
v


A
 (23)
5. Numerical Experiments
The data used in our experiments is the simulated coor-
dinates of GPS stations distributed in 2000 square kilo-
meters. We use five GPS points as control points, and
apply “true” coordinate transformation parameters to get
corresponding “true” coordinates in the local coordinate
system. From initial investigations, we know that if
points locate in the interior of a network composed by
control points, we can get their corresponding coordi-
nates with acceptable precision in a local coordinate sys-
tem by coordinate transformation parameters even using
the classical least squares method. Moreover, the results
of coordinate transformation are also acceptable when
coordinates of control points have smaller noise in both
coordinate systems. So, the coordinates of points outside
the region of the control network will be used to check
the precision of coordinate transformation by different
methods.
In our experiments, we give five centimeters perturba-
tions to coordinates of control points in both systems.
Firstly, we use the “true” transformation parameters to
get some “true” coordinates outside the control network
in both coordinate systems. Secondly, we transform co-
ordinates of those outside points to the local system with
different coordinate transformation parameters which ob-
tained by different regularization methods, and compare
each of them with their “true” coordinates.
Root mean square error (RMSE) of points by least
squares method is presented in Figure 2. Here, we simu-
lated coordinates of 100 points in exterior area with a
white noise with zero means and standard deviations of 5
Figure 2. RMSE of transformed coordinates solving by least
squares method.
centimeters, and 500 coordinate transformation experi-
ments have been repeated for obtaining mean of RMSE
with statistical significance. Figure 2 shows that the
mean of RMSE is clearly larger than the noise has been
imposed in the coordinates, and the results have large
oscillations.
The results solving by TSVD, Tikhonov regularization
with L-curve, and MSE methods are presented in Fig-
ures 3(a)-(c) respectively. Clearly, the means of RMSE
are smaller than the result in Figure 2, especially the
mean value by MSE is approximate to the error which has
been given before, and also has stronger robustness.
Figure 4 presents the results solving by our MSV
method with the modified “new” singular values. Obvi-
ously, the results solving by MSV method have the
smallest mean of RMSE and stronger robustness. Table
1 summarizes means of RMSE and their corresponding
standard deviations of different methods.
The performance about two of those points in 500 tests
by MSV method are presented in Figure 5. Clearly, The
No. 68 point (red) is more seriously corrupted by noise
than the No. 28 point (black). In order to present some
good properties of our new method, the results of the No.
68 point by MSV, TSVD, and Tikhovon regularization
with L-curve are drawn in Figures 6(a)-(c), respectively.
Table 2 summarizes mean of RMSE and their corre-
sponding standard deviations of No. 68 points through
using different methods. Clearly, when the point has poor
precision, our MSV method can balance the point preci-
sion and robustness well.
6. Conclusions
It is well known that regularization has been successfully
applied to solve ill-posed problems by significantly im-
proving the condition number of ill-condition matrix. A
very large condition number is usually caused by the
small singular values of matrix, so we propose a new me-
Table 1. Mean of RMSE and standard deviations.
Mean of RMSE (m) Standard Deviations
LS 0.1930 0.04989
TSVD 0.1408 0.04648
L-curve 0.1347 0.03743
MES 0.1351 0.02858
MSV 0.1116 0.03384
Table 2. Mean of RMSE and standard deviations of No. 68
point.
Mean of RMSE (m) Standard Deviations
MSV 0.1561 0.08363
TSVD 0.2076 0.1034
L-curve 0.1634 0.0910

X. M. GE ET AL. 19
(a)
(b)
(c)
Figure 3. RMSE of transformed coordinates solving by TSVD
(a); Tikhonov regularization with L-curve (b); and MSE (c)
methods.
Figure 4. RMSE of transformed coor dinates solving by MSV
with “new” singular values by new method.
Figure 5. Point error of the No. 28 (black) and No. 68 (red)
solving by new method.
Figure 6. Point error of the No.68 solving by new method
(a); TSVD (b); and Tikhonov regularization with L-curve
(c).

X. M. GE ET AL. 20
thod for solving ill-posed problems in coordinate trans-
formation through modifying small singular values. The
numerical experiments have demons-trated that the new
method can solve these kinds of ill-posed problems, and
gain an applicable precision, moreover, perform stronger
robustness. In practical coordinate transformation prob-
lems, we do not know the true and error Fourier coeffi-
cients, the total Fourier coefficients can be used to mod-
ify the singular values so as to implement the MSV
method.
7. Acknowledgements
This paper is sponsored by the Natural Science Founda-
tion of China (No. 41074019) and International coopera-
tion project of Chinese ministry of Science (No. 2010-
DFB20190).
REFERENCES
[1] E. W. Grafarend, F. Krumm and F. Okeke, “Curvilinear
Geodetic Datum Transformations,” Z Vermessungswesen,
Vol. 120, No. 4, 1995, pp. 334-350.
[2] P. Vanicek and R. R. Steeves, “Transformation of Coor-
dinates between Two Horizontal Geodetic Datum,” Jour-
nal of Geodesy, Vol. 70, No. 11, 1996, pp. 740-745.
[3] P. Vanicek, P. Novak and R. Craymerm, “On the Correct
Determination of Transformation Parameters of Horizon-
tal Geodetic Datum,” Geomatica, Vol. 56, No. 4, 2002, pp.
329-340.
[4] Y. Yang, “Robust Estimation of Geodetic Datum Trans-
formation,” Journal of Geodesy, Vol. 73, No. 9, 1999, pp.
268-274. doi:10.1007/s001900050243
[5] E. W. Grafarend and L. J. Awange, “Nonlinear Analysis
of the Transformational Datum Transformation,” Journal
of Geodesy, Vol. 77, 2003, pp. 66-76.
doi:10.1007/s00190-002-0299-9
[6] A. N. Tikhonov, “Regularization of Ill-Posed Problems,”
Doklady Akademi Nauk, Vol. 151, No. 1, 1963, pp. 49-
52.
[7] A. N. Tikhonov, “Solution of Incurrectly Formulated Pro-
blems and the Regularization Method,” Doklady Akade-
mi Nauk, Vol. 151, No. 3, 1963, pp. 501-504.
[8] G. H. Golub and C. Reinsch, “Singular Value Decompo-
sition and Least Squares Solutions,” Numerical Mathe-
matics, Vol. 14, 1970, pp. 403-420.
doi:10.1007/BF02163027
[9] P. C. Hansen, “The Truncated SVD as a Method for Re-
gularization,” BIT, Vol. 27, 1987, pp. 534-553.
doi:10.1007/BF01937276
[10] P. L. Xu and R. Rummel, “Generalized Ridge Regression
with Applications in Determination of Potential Fields,”
Manuscripta Geodaetica, Vol. 20, No. 1, 1994, pp. 8-20.
[11] P. L. Xu, Y. Fukuda and Y. Liu, “Multiple Parameter
Regularization: Numerical Solution and Application to
the Determination of Geopotential from Precise Satellite
Orbits,” Journal of Geodesy, Vol. 80, No. 1, 2006, pp.
17-27. doi:10.1007/s00190-006-0025-0
[12] G. H. Golub and C. F. Van Loan, “Matrix Computation,”
3rd Edition, The Johns Hopkins University Press, Balti-
more, 1996.
[13] P. C. Hansen, “Rank-Deficient and Discrete Ill-Posed Pro-
blems,” SIAM, Philadelphia, 1998.
doi:10.1137/1.9780898719697
[14] P. Tarantola, “Inverse Problem Theory,” SIAM, Phila-
delphia, 2005.
[15] P. C. Hansen, “The Discrete Picard Condition for Dis-
crete Ill-Posed Problems,” BIT, Vol. 30, No. 4, 1990, pp.
658-672. doi:10.1007/BF01933214
[16] P. C. Hansen, “Truncated SVD Solutions to Discrete Ill-
Posed Problems with Ill-Determined Numerical Rank,”
Journal on Scientific and Statistical Computing, Vol. 11,
1990, pp. 503-518. doi:10.1137/0911028
[17] P. C. Hansen, “Analysis of Discrete Ill-Posed Problems
by Means of the L-Curve,” SIAM Review, Vol. 34, No. 4,
1992, pp. 561-580. doi:10.1137/1034115
[18] M. Hanke, “Limitations of the L-Curve Method in Ill-
Posed Problems,” BIT, Vol. 36, No. 2, 1996, pp. 287-301.
doi:10.1007/BF01731984
[19] P. L. Xu, “Truncated SVD Methods for Discrete Linear
Ill-Posed Problems,” Geophysical Journal International,
Vol. 135, No. 2, 1998, pp. 505-514.
doi:10.1046/j.1365-246X.1998.00652.x
[20] Y. Z. Shen and B. F. Li, “Regularized Solution to Fast
GPS Ambiguity Resolution,” Journal of Surveying Engi-
neering, Vol. 133, No. 4, 2007, pp. 168-172.
doi:10.1061/(ASCE)0733-9453(2007)133:4(168)
[21] T. Regińska, “Regularization of Discrete Ill-Posed Prob-
lems,” BIT, Vol. 44, No. 3, 2004, pp. 119-133.
[22] G. H. Golub, M. T. Heath and G. Wahba, “Generalized
Cross-Validation as a Method for Choosing a Good Ridge
Parameter,” Technometrics, Vol. 21, No. 2, 1979, pp. 215-
223. doi:10.2307/1268518