On the extension of Helmert transform
The article deals with the method of comparison of the coordinate systems used by two quite different scientific disciplines: stellar astronomy and the geodesy. Geodesic Helmert transform is analysed along with a series of stellar astronomy kinematic models: Kovalsky-Airy, Lindbladt-Oort and Ogorodn...
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irk-123456789-1198022017-06-10T03:03:09Z On the extension of Helmert transform Choliy, V.Ya. The article deals with the method of comparison of the coordinate systems used by two quite different scientific disciplines: stellar astronomy and the geodesy. Geodesic Helmert transform is analysed along with a series of stellar astronomy kinematic models: Kovalsky-Airy, Lindbladt-Oort and Ogorodnikov-Milne. An analogy was built allowing us to propose an extension to the Helmert transform. In the second part of the article, three different approaches to the solution of the correlation problem are compared, and the results of the numerical experiment are presented. 2014 Article On the extension of Helmert transform / V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2014. — Т. 4., вип. 1-2. — С. 15-19. — Бібліогр.: 12 назв. — англ. 2227-1481 DOI: 10.17721/2227-1481.4.15-19 http://dspace.nbuv.gov.ua/handle/123456789/119802 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України |
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The article deals with the method of comparison of the coordinate systems used by two quite different scientific disciplines: stellar astronomy and the geodesy. Geodesic Helmert transform is analysed along with a series of stellar astronomy kinematic models: Kovalsky-Airy, Lindbladt-Oort and Ogorodnikov-Milne. An analogy was built allowing us to propose an extension to the Helmert transform. In the second part of the article, three different approaches to the solution of the correlation problem are compared, and the results of the numerical experiment are presented. |
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Choliy, V.Ya. On the extension of Helmert transform Advances in Astronomy and Space Physics |
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Choliy, V.Ya. |
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On the extension of Helmert transform |
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On the extension of Helmert transform |
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On the extension of Helmert transform |
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On the extension of Helmert transform |
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On the extension of Helmert transform |
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on the extension of helmert transform |
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On the extension of Helmert transform / V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2014. — Т. 4., вип. 1-2. — С. 15-19. — Бібліогр.: 12 назв. — англ. |
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On the extension of Helmert transform
V.Ya.Choliy1,2∗
Advances in Astronomy and Space Physics, 4, 15-19 (2014)
© V.Ya.Choliy, 2014
1Faculty of Physics, Taras Shevchenko National University of Kyiv, Glushkova ave., 4, Kyiv 03127, Ukraine
2Main Astronomical Observatory of the NAS of Ukraine, Akademika Zabolotnoho Str. 27, Kyiv 03680, Ukraine
The article deals with the method of comparison of the coordinate systems used by two quite di�erent scienti�c
disciplines: stellar astronomy and the geodesy. Geodesic Helmert transform is analysed along with a series of
stellar astronomy kinematic models: Kovalsky-Airy, Lindbladt-Oort and Ogorodnikov-Milne. An analogy was built
allowing us to propose an extension to the Helmert transform. In the second part of the article, three di�erent
approaches to the solution of the correlation problem are compared, and the results of the numerical experiment
are presented.
Key words: stellar astronomy, geodesy, mathematical methods
introduction
Let us state several de�nitions �rst. The con-
struction of a coordinate systems for the sky (galax-
ies, quasars) or for the Earth (cities, geodetic mark-
ers, GPS navigators) leads to a catalogue of coordi-
nates of objects. Thus, the word �catalogue� in this
article always refers to a set of object coordinates,
while catalogue objects are referred as points.
An essential method for building catalogues is
through observations and subsequent processing.
There are no possibilities to build absolute cata-
logues, as objects tend to change their positions.
There are individual object velocities, due to galactic
rotation in the case of celestial coordinate systems,
and due to tectonic movements and secular tidal ef-
fects in the case of Earth coordinate systems. That
is why one should always specify the catalogue epoch
and precision, meaning its systematic (averaged over
all objects) and random (every object has own error)
precisions. The only way to determine such errors is
through the comparison of di�erent catalogues.
Furthermore, a catalogue should not be inter-
preted as just a list of point coordinates. Every cata-
logue de�nes its own coordinate system. This should
be kept in mind when using the catalogue. Through-
out the years, geodesy has constructed of numerous
catalogues of terrestrial objects, and has de�ned nu-
merous Terrestrial Reference Frames (TRF). Astron-
omy has done the same with respect to catalogues of
celestial bodies, and has similarly de�ned numerous
Celestial Reference Frames (CRF). Space geodynam-
ics makes use of both types of catalogues. Determi-
nation of the transformation between CRF and TRF
is its main task. In this article we will not anal-
yse geodynamic transformations, but the TRFs and
CRFs themselves only.
Let ~ri be the coordinate of some object given in
i-th xRF (x might be C or T, but C and T may
never be present simultaneously in the same for-
mula). Spherical coordinates in TRF are λ � longi-
tude, and ϕ � lattitude. If ri if the distance from
the centre, then:
~ri = ri (cosλ cosϕ, sinλ cosϕ, sinϕ)
T .
For TRF ri ≈ R⊕, the Earth' radius.
Spherical coordinates in CRF are α � right as-
cension and δ � declination. Following the spherical
astronomy de�nition where all sources are placed on
the same celestial sphere of any useful radius R� we
will have the same coordinate de�nitions:
~ri = ri (cosα cos δ, sinα cos δ, sin δ)T .
For CRF ri = R�, usually 1.
Helmert transform was proposed for the compar-
ison of di�erent geodesic catalogues (read: TRFs)
by Friedrich Robert Helmert (1843�1917), director
of Potsdam geodetic institute and Professor at the
University of Berlin [11]. He introduced shift plus
rotation transform. It is widely used until now, see
for example [2, 3, 10], and not only for TRF/CRF
comparison [4].
But there are other possibilities to be analysed.
shifted centres, parallel axes
In this case for any point:
~r2 = ~r1 +~b, (1)
∗
charlie@univ.kiev.ua
15
Advances in Astronomy and Space Physics V.Ya.Choliy
where ~b is the shift of the centres. In stellar astron-
omy we may suppose that ~r1 and ~r2 are coordinates
in catalogues built for two distinct epochs with time
interval ∆t between them. That is:
~b = ~V∆t = ~r2 − ~r1.
Point coordinates may change:
~r2 = ~r1 +
d~r
dt
∣∣∣∣
1
∆t = ~r1 + ~µ∆t,
where ~µ is a proper motion vector.
In spherical coordinates:
~V∆t =
d~r
dt
∣∣∣∣
1
∆t,
or
~V =
ṙ cosα cos δ − r sinαα̇ cos δ − r cosα sin δδ̇
ṙ sinα cos δ + r cosαα̇ cos δ − r sinα sin δδ̇
ṙ sin δ + r cos δδ̇
.
In matrix notations:
~V =
(
X
Y
Z
)
=
=
(
r cosα cos δ −r sinα −r cosα sin δ
r sinα cos δ r cosα −r sinα sin δ
r sin δ 0 r cos δ
)
×
×
(
ṙ/r
α̇ cos δ
δ̇
)
. (2)
With generally used stellar astronomy designations:
ṙ = Vr, α̇ cos δ = µα, δ̇ = µδ having in mind
∆t = 1 year and after inverting the matrix in (2):(
Vr/r
µα
µδ
)
=
(
cosα cos δ sinα cos δ sin δ
sinα − cosα 0
− cosα sin δ sinα sin δ cos δ
)
×
×
(
X/r
Y/r
Z/r
)
. (3)
Or without matrices:{
Vr/r = X/r cosα cos δ + Y/r sinα cos δ + Z/r sin δ,
µα = X/r sinα− Y/r cosα,
µδ = −X/r cosα sin δ + Y/r sinα sin δ + Z/r cos δ,
(4)
which is more essential for classical classroom pre-
sentation.
The two last equations in (4) are identical to the
Kovalsky-Airy model. The �rst one is used to deter-
mine the Sun's velocity from radial stellar velocities
only [9], here (X,Y, Z)T = ~V�. In geodesy we may
use (1), or the Kovalsky-Airy model (3), for exam-
ple for comparison of geocentre positions determined
from VLBI and SLR. Both relations are identical and
di�er only in mathematical writing style.
shifted centres,
non-parallel axes
Following Helmert we imply that two RFs have
di�erent centres and their axes might coincide
through three rotations around three coordinate
axes:
~r2 = B~r1 +~b = R(n)Q(m)P(l)~r1 +~b, (5)
where P,Q,R are three elementary rotation matrices
around x, y, z axes respectively. In the general case,
angles l,m, n are small and their cosines might be re-
placed with unity, and sinuses with their arguments.
It gives us:
B =
(
1 l 0
−l 1 0
0 0 1
)(
1 0 −m
0 1 0
m 0 1
)(
1 0 0
0 1 n
0 −n 1
)
,
and:
~r2 =
(
1 l −m
−l 1 n
m −n 1
)
~r1 +~b. (6)
While the angles remain small, the result does not
depend upon the order of multiplications.
For the next step, let us extract a unity matrix
from the right side, move it to the left side:
~r2 − ~r1 =
(
0 l −m
−l 0 n
m −n 0
)
~r1 +~b = A~r1 +~b,
which may be rewritten in yet another form (here
x1, y1, z1 are components of the vector ~r1):
~r2 − ~r1 =
(
ly1 −mz1
nz1 − lx1
mx1 − ny1
)
+~b = [~ω × ~r1] +~b, (7)
where ~ω = (n,m, l)T = (ω1, ω2, ω3)
T is an angular
velocity vector which is generally used to explain ro-
tation from RF1 to RF2. It is a well-known result,
as any rotations around the main coordinate axes
can be replaced with only one around some specially
selected axis.
Inserting sphericals into (7) gives:
(
r cosα cos δ −r sinα −r cosα sin δ
r sinα cos δ r cosα −r sinα sin δ
r sin δ 0 r cos δ
)
×
×
(
Vr/r
µα
µδ
)
=
(
ω2z − ω3y
ω3x− ω1z
ω1y − ω2x
)
+
(
X
Y
Z
)
,
16
Advances in Astronomy and Space Physics V.Ya.Choliy
or after matrix inversion, rewriting without matrices:
Vr/r = X/r cosα cos δ + Y/r sinα cos δ + Z/r sin δ,
µα = −X/r sinα+ Y/r cosα− ω1 cosα sin δ−
−ω2 sinα sin δ + ω3 cos δ,
µδ = −X/r cosα sin δ − Y/r sinα sin δ + Z/r cos δ+
+ω1 sinα− ω2 cosα.
(8)
Thus, starting from (5) and supposing that rota-
tions are small, we came to the Lindblat-Oort model
of stellar astronomy [8].
Let us have a look at (6) again, where [~ω × ~r1] =
A~r1, A is an antisymmetric matrix, and A~r1 or al-
ternatively [~ω × ~r1] de�nes the velocity �eld essential
for the �rst catalogue. That velocity �eld moves the
points of the �rst catalogue to the positions of the
second one in full analogy with liquid �ow, with the
points immersed into that liquid. The points are the
partices �owing from the �rst catalogue positions to
the second catalogue positions. This analogy lead us
to the third model.
deformable velocity field
with shifted centres
and non-parallel axes
If the velocity �led is de�ned by matrix (or ten-
sor)A, can we add another part to that tensor? Since
the transformation matrix (or velocity �eld tensor)
A is antisymmetric, let us add the symmetric part S
as well. A new model arises:
~r2 − ~r1 = [~ω × ~r1] +~b+ S~r1. (9)
A symmetric matrix explains additional deforma-
tions of the velocity �eld. Interpretation of these
values are as follows: diagonal elements explain scal-
ing along their axes, while non-diagonal elements ex-
plain the pressure e�ects within their planes.
Substitution of stellar astronomy formulas into
(9) leads to the classical Ogoridnikov-Milne model
[6, 7] in matrix form:
(
r cosα cos δ −r sinα −r cosα sin δ
r sinα cos δ r cosα −r sinα sin δ
r sin δ 0 r cos δ
)
×
×
(
Vr/r
µα
µδ
)
=
(
ω2z − ω3y
ω3x− ω1z
ω1y − ω2x
)
+
(
X
Y
Z
)
+
+
(
S11x+ S12y + S13z
S12x+ S22y + S23z
S13x+ S23y + S33z
)
,
or without matrices:
Vr/r = X/r cosα cos δ + Y/r sinα cos δ+
+Z/r sin δ + S12 sin 2α cos2 δ+
+S13 cosα sin 2δ + S23 sinα sin 2δ+
+S11 cos
2 α cos2 δ + S22 sin
2 α cos2 δ+
+S33 sin
2 δ,
µα = −X/r sinα+ Y/r cosα−
−ω1 cosα sin δ − ω2 sinα sin δ+
+ω3 cos δ + S12 cos 2α cos δ−
−S13 sinα+ S23 cosα sin δ−
−(S11 − S22)/2 sin 2α cos δ,
µδ = −X/r cosα sin δ − Y/r sinα sin δ+
+Z/r cos δ + ω1 sinα− ω2 cosα−
−S12 sin 2α sin 2δ/2+
+S13 cosα cos 2δ + S23 sinα cos 2δ−
−(S11 − S22)/2 cos
2 α sin 2δ+
+(S33 − S22)/2 sin 2δ.
A geodesic transformation of type (9) is an ex-
tended Helmert transform.
We should understand �deformation� in a very
weak sense. There is no real deformation of the space
with embedded sources. The only deformed �eld is
the virtual �eld of velocities, which moves the ob-
jects of the catalogue to their positions in another
catalogue, in the best way.
case of correlated
random errors
Typically, the Helmert transform is used as the
�rst step of a multiple-step procedure of constructing
the combined catalogue. Parameters of (9) together
form the model of systematic errors between RFi and
RFj . Random errors might be estimated as a total
residual error after removing of the systematic one.
There is some probability that the combined cata-
logue will have a lower level of random errors, as
compared with the raw ones. Let us de�ne RF0 as a
combined catalogue. The classical method of build-
ing them from M raw ones postulates that [12]:
~r0 =
M∑
i=1
pi~ri
M∑
i=1
pi
, σ2
0 =
1
M∑
i=1
pi
,
where ~r0 is the best position, estimated with error
σ2
0, pi are weights.
To estimate the random errors let us suppose that
all numbers in (9) are already determined with Least
Squares procedure applied to N shared points of RFi
and RF0. It means that now we are able to trans-
form all the shared points from RFi to RF0 applying
(9) to points coordinates in RFi. It leads us to:
~r
(∗)
i = [ω × ~ri] + ~ri +~b+ S~ri. (10)
17
Advances in Astronomy and Space Physics V.Ya.Choliy
The total residual dispersion is then:
σ2
i0 =
∑(
~r
(∗)
i − ~r0
)2
/N, (11)
and it is the mean random error of the RFi and RF0.
According to statistics for any two RFi and RFj :
σ2
ij = σ2
i + σ2
j − 2ρijσiσj , (12)
where σ2
i is the dispersion of random errors of the
RFi, and ρij is the correlation of random errors of
two RF ′s.
There are numerous unknowns in (12) which
make the equation system underdetermined. In the
case of comparison of M di�erent RF ′s, there are
M(M − 1) equations with M(M − 1) unknown cor-
relation coe�cients ρij and M unknown σi. The so-
lution of these equations is possible only with addi-
tional assumptions.
To solve (12) one needs to determine the correla-
tions ρij in some way. It might be done directly:
ρij =
∑(
~ri − ~̂ri
)(
~rj − ~̂rj
)
σiσj
, (13)
or by the method proposed in [1], where ρij are
the solutions of a linear equation system, built on
(RFi−RFj) and (RFi+RFj) dispersions. Here ~r
(∗)
i
are points of RFi after applying systematic correc-
tion between RFi and combined RF0. If d2ij is the
estimation of the di�erence between the dispersions
of the catalogues (RFi −RFj), and s2ij is an estima-
tion of the sum of the dispersions of the catalogues
(RFi+RFj). For three catalogues [1] gives the linear
equations: σ2
1 = (s212 + s213 − s223 − σ2
12 − σ2
13 + σ2
23)/4,
σ2
2 = (s223 + s212 − s213 − σ2
23 − σ2
12 + σ2
13)/4,
σ2
3 = (s223 + s213 − s212 − σ2
23 − σ2
13 + σ2
12)/4,
(14)
and
ρij =
s2ij − d2ij
2σiσj
.
Yet another solution method of (13) uses the op-
timization procedure in the space of ρ′s. For the case
of M = 3 let us build a three-dimensional space (~ρ-
space) where ρ12, ρ23, ρ31 are coordinates. Then, let
us rewrite (12) as a 3-dimensional vector equation:
~f =
σ2
1 + σ2
2 − 2ρ12σ1σ2 − σ2
12
σ2
2 + σ2
3 − 2ρ23σ2σ3 − σ2
23
σ2
3 + σ2
1 − 2ρ31σ3σ1 − σ2
31
= ~0, (15)
and proceed with the solution of (15) for σ′s on the
lattice of ρ′s in ~ρ-space. Newton method of tangents:
~σnew = ~σold − J−1 ~f(~σold),
is fully su�cient with starting value ~σ0 and Jakobi
matrix J:
~σ0 =
(
σ12
σ23
σ31
)
, J =
[
∂ ~f
∂~σ
]
.
Unfortunately, the proposed method generates a
large number of solutions. For example, if we built
a lattice in ~ρ-space with 0.1 step, we will have
213 = 9261 solutions.
Since the Newton method is unrestricted, some
solutions have negative σi and therefore should be
dropped. Other solutions demonstrate known be-
haviour: the greater negative ρ′s are, the lower σ′s
are as a result. This is why we apply an additional
restriction: the solution of (12) in ~ρ−space should
be a minimum-length positive component vector ~σ
corresponding to the minimum value of ρ′s modules.
shifted correlated
random errors
Let us return to (12) and (10), which are the def-
inition of σij . If one estimates Helmert parameters
from (11), then calculates σij for transformation RFi
to RFj , and then calculates σ2
ji for transformation
from RFj to RFi, then one will have σ2
ij 6= σ2
ji and
even ρij 6= ρji in general.
Our explanation of this fact is that there is an
uncompensated systematic error still present in the
data (but not accounted with (9)) which distorts and
shifts the dispersion estimation (12). If we suppose
that the systematic part is not correlated with ran-
dom one, we can rewrite (12) in an another form:
{
σ2
ij = σ2
i + σ2
j + kij − 2ρijσiσj ,
σ2
ji = σ2
i + σ2
j − kij − 2ρijσiσj ,
(16)
where kij is an additional member, while the dis-
persion of a portion of the systematic errors is still
present in random residuals.
It is thus evident that:
kij =
(
σ2
ij − σ2
ji
)
/2.
This term is absolutely arti�cial and its inclusion in
(16) still needs mathematical approvement, however
its value might serve as goodness-of-�t criteria of the
applicability of Helmert transform.
18
Advances in Astronomy and Space Physics V.Ya.Choliy
numerical experiment
The specially assigned software was created for
the purpose of numerical experiments with the
Helmert transform. The software builds three ar-
ti�cial catalogues consisting of points on the Earth's
surface, equally distributed along longitude and lat-
itude with 5◦ steps; each catalogue containing 2664
points. Then Gaussian correlated random noise was
added to each points in the catalogues. To generate
Gaussian random noise values, the following well-
known transformation was used:
(u1, u2) →
→ (n1 = u1
√
−2 log s/s, n2 = u2
√
−2 log s/s),
where s = u21 + u22, and 0 < s < 1 with u1 and u2
are random values uniformly distributed on [−1, 1].
Before they were applied, the values of (n1, n2) were
correlated with one another according to the rule:
(n1, n2) → (n1, n1ρ+ n2
√
1− ρ2)
with prede�ned ρ. Generation of uniformly dis-
tributed values was conducted with x128() algorithm
[5].
After comparison of the catalogues, building the
combined one, values of errors and correlations were
calculated using three di�erent approaches. These
are a) standard formulae (11) and (13); b) linear so-
lution from [1]; c) optimization in ~ρ-space.
The most interesting for us are values of correla-
tions, calculated through di�erent methods. In most
cases the correlations according to (14) are 5%�10%
less than correlations according to (13). In contrary
to this, values of σ do not show any similarities.
The following example might show the correla-
tion di�erences: (0.261, 0.677, 0.383) from (14) and
(0.331, 0.700, 0.442) from (13), whilst sigmas are
(7.68, 1.16, 4.79) with (14) correlations and (3.72,
4.72, 0.89) with (13) ones. The most surprising third
method, as it can converge to both of the results de-
pending of starting conditions.
conclusion
Extended Helmert transform was used for the
comparison of model TRF catalogues, and it was
shown that systematic errors are accounted for in
a more precise way than with the classical Helmert
transform.
Can the extended transform be extended once
again? We may add spherical functions to the right
side of (9), like in many astrometric texts on cata-
logue comparison, e. g. [12].
However, as we used throughout the article the
analogy between geodesy and stellar astronomy, we
can suppose that there is a time to start searching
for �tectonic plates� in the sky, like it was done in
geodesy years ago.
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