The integrated approach to the geodetic
determination of crustal deformation
Athanasios Dermanis
Department of Geodesy and Surveying
University of Thessaloniki, Greece
1. Introduction
geodesy to what could be justly called integrated geophysics.
The importance of the study of crustal deformation needs
hardly to be stressed, especially in those parts of the world
where tectonic activity is the cause of earthquakes that affect the lives of millions of people. Geodesy, being the science which is concerned with the determination of the shape
and gravity field of the earth, can contribute to the assessment of crustal motion in two ways: from the determination
of the deformation of the surface of the earth and from the
determination of the temporal variation of the gravity field
which reflects also the redistribution of internal masses.
The basic techniques described here are based in the work
of Rossikopoulos (1986), Dermanis and Rossikopoulos
(1988) and Dermanis and Rossikopoulos (1991). A review
of work in classical integrated geodesy, following the pioneering ideas of T. Krarup (Eeg and Krarup, 1975), can be
found in Hein (1986). First applications of relevant concepts
in the study of deformation have been presented by Dermanis et al. (1981), Bencini et al. (1982) for horizontal networks, Kanngieser (1983) for leveling networks.
Classical geodetic techniques treat such information independently by geometric surveys (horizontal networks and
leveling) and gravimetric ones. The standard approaches
suffer furthermore from the somewhat arbitrary separation
of the horizontal from the vertical part, as well as, from the
use of numerous modeling approximations and data processing strategies whose validity is questionable in cases
where observations of the highest possible accuracy are
analyzed for the determination of physical processes with
magnitudes not always well above the level of observational
noise.
2. The description of crustal deformation
Crustal deformation is usually described by separating the
horizontal (planar) from the vertical component of displacement, following the traditional separation of geodetic
work in leveling and horizontal networks. In addition to the
2-dimensional "horizontal", crustal deformation can be described in a less fictitious 3-dimensional way, corresponding
to the actual deformation of the 3-dimensional earth, by
either combining the results from leveling and horizontal
networks, or by direct use of 3-dimensional networks. However, the fact that relevant information is limited to points
on the earth surface, which is a 2-dimensional medium
though not flat or horizontal, makes the study of plane deformation still attractive and perhaps more convenient. In
this approach the surface of the earth is mapped on a plane
(or a sphere or ellipsoid for larger areas) and the deformation of the imaginary surface produced point-wise by such a
"vertical projection" mapping is replacing the actual deformation of the 3-dimensional surface.
Integrated geodesy has been introduced for the unified and
rigorous adjustment of observations with both geometric
and gravimetric information, using clearly defined optimality criteria and precise mathematical models. In recent years
integrated geodesy has been considered in its more abstract
form as a method for the adjustment of observations depending not only on discrete parameters but also on unknown functions. This generalization allowed its proper
application to the study of crustal estimation, utilizing all
available geodetic observations depending not only on the
(static) gravity field of the earth (classical integrated geodesy) but on a gravity potential which is a function of both
time and position, as well as, on network point positions
which are functions of time. The observations are, in this
context, discrete not only with respect to space but also with
respect to time, thus allowing a departure from the classical
scheme of comparison between two or more different epochs.
With respect to the time component, two possible approaches can be distinguished. The first is the comparison of
the change of the shape of the earth crust between two epochs t and t ′ . Material points are identified by their position in one of the two epochs usually the first one t (Lagrangean description). Deformation is then described by
means of invariants of the strain tensor, which describes for
each point the alteration of shape in a small (infinitesimal)
neighborhood of the point. Invariance is an essential property, since deformation is a concept independent of any particular choice of coordinate systems used for the shake of
mathematical convenience. The required information is the
shape of the deforming medium in the two epochs, which
means that the position of every point should be known at
both epochs with respect to some arbitrary frame of refer-
It is the aim of this work to describe the principles, techniques and some special applications of immediate interest
of integrated four-dimensional geodesy. Of course the concept of integration can be expanded beyond the limits of
geodesy with the, more or less straightforward, inclusion of
non-geodetic observations, such as those from strain-meters,
tilt-meters, seismic sounding, etc., leading from integrated
1
ence, i.e. continuous spatial information is needed for this
type of description.
3. The concept of integrated geodesy as applied
to the study of crustal deformation
The second approach refers not to the comparison of any
two epochs but to the "rate of change of shape" at any particular epoch t. Deformation is in this case described by invariants of the time derivative of the strain tensor (strainrate tensor). The required information is not only continuous
in space as in the previous case, but it should also be continuous in time, i.e. the positions of material points should
be known for all epochs at least in a small time-neighborhood of the particular epoch of interest. The positions must
refer to a reference frame which is defined arbitrarily for
every epoch with the essential restriction of being time-wise
smooth, a concept referred to as "framing" (Truesdell,
1977). In classical continuum mechanics there exists always
an undeformable environment (the laboratory) which provides the means for the definition of a time independent
arbitrary reference frame for the study of the deformation of
material bodies. On the contrary geodetic work is confronted with a "sea" of moving points and a reference frame
must be defined arbitrarily at every epoch. These epochwise frame definitions constitute a framing and the study of
crustal deformation must be addressed to quantities invariant with respect to changes of framings, unlike the case of
classical "tensorial" invariants where invariance refers to a
change of reference common for all points. This is indeed a
critical point not always completely understood from the
viewpoint of classical physics. A study of the invariance of
such parameters, which are related to the concept of their
estimability from framing invariant geodetic observables
has been carried out by Dermanis (1981), Dermanis (1985a)
and Dermanis and Grafarend (1992).
Integrated geodesy has originally been introduced by Krarup
(Eeg and Krarup, 1975) as a concept for the integrated
analysis of all data related to the gravity field of the earth,
even those in traditional "geometric" geodetic observations
which relate to gravity through the local direction of the
vertical. An essential aspect of the approach was the incorporation of the solution not only for the gravity related parameters (signals) but also for the unknown gravity potential
function itself, which in turn allows the determination of the
values of any gravity related parameters (new signals) other
than those present in the observations. This in fact extended
the classical least squares adjustment with a solution to the
interpolation problem for the gravity field. This problem
was solved by Krarup in his celebrated "contribution"
(Krarup, 1969) by the now famous "collocation" method.
The resulting adjustment algorithm is sometimes referred to
as "least-squares collocation" (Moritz, 1973).
There are two independent but equivalent interpretations of
the algorithm for integrated geodesy. In the deterministic
interpretation the least squares principle for the observation
errors is extended to include the square of the norm of the
unknown function which is considered to belong to an appropriate class of functions (a Hilbert space with reproducing kernel). The solution is attaining a balance between the
requirement for small errors and a smooth interpolated
function. In the statistical interpretation the unknown function is modeled as a stochastic process (random function)
and the proper model for the adjustment is the linear mixed
model with both deterministic (e.g., coordinates) and random parameters (signals). The determination of new signals
is achieved by the well-known solution of the prediction
problem (determination of outcomes of random variables
from the known outcomes of other random variables). Parameters are estimated and signals are predicted in an optimal way by minimizing the mean square error (mean over
all possible outcomes) of such estimates or predictions
within the class of the linear and unbiased ones. More details on these two interpretations can be found in Krarup
(1969), Lauritzen (1973), Dermanis (1976, 1978, 1983),
Moritz (1978, 1980), 6DQV Although continuous spatial or continuous spatial-temporal
shape information is required, according to the type of deformation description followed, geodetic observations are
discrete both in space and time. Even in the two epoch description approach, the possibility to have observations referring to the two epochs only (in fact in two small time
periods around these epochs within which deformation can
be neglected) can be realized only for horizontal networks
and not for the time consuming process of leveling.
The contrast between the discrete character of the information provided by geodetic observations and the required
continuous information required for deformation description, makes it clear that an appropriate approach to the
analysis of geodetic data related to crustal motion detection,
must not be confined to the classical adjustment techniques
for parameter estimation but should in some way incorporate a solution to the implicit interpolation problem. Interpolation is not only a mathematical necessity for the determination of deformation at the points of observation, but
also for obtaining information at all points of a tectonic active area, a matter of obvious geophysical importance.
Although the deterministic interpretation is easier to accept
(there is hardly anything random about the gravity field of
the earth), the stochastic-statistical exposition has gained
more popularity, because it does not require the rather deep
mathematical foundation of the deterministic one.
Integrated geodesy becomes a more general and more powerful technique by the process of abstraction: From a
method for the analysis of data related to the unknown
gravity potential function, it can be conceived as a method
for analyzing any data related to one or even more unknown
functions. It is in this framework that it qualifies as a natural
tool for the study of crustal deformations.
It is for these reasons that the concept of integrated geodesy
provides, under proper extension and modification, the most
valuable tool for the integrated analysis of all data relevant
to crustal motion, not only the purely geometric ones, but
also those related to the gravity field of the earth, whose
temporal variation is directly connected to crustal motion.
The unknown functions in crustal deformation analysis are
the gravity potential and the displacements of material
points, both being functions of space, as well as, of time.
The relevance of the gravity field is not only due to the fact
that some geodetic observations are sensitive to it, but
mainly to that crustal deformation causes changes of the
local gravity vector in two ways: from the motion of the
2
points within the field domain and directly from the change
of the field itself due to mass redistribution.
is used, where φ1 , φ 2 , …, φ k , are known functions (base
functions).
4. Fundamentals of integrated geodesy
The (purely) stochastic approach: In this case the mean
function E{ f } of f is assumed to be completely known,
4.1. Function modeling approaches
i.e. coinciding with the normal function f 0 .
f = f0 + fs
Of fundamental importance in the integrated geodesy approach is the modeling of the unknown functions, which
give rise to the specific signals (in the sense of discrete
function-dependent parameters) which appear in the
mathematical models for the observable quantities. Every
unknown function f may be in general considered to consist of the following parts:
f = f 0 + δf = f 0 + m + f s
The stochastic part f s has therefore zero mean, E{ f s }= 0 ,
and a finite covariance function
C ( P, Q ) = E{ f s ( P) f s (Q )}
(4)
which is assumed to be known, where P and Q are any
two of the function arguments.
(1)
An equivalent model is to consider f s as a linear combina-
where f 0 is the known (normal) part of the function, δf is
the unknown part which in turn consists of two parts, the
deterministic part m = E{δf } , which is the "trend" of the
unknown part and the zero-mean stochastic part
(3)
tion of known base functions ψ i
f s = b1ψ 1 + b 2ψ 2 ++ biψ i +
fs
( E{ f s }= 0 ). It is not necessary to include all the above three
parts in the model. It is not even necessary to take into account the dependence of the signals on an unknown function. In the latter case the signals are treated as purely independent deterministic unknown parameters along with the
usual geodetic parameters. This is in fact the approach in the
three-dimensional network adjustment according to the
Bruns' polyhedron concept (Bruns, 1878, Heiskanen and
Moritz, 1967).
(5)
where b1 , b2 , …, bi , … are uncorrelated stochastic parameters with
E{bi b j }= 0
for i ≠ j ,
E{bi2 }=σ i2
(6)
which leads to the same model as (4) through the equivalence relation (Dermanis, 1983)
C ( P, Q ) =
∑σ i2ψ i ( P)ψ i (Q) .
When the dependence of the signals on an unknown function is taken into account, different approaches may arise
depending on which of the three parts f 0 , m , f s , are included in the function model (Dermanis & Rossikopoulos,
1988):
Note that even an infinite number of terms are allowed in
the representation (5) provided that the sum in (7) is finite.
The classical method of reduced observations: In this case it
is assumed that f = f 0 = known , by taking advantage of an
The combined (analytical-stochastic) approach: This is the
more general case f = f 0 + m + f s , where the trend m is usu-
existing estimate f 0 of f , and the signals become also
known parameters. Their effect can be subtracted from the
observations to produce "reduced observations". An example is the reduction of theodolite observations from the local
vertical to the local normal to the ellipsoid, using existing
estimates of the deflections of the vertical. The result is a
purely geometric three-dimensional adjustment, or even a
two-dimensional one on the ellipsoid by using only horizontal observations further reduced from the actual point to
its projection on the reference ellipsoid.
ally modeled according to equation (2) and f s has the covariance function (4).
f = f 0 + a1φ1 + a 2φ 2 ++ a k φ k + f s ,
(8)
E{ f s }= 0 ,
E{ f s ( P ) f s (Q)}= C ( P, Q)
Let it be noted however that for functions of both space and
time, it is possible to follow a different modeling approach
with respect to each argument (different treatment with respect to different variables). A typical example is the timeanalytical & space-stochastic approach: a space-time function f (t , P) is treated analytically with respect to time, e.g.
by a model of type (2)
The analytical (deterministic) approach: In this case
f = f 0 + m , where the trend function m = m(a1 ,, a k ) is
further modeled to depend on k k deterministic unknown
parameters a1 ,, a k . Usually a linear combination model
of the form
m = a1φ1 + a 2φ 2 + + a k φ k
(7)
i
f (t , P ) = f 0 ( P) + f 1 ( P)(t − t 0 ) ++ f n ( P)(t − t 0 ) n
(2)
(9)
and in turn the coefficient space-functions are modeled independently by one of the approaches described above).
Usually the f i (P) , i =1, 2, , n , are considered to be zero3
mean stochastic functions with known covariance functions
C i ( P, Q) = E{ f i ( P ) f i (Q)} . The zero term f 0 ( P) = f ( P, t 0 )
can be treated either as a stochastic space-function (case of
gravity potential W ( P, t 0 ) ) or as a deterministic function
yielding signals which are treated as independent deterministic unknowns (case of coordinate function x( P, t 0 ) . More
than one unknown function may be simultaneously present
in the model, each one treated by a different modeling approach.
achieved using known approximate values x 0 for the parameters x a , and a known approximate function f 0 for f ,
which gives rise to approximate signals s 0 = s a ( f 0 ) , using
the known relation in (10) between f and the signals s a .
The relation of the signals to the unknown function is typically a linear one, i.e. s a ( f 0 + δf ) = s a ( f 0 ) + s a (δf ) . In fact
the signals arising in geodesy are either the values of the
unknown function, or of its derivatives with respect to space
and/or time, at a specific point and/or epoch. Setting
The question of which model is better to use in a specific
problem is a difficult one and cannot certainly be answered
a-priori. A trial and error approach will give an insight into
the capabilities of specific model choices. Statistical hypothesis testing can provide criteria for choosing between
alternative models (see e.g. Dermanis and Rossikopoulos,
1991), although a theoretically complete and universally
applicable method is still missing. However some simple
rules can give a hint to the proper model choice: A slowly
varying physical process can be better modeled by simple
analytic models. A process which exhibits an erratic but still
smooth variation around zero (or around an already modeled
simple trend), can be better described by a stochastic model.
In crustal deformation analysis, analytical models are best
suited for time-variation and stochastic ones for spatial
variation. Even when a stochastic model is used it is still
possible to use an analytic model for its non-zero trend, e.g.,
by allowing some of the parameters of the normal potential
to be treated as unknowns for best local fit. The standard
normal potential, of either the Somigliana-Pizzetti type (i.e.
with reference ellipsoid as equipotential surface) or given by
one of the available spherical harmonic models, is designed
to achieve a best global fit.
xa =x0 +x ,
s a =s 0 + s ,
s0 =s a ( f 0 ) ,
δs = s a (δf ) ,
(12)
b=yb −y 0 ,
y 0 = y a (x 0 , s 0 ) ,
(13)
equation (10) is transformed into the linear form
b = Ax + Gδs + v ,
(14)
where
∂y a
∂x a
A=
A=
,
( x 0 ,s 0 )
∂y a
∂s a
.
Further development depends on the specific approach taken
for the modeling of the unknown function f . As an example we take the more general case of the combined analytical-stochastic approach according to equation (8). The linear
signal-to-function relation s a ( f ) applied to each specific
4.2. Basic form of mathematical model - Signals
signal s ia becomes
The unknown function f which affect the observed quantities does not appear directly in the mathematical model, but
its presence is manifested through discrete unknown parameters which in turn depend on f and are called signals.
s ia = s ia ( f ) = s ia ( f 0 + a1φ1 + + a k φ k + f s ) =
= s ia ( f 0 ) + a1 s ia (φ1 ) ++ a k s ia (φ k ) + s ia ( f s ) =
The mathematical model relating observables y a to un-
= s i0 + a1 hi1 ++ a k hik + δs i ,
known parameters x a and signals s a has the general nonlinear form
y a = y a (x a , s a ) ,
s a =s a ( f ) .
(16)
with
(10)
s i0 = s ia ( f 0 ) ,
The observations
y b = y a (x a , s a ) + v ,
(15)
( x 0 ,s 0 )
hij = s ia (φ j ) ,
s i = s ia ( f s )
(17)
or in matrix notation
E{v}= 0 ,
E{vv T } = C v
(11)
s a = s 0 + δs = s 0 + Ha + s .
are typically affected by zero mean random errors v with
known covariance matrix C v . It is further assumed that
signals and observational errors are uncorrelated so that
C sv = 0 .
(18)
Replacing δs = Ha + s in (14) we obtain the more general
linearized model
x 
b = Ax + GHa + Gs + v =[ A GH ]  + Gs + v
a 
4.3. General linearization scheme
(19)
where the deterministic parameters (x, a) have been separated from the stochastic signals s and the observational
errors v.
An essential step for the analysis of the available observations using standard estimation and prediction techniques is
the linearization of the mathematical model, which is
4
and/or epoch. This means that in fact the function itself is
reconstructed from the available observations thus also
solving an interpolation problem. The interpolation approach can be used as an alternative to the above stochastic
formulation, especially when the unknown function is
known to be of a deterministic rather than a stochastic nature (see e.g. Dermanis, 1987). In this case the least squares
principal (21) is replaced by
Different linearized models arise when different approaches
are followed to the modeling of the unknown function. In
the
analytical
approach,
the
model
becomes
b = Ax + GHa + v , and in the purely stochastic one
b = Ax + Gs + v .
4.4. The adjustment of the observations for parameter
and signal estimation
|| f s || 2 + v T C −v1 v = min ,
After the linearization has been performed the mathematical
model has the general form
b = Ax + Gs + v ,
where || f s || is the norm of the "disturbing" function f s ,
assumed to belong to an appropriate function space. If the
function admits a representation of the form (5), though
with deterministic coefficients bi , it turns out that the norm
in (23) has the form of a sum of squares
(20)
where x contains all possible deterministic parameters
(original plus signal trend parameters if so modeled). This
model is usually called the mixed linear model in the statistical literature.
|| f s || 2 =
The adjustment of the observations is carried out by applying the least squares principle
s T C s−1s + v T C −v1 v = min
(23)
∑ pi bi2
(24)
i
which means that a smoothness condition is imposed on f s ,
simultaneously with the minimization of the sum of
weighted residuals. The connection between the deterministic (interpolation) and the stochastic (prediction) is established by letting p i =1 / σ i2 , σ i2 being the variances of the
(21)
which leads to best linear unbiased estimates for the deterministic parameters x , a and best linear unbiased predictions for the stochastic ones s , v (best in the sense of
minimizing the mean square error, see Dermanis, 1991).
coefficients bi according to (6), i.e. in a way similar to that
in relating deterministic least squares adjustment with best
linear unbiased estimation. For more details on this duality
see Dermanis (1976, 1978, 1983, 1987).
The covariance matrix C s = E{ss T } of the signals is obtained from C ( P, Q ) by applying the law of covariance
propagation to the relations s = s a ( f s ) .
The algorithms which can be used for the adjustment of (20)
are presented in Appendix A, summarized from Dermanis
and Fotiou (1992), see also Koch (1987).
5.
4.5. Prediction of new signals - Interpolation
The observations we shall consider here for inclusion in an
adjustment according to the integrated approach will be
limited to the usual observations in geodetic networks as
well those of leveling and gravity surveys. Other types of
geometric observations, such as those from strain-meters or
tilt-meters, can also be included in a straightforward way.
5.1. Basic models of geodetic observables
An essential additional feature of the algorithm is the possibility to predict the outcomes of stochastic signals s ′ others
than those s appearing in the model. When s is directly
observed s ′ can be predicted using the well known predic−1s , provided that E{s ′} = 0 , E{s} = 0
tion formula sˆ ′ = C s′s C ss
and the relevant cross-covariance and covariance matrices
C s′s = E{s ′s T } and C ss = E{ss T } are known, except possibly for a common scalar factor which cancels out. When s
is not directly observed but appears in the linearized model
(20) the (best linear unbiased) prediction turns out to be
equivalent to a two-step procedure: First estimates ŝ are
obtained (see Appendix C) and the prediction is carried out
next according to
−1sˆ .
sˆ ′ = C s′s C ss
Application of integrated geodesy
In the more general case an observation y carried out at a
point P depends on the position of P , expressed through
global cartesian coordinates x , the gravity vector
g = gradW at the same point and possibly the position of
other (target) points. The linearization of the mathematical
model of the form y = y (x, g,) is based on approximate
coordinates x 0 and the normal potential U approximating
the actual gravity potential W , which gives rise to the normal gravity vector = gradU . Linearization follows the
scheme
(22)
(
)
 ∂y ∂y ∂g 
 (x − x 0 ) +
y − y x 0 , (x 0 ), =  +
 ∂x ∂g ∂x 0
Prediction is an excellent tool for the direct determination,
at any required point and epoch, of exactly those parameters, which are the most appropriate for the description of
crustal deformation.
 ∂y 
+   gradT (x 0 ) +
 ∂g 0
In particular it is possible to predict signals s ′ , which are
the values of the unknown function f at any desired point
5
(25)
where T =W −U is the disturbing potential and the relation
gradT = g − has been used. The partial derivatives are
g = g (g ) = g 12 + g 22 + g 32
evaluated using x 0 and U , in place of x and W , i.e. by
replacing g with , gravity g =|g | with normal gravity
Λ = Λ (g ) = arctan
γ =| | , astronomic longitude Λ and latitude Φ with their
normal counterparts λ and φ .
Φ = Φ (g ) = arctan
Theodolite observations at a point P depend not directly on
g , but on the unit vector n in the opposite direction
1
n=− ,
g
g =[cos Φ cos Λ cos Φ sin Λ sin Φ ]
T
(31)
g2
g1
(32)
−g3
(32)
g 12 + g 22
and they can be linearized according to the simplified general scheme
(26)
 ∂y ∂g 
 ∂y 
 (x − x 0 ) +   gradT (x 0 )
y − y ( (x 0 ) )= 
∂
∂
g
x

0
 ∂g  0
and they are more directly expressed in terms of the pointto-target vector x Q − x P expressed in the local astronomic
frame at P
The resulting linear models have the following form:
= R ( x Q − x P ) = R 1 (90 $ − Φ P ) R 3 (90 $ + Λ P )(x Q − x P )
Two point observations (from x P to x Q ):
(34)
(27a)
α −α 0 = a αT P (x P − x 0P ) + a αTQ (x Q − x Q0 ) + c αT P gradT (x 0P ) (35)
or explicitly
ζ −ζ 0 = a ζTP (x P − x 0P ) + a ζTQ (x Q − x Q0 ) + c ζTP gradT (x 0P ) (36)
0
0
ξ1  1

ξ  = 0 cos(90$ − Φ ) sin(90$ − Φ )  ×
 2 

ξ3  0 − sin(90$ − Φ ) cos(90$ − Φ )
s − s 0 = a TsP (x P − x 0P ) + a TsQ (x Q − x Q0 ) .
(27b)
Single point observations (at point x ):
 cos(90 + Λ ) sin(90 + Λ ) 0  ( xQ − xP ) 



× − sin(90$ + Λ ) cos(90$ + Λ ) 0 ( yQ − yP )

0
0
1   ( zQ − z P ) 


$
$
The mathematical models for the azimuth α the zenith distance ζ and the spatial distance s from point P to point
Q are
ξ
α PQ =α (x P , x Q , g P ) = arctan 1
ξ2
ζ PQ =ζ (x P , x Q , g P ) = arctan
sPQ = s (x P , xQ ) = (x P − xQ )T (x P − xQ ) =
(30)
= ( x P − xQ ) + ( y P − y Q ) + ( z P − z Q )
2
2
(38)
Λ − λ 0 = a TΛ (x − x 0 ) + c TΛ gradT (x 0 )
(39)
Φ −φ 0 = a TΦ (x − x 0 ) + c TΦ gradT (x 0 ) .
(40)
Instead of cartesian coordinates x , it is possible to use any
other type of curvilinear coordinates, of which more convenient are the geodetic coordinates z =[b λ H ]T . Geodetic
longitude λ coincides with the normal longitude since the
adopted normal field U is rotationally symmetric. H is the
geometric height above the reference ellipsoid and does not
relate directly to the dynamic heights computed by standard
leveling procedures.
(29)
ξ3
g − γ 0 = a Tg (x − x 0 ) + c Tg gradT (x 0 )
Expressions for the coefficients in the above relations are
also given in Appendix A.
(28)
ξ 12 + ξ 22
(37)
To introduce geodetic coordinates in the linear models it is
sufficient to replace
2
 ∂x 
x − x 0 =   (z − z 0 ) ,
 ∂z  0
Horizontal angles θ and directions δ are not considered
explicitly here, because they are directly related to azimuths,
i.e. θ PQR =α PR −α PQ , δ PQ =α PQ −α 0 , α 0 being the un-
(41)
T
 ∂x 
gradT (x 0 ) =  
 ∂z  0
known orientation constant.
Observations of gravity g and astronomic observations of
longitude Λ and latitude Φ depend only on the gravity
vector g =[ g 1 g 2 g 3 ]T and have the form
T
 ∂T ∂T ∂T 
 ∂ ∂b ∂H  0

z
where the required derivatives follow from differentiation of
the well known relation
6
 ( N + H ) cos b cos λ 
x = x(z ) =  ( N + H ) cos b sin λ  ,
[(1− e 2 ) N + H ]sin b 
N=
a
H is eliminated from combination with the observed potential W (H ) to obtain the gravity anomaly ∆g = g (x) −γ (x 0 ) :
(42)
1− e 2 sin 2 b
∆g = −n T0 gradT −
a being the semi-major axis and e 2 the eccentricity of the
reference ellipsoid, which yields
− ( N + H ) cos b sin λ
=  ( N + H ) cos b cos λ

0
(43)
− ( M + H ) sin b cos λ
− ( M + H ) sin b sin λ
( M + H ) cos b
1
n0 =− ,
γ
pseudo-observations, which are already in a simple linear
form
(44)
δx ij − x 0j + x i0 = (x j − x 0j ) − (x i − x 0i ) + v δxij
Note that
(48)
when more than one campaign is involved care should be
taken for the different orientation of the reference frames
involved (absolute position plays no role in the relative
mode). The model for coordinate differences in a satellite
frame is δx ijs = R (α , β ,γ )(x j − x i ) , where α , β , γ are the
(45)
is the unit vector normal to the reference ellipsoid and
cos b cos λ 
 N cos b cos λ 


x =  N cos b sin λ  + H  cos b sin λ  = x E + Hm
 sin b 
(1− e 2 ) N sin b 
∂
,
∂x
GPS observations can be included after their independent
preprocessing in order to determine and remove systematic
effects. The results are estimates of coordinate differences
δx ij = x j − x i , between points Pi and P j to be used as
with
cos b cos λ 
∂x 
= cos b sin λ 
m=
∂H 
 sin b 
,
(all terms evaluated at x 0 ) which is a more precise form of
the usual "spherical approximation" ∆g = − Tr − 2r T .
cos b cos λ 
cos b sin λ 
sin b 
M = a (1− e 2 )(1− e 2 sin 2 b) −3/ 2 .
n T0 m γ
(47)
M=
∂x  ∂x ∂x ∂x 
=
=
∂z  ∂λ ∂b ∂H 
n T0 Mm T
three parameters of rotation relating the earth frame to the
satellite one. The linearized form is
(46)
δx ijs − R 0 (x 0j − x i0 ) = R 0 (x j − x 0j ) − R 0 (x i − x i0 ) +
(49)
+ R α0 (x 0j − x i0 )δα + R 0β (x 0j − x 0i )δβ + R γ0 (x 0j − x i0 )δγ + v δx ij
where x E is the projection of x on the reference ellipsoid.
Not all the terms in the linearized equations should be retained in the adjustment procedure. The inclusion of a certain parameter in an observation equation is meaningful
only when the relevant observation contains sufficient information for the determination of the parameter.
where δα =α −α 0 , δβ = β − β 0 , δγ = γ − γ 0 , R α = ∂ R ,
∂α
∂
∂
Rβ =
R , R γ = R , and the zero index denotes
∂β
∂γ
evaluation with the approximate values α 0 , β 0 , γ 0 .
The terms of (x − x 0 ) in single point observations of g , Λ
and Φ should be included only when carried out at network
points. In isolated points the solution tends to minimize the
signals in gradT (x 0 ) even to the point of completely elimi-
The signals appearing in the linearized observation equations are the values at specific known points x 0 , of the disturbing potential T and of its gradient gradT , alternatively
expressed by the derivatives with respect to geodetic coordinates ∂T , ∂T , ∂T .
∂λ ∂b ∂H
nating them by shifting the point position from x 0 to an
appropriate point x such that g (x) = (x 0 ) . Elimination of
these coordinate terms is equivalent to setting x − x 0 = 0 ,
which means that it is essentially assumed that x = x 0 is
exactly known.
5.2. The treatment of leveling observations
There are two ways to treat and model leveling observations. In the classical approach observed height differences
∆h are converted to potential differences ∆W = − g∆h ,
utilizing observed or independently predicted gravity values,
the conversion being a finite approximation to the relation
g = − ∂W , where h is height along the plumb line. These
∂h
differences are summed along each leveling route to produce an "observed" potential difference ∆W =W B −W A
A different linearization approach is based on the concept of
telluroid mappings (Grafarend, 1978) which utilize observations at x to obtain the approximate coordinates x 0 . The
procedure corresponds to the elimination of the term
(x − x 0 ) from a set of observations to obtain a "reduced"
observations that depends only on T and gradT . We shall
give only the application to observations of gravity where
the horizontal position is held fixed ( λ = λ 0 , b = b 0 ) while
7
between the two endpoints A and B . The observation is
then related to the disturbing potential by
These can be solved for rP , rQ to produce the observed
"height" difference
∆W − ∆W 0 =
= T (x0B )(x B − x0B ) − T (x0A )(x A − x0A ) + T (x0B ) − T (x0A ) ,
(50)
∆W 0 =U (x 0B ) −U (x 0A )
∆hPQ = rP − rQ =
T
nTL (x L − x P ) n L (x L − xQ )
−
=
nTL n P
nTL nQ
 1
nTL xQE nTL x EP
1 
= T
− T nTL x L + T
−
 n L n P n L nQ 
n L nQ nTL n P


If leveling endpoints are isolated, i.e. not part of a 3-dimensional geodetic network the (x A − x 0A ) and (x B − x 0B ) terms
should be related to geometric heights using fixed values
λ0 , b 0 , for geodetic longitude and latitude, in which case
m A = m A (λ0 , b 0 ) , m B = m B (λ0 , b 0 ) and
(55)
This model is too complicated to be practical and furthermore contains the position vector of the level x L , which
may not be even approximately known. It can be simplified
by assuming that n L has the direction of 12 (n P + n Q ) , im-
x A − x 0A = ( H A − H A0 )m A ,
plying that
(51)
n TL n P = n TL n Q , and leading to the simpler
x B − x 0B = ( H B − H B0 )m B ,
model
A more precise model for leveling can be introduced on the
basis of the individual height differences produced by setting the level at point x L , and the staff at points x P and
∆h PQ =
xQ .
If x EP , x QE are the projections on the reference ellipsoid of
1
(n P + n Q ) T ( x Q − x P )
1+ n TP n Q
(56)
x P , x Q , respectively, H P , H Q are the geometric heights
(above the ellipsoid) and m P , m Q the corresponding unit
vectors normal to ellipsoid, it holds that
xr
x P = x EP + H P m P ,
r
x Q = x QE + H Q m Q
(57)
n
and the model becomes
x
∆h PQ =
H
m
1
(n P + n Q ) T (x QE + H Q m Q − x EP − H P m P )
1+ n TP n Q
(58)
xE
which has the form ∆h PQ = ∆h PQ (n P , n Q , H P , H Q ) . Its
linearized form derived in appendix B is
∆h − ∆h 0 = ahP ( H P − H P0 ) + ahQ ( H Q − H Q0 ) +
If the corresponding vertical unit vectors are n L , n P and
+ cThP gradT (x0P ) + cThQgradT (x0Q ) .
n Q , and rP and rQ are the readings on the staff, the position vectors of the reading points are
x rP = x P + rP n P ,
x rQ = x Q + rQ n Q .
Perhaps a more reasonable assumption is that
g L = 12 (g P + g Q ) but the choice n L = 12 (n P + n Q ) made
(52)
here has the advantage that eliminates x L from the model.
A similar linearization scheme may be found in Milbert
(1988).
Taking into account the fact that the lines of sight from x L
to the reading points x rP , x rQ should be horizontal, i.e.,
perpendicular to vertical n L , we obtain the orthogonality
conditions
+ rP n P − x L ) = 0
(53)
n TL (x rQ − x L ) = n TL (x Q + rQ n Q − x L ) = 0
(54)
n TL (x rP
− x L ) = n TL (x P
(59)
5.3. Dynamic modeling
The passing from the static to dynamic modeling can be
more conveniently performed after the linearization, by
noting that position vectors x(t ) , geometric heights H (t ) ,
are functions of time t in a deformable earth, while the
8
distance d PQ when a planar approximation is used within a
disturbing potential T (x, t ) and its gradient gradT (x, t ) are
functions of both space and time. By setting
small area.
The gravity related signals T (x 0 , t ) and gradT (x 0 , t ) depend on the modeling of the time varying disturbing potential function T (x, t ) . This can be treated in a stochastic way
for both the time and the space argument, with the introduction of a space-time covariance function
x P (t ) = x P (t 0 ) + δ t x P (t ) ,
(60)
H P (t ) = H P (t 0 ) + δ t H P (t ) ,
where t 0 is a chosen reference epoch, position information
is separated into two parts: A deterministic unknown part
x P (t 0 ) , H P (t 0 ) and a displacement signal δ t x P (t ) ,
CT ( P, Q, t , t ′) = C (ψ PQ , | t ′ − t |) .
δ t H P (t ) . The three-dimensional displacements or the
height displacements have the character of signals because
they depend on corresponding displacement functions
A simpler and perhaps more appropriate choice is to follow
the same procedure as for the displacements and set
T (x, t ) = T (x, t 0 ) + T (x)(t − t 0 ) + δ t x( P, t ) = δ t x(x P (t 0 ), t ) ,
(61)
δ t H ( P, t ) = δ t H (x P (t 0 ), t ) ,
(66)
(67)
where T (x, t 0 ) and T (x) are treated as independent stochastic functions with known covariance functions
C T ( P, Q) and C T ( P, Q) .
where points are identified by their coordinates at the reference epoch t 0 .
Summarizing, the signals present in the linearized models
are of three types: "displacement signals" d related to the
Further model development depends on how the displacement functions are modeled. Since crustal motion is a slow
process (apart from the occurrence of earthquakes), it can be
sufficiently modeled in an analytical way with respect to
time, e.g.,
functions x (P ) or H (x) , "gravity at reference epoch" signals g related to the disturbing potential T (x, t 0 ) and
"gravity variation" signals q related to the rate of the dis-
δ t x(x, t ) = δ t x0 (x, t ) + δ t x1 (x, t )(t − t0 ) +
turbing potential T (x) . When all signals are treated as stochastic parameters the mixed linear model (20) takes the
form
+ δ t x 2 (x, t )(t − t 0 ) 2 + =
= x (x)(t − t 0 ) + x(x)(t − t 0 ) 2 +
b = Ax + Gs + v = Ax + G d d + G g g + G q q + v ,
(62)
where the first time-independent term has been eliminated
since it is already contained in x = x(t 0 ) . Usually a piecewise linear trend between earthquake events is sufficient
and only the x (x) function is preserved. This function is
characterized by a strong similarity between neighboring
points (not separated by faults) since crustal motion is
known to be a spatially smooth process. This similarity in
proximity can be best described by a spatially correlated
stochastic model for the function x (x) . We assume zero
mean E{x (x)}= 0 , and a known matrix of covariance functions between the components of x ,
C x ( P, Q ) = E{x ( P )x (Q) T } .
G = [G d G g G q ] ,
δt
0
( x)(t − t
)+ H
5.4. Adjustment strategies
The linearized observations equations for the study of crustal deformation are formulated from the static linearized
models by introducing time dependence, replacing observables y with observed values y b and adding the observa-
(63)
0
)2
+
(68)
Although other modeling possibilities are possible we will
restrict ourselves to the above ones for the shake of demonstration.
tional errors v y :
A completely similar approach is taken for the vertical displacements
H (x, t ) = H (x)(t − t
d 
 
s = g 
q 
α b − α 0 = a αT P (x P − x 0P ) + (t − t 0 )a αT P x (x 0P ) +
(64)
+ a αTQ (x Q − x 0Q ) + (t − t 0 )a αT Q x (x Q0 ) +
E{H ( P)}= 0 ,
C H ( P, Q ) = E{H ( P ) H (Q )} .
(65)
+ c αT P gradT (x 0P ) + (t − t 0 )c αTP gradT (x 0P ) + vα
The covariance functions C x and C H are assumed to be
stationary and isotropic, i.e. to depend only on the spherical
distance ψ PQ between the two points, or their horizontal
ζ b − ζ 0 = a ζTP (x P − x 0P ) + (t − t 0 )a ζTP x (x 0P ) +
+ a ζTQ (x Q − x Q0 ) + (t − t 0 )a ζTQ x (x Q0 ) +
9
(69)
+ c ζTP gradT (x 0P ) + (t − t 0 )c ζTP gradT (x 0P ) + vζ
This means that coordinates of isolated observation points
cannot be determined like the ones in a geodetic network.
The same holds for heights at points not belonging in a leveling network. The signals do not impose a determinability
problem, in theory at least, but a very crucial point from the
practical point of view is whether their predicted values are
obtained from information contained mainly in the observations or in their adopted covariance functions. A related
problem is the separability of various different signals ( d ,
g , q ) present in the same observations, namely whether the
separation is a result of the experimental design (i.e. the
form of the matrices G d , G g , G q ) or of their adopted
(70)
s b − s 0 = a TsP (x P − x 0P ) + (t − t 0 )a TsP x (x 0P ) +
+ a TsQ (x Q − x Q0 ) + (t − t 0 )a TsQ x (x Q0 ) + v s
(71)
g b − g 0 = a Tg (x − x 0 ) + (t − t 0 )a Tg x (x 0 ) +
+ c Tg gradT (x 0 ) + (t − t 0 )c Tg gradT (x 0 ) + v g
(72a)
covariance functions. The choice of covariance functions is
a critical problem, because they cannot be determined by
sampling from directly observed signal values (available
observations depend simultaneously on all of them).
or when a telluroid mapping x → x 0 with W (x) = U (x 0 ) is
involved
∆g b = −n T0 gradT − (t − t 0 )n T0 gradT −
−
n T0 Mm T
n T0 Mm T
−
−
+ vg
(
t
t
)
0
n T0 m γ
n T0 m γ
The computational burden should be also taken into account. The simultaneous treatment of all available observations leads to extensive computations involving the inversion of large matrices. Sequential treatment of the data and
deviations from the rigorous sequential solution are sometimes adopted for computational convenience in the hope
that the derived solution is close to the theoretically optimal
one.
(72b)
Λb − λ 0 = a TΛ (x − x 0 ) + (t − t 0 )a TΛ x (x 0 ) +
+ c TΛ gradT (x 0 ) + (t − t 0 )c TΛ gradT (x 0 ) + v λ
(73)
Two different sets of observations play a different role in
the data analysis. Observations at isolated but densely distributed points in the studied area contribute to the determination by interpolation of the functions giving rise to the
signals. Observations at network points contribute to the
determination of position and position variation. This leads
to a question with respect to the displacement signals. These
can be treated as independent deterministic signals at network points, ignoring their dependence on a smooth displacement function, but their interpolation outside the network or their treatment at isolated points calls for the use of
a stochastic model. The simultaneous use of two different
models for the same displacement signals is very unsatisfactory (not to say unacceptable) from a theoretical point of
view but it has been implemented in practice, even indirectly by the use of independent coordinate unknowns at the
different epochs of repeated observational campaigns.
Φ b − φ 0 = a TΦ (x − x 0 ) + (t − t 0 )a TΦ x (x 0 ) +
+ c TΦ gradT (x 0 ) + (t − t 0 )c TΦ gradT (x 0 ) + vφ
(74)
∆W b − ∆W 0 = T (x 0B )(x B − x 0B ) + (t − t 0 ) T (x 0B )x (x 0B ) −
− T (x 0A )(x A − x 0A ) + (t − t 0 ) T (x 0A )x (x 0A ) −
(75)
+ T (x 0B ) − T (x 0A ) + (t − t 0 )T (x 0B ) − (t − t 0 )T (x 0A ) + vW
or
∆h b − ∆h 0 = a hP ( H P − H P0 ) + (t − t 0 )a hP H (x 0P ) +
+ a hQ ( H Q − H Q0 ) + (t − t 0 )a hQ H (x 0Q ) +
Coming to the signal separability problem, we note that displacements d and gravity variations q are physically interrelated and they cannot be effectively separated since they
have similar effects on observations: It is impossible to say
whether the observed change is due to the motion of a point
within an invariant gravity field, or to the variation of the
gravity field at a point which does not move at all, or to a
combination of the two effect. The actual modeling of the
interrelation between the two requires knowledge of the
internal masses and (in principle) global data coverage. An
approach that has been suggested is to consider an invariant
gravity field with respect not to an earth-fixed global reference frame but with respect to a local one moving along
with the network and the surrounding area (Reilly, 1981).
+ c ThP gradT (x 0P ) + (t − t 0 )c ThP gradT (x 0P ) +
+ c ThQ gradT (x Q0 ) + (t − t 0 )c ThQ gradT (x Q0 ) + v h .
(76)
In all the above relations, coordinates (x P , x Q , x) refer to
the reference epoch t 0 .
The derivation of the linearized observation equations (68)
does not lead to a straightforward adjustment, even if the
problem of choosing the proper models for the functions T ,
T , x or H has been effectively solved. Here we will assume a purely stochastic model for these functions. We shall
point out and discuss the main problems involved.
Keeping all the above problems in mind we shall give a rigorous sequential approach to the adjustment of observations
at a network (horizontal, three-dimensional or vertical) in
combination with observations in a dense network of isolated points in the sane area. The coordinates of isolated
points are assumed known with sufficient accuracy so that
Deterministic parameters should be included only when
their expected influence on the observations is significant
(above the noise level) and they could be determined by
standard least squares if the signals were exactly known.
10
they do not appear as unknowns. The same can be done for
the displacement signals at isolated points, though this approach is not followed here in the demonstrated algorithm.
~
b1 = A 1 x + G 1δs 1 + v 1 ,
The structure of the observation equations (98) can be more
easily seen when they are separated into two sets, one b 1
0
δs1 = s1 − sˆ 10 ~ δQ 11 = Q 11 − Q11
.
for network points and one b 2 for isolated points:
STEP 4: Solution for the reduced model :
b 1 = A 1 x + G 1d d 1 + G 1g g 1 + G 1q q 1 + v 1 =
~
T
M 11 = G 1δQ11G 1T + P1−1 = M 11 − M 12 M −221 M 12
d 1 
1
1
1 
= A 1 x + [G d G g G q ]g 1  + v 1 = A 1 x + G 1s 1 + v 1
q 1 
(82)
(83)
where
(77)
M 11 = G 1Q11G 1T + P1−1 ,
(
~ −1
xˆ = A 1T M 11
A1
b 2 = G 2d d 2 + G 2g g 2 + G 2q q 2 + v 2 =
)
−1
(78)
M 12 = G 1Q 12 G T2 ,
(84)
~ −1 ~
A 1T M 11
b1
~
δsˆ 1 = δQ 11G 1T M −1 (b1
d 2 
= [G d2 G 2g G q2 ]g 2  + v 2 = G 2 s 2 + v 2
q 2 
~
b1 = b1 − G 1sˆ 10 ,
(85)
− A 1 xˆ ) .
STEP 5: Prediction of residual signal for s 2 (contribution
~
from b1 :
where d , g and q are the vectors of displacement, gravity
and gravity variation signals respectively.
T
T
− Q 22 G T2 (G 2 Q 22 G T2 + P2−1 )G 2 Q12
Q δs 2δs1 = Q 12
(86)
The two sets contain no common unknowns or signals, but
they cannot be treated independently due to the existing non
zero cross covariance matrices C d1d 2 , C g1g 2 , C q1q 2 . It is
~ −1 ~
−1
δsˆ 2 = Q δs 2δs1 δQ 11
δsˆ 1 = Q δs 2δs1 G 1T M 11
(b1 − A 1 xˆ ) .
(87)
this correlation which allows information about the underlying functions x , T , T to pass from the isolated to the
network points. The observation equations have the form of
the mixed linear model b = Ax + Gs + v , s ~ σ 2 Q ,
STEP 6: Reconstruction of model signals:
sˆ 1 = sˆ 10 + δsˆ 1 ,
(88)
STEP 7: Prediction of new signals:
v ~ σ 2 Q v = σ 2 P −1 , with solution presented in appendix C.
Taking into account the special form of the model (77), (78)
which is of the form
 b 1   A 1  G 1
  =  x + 
b 2   0   0
sˆ 2 = sˆ 02 + δsˆ 2 .
−1
T
δsˆ 1 =
sˆ ′ = Q s′s 2 Q −221 sˆ 02 + (Q s′s1 − Q 22 G T2 Q 12
)δQ 11
(89)
= Q s′s 2 G T2 M −221 b 2 +
0   s1   v1 
  +   ,
G 2  s 2   v 2 
~
T
+ (Q s′s1 − Q 22 G T2 Q12
)G T M −1 (b1 − A 1 xˆ ) .
(79)
 s 1  Q 11
 ~ T
s 2  Q 12
Q 12 
,
Q 22 
 v 1  P1−1
v  ~ 
 2   0
0 

P2−1 
The above solution is a "rigorous sequential solution"
(Dermanis, 1986) identical with the non-sequential solution
from the algorithms described in appendix C.
the solution can be described in a sequential scheme as follows:
An alternative to the optimal common adjustment that has
been commonly followed is to arrive at a suboptimal solution in a two step procedure: In the first step the b 2 obser-
STEP 1: Prediction of s 2 from b 2 only:
vations are used to obtain estimates of the signals s 2 (in
fact only g 2 , q 2 because d 2 cannot be detected in isolated
M 22 =
G 2 Q 22 G T2
+ P2−1 ,
sˆ 02
=
Q 22 G T2 M −221 b 2
.
points and d 2 = 0 is assumed), as well as to predict the
(80)
"new" signals s 1 (i.e. g 1 , q 1 ), at the network points, thus
STEP 2: Prediction of s 1 from ŝ 02 :
ignoring the availability of the b 1 observations which also
contain information on s 1 . In the second step the signals s 1 ,
are fixed to their predicted values from the previous step,
their effect is subtracted from the observed values b 1 , and
the adjustment is carried out on the basis of the reduced observations
sˆ 10 = Q 12 Q −221 sˆ 02 = Q 12 G T2 M −221 b 2 ,
(81)
0
T
Q11
= Q sˆ 0sˆ 0 = Q 12 G T2 M −221 G 2 Q12
.
1 1
STEP 3: Reduction of the model for b 1 to the new form
11
b 1R = b 1 − G 1g g 1 − G 1q q 1 = A 1 x + G 1d d 1 + v 1
(90)
Since horizontal position cannot be determined from such
observations we will assume in eq. (91) that
This approach is computationally attractive but it suffers
from two defects: (a) it leads to suboptimal solutions (i.e.
with larger mean square errors) (b) the models used in the
two steps for the signals (stochastic in the first, deterministic
in the second) are inconsistent.
x − x 0 = m( H − H 0 ) = m∆H ,
(92)
in which case the observation equation for gravity becomes
In conclusion, the answers to problems related to modeling,
adjustment strategy and adoption of computationally attractive compromises, cannot be answered by theory, but
mainly from repeated practical experience obtained from
trial and error.
g b − g 0 = a g ∆H + (t − t 0 )a g H (x 0 ) +
+ c Tg gradT (x 0 ) + (t − t 0 )c Tg gradT (x 0 ) + v g
(93)
with
5.5. An example: Combination of leveling and gravity
a g = a Tg m = c Tg
Geometric uplift of the earth surface can be separated into
two parts. One part corresponds to the variation of orthometric height, i.e. to motion with respect to the geoid,
while the other refers to the variation of the geoid undulation, i.e. to the motion of the geoid with respect to the reference ellipsoid. The corresponding geodetic techniques for
the determination of these two components of vertical displacement are, respectively, repeated leveling and repeated
gravity observations. Strictly speaking, dynamic leveling
provides potential values W , which are converted to
"heights" when divided by a constant mean value γ 0 of gravity in the area. These (dynamic) heights are only an
approximation to orthometric heights and heights above the
geoid along the ellipsoidal normal.
∂
∂
m = −n T0
m = −n T0 Mm ,
∂x
∂x
(94)
∂
,
M=
∂x
where the values of a Tg and c Tg from appendix A have been
used.
Eq. (93) is appropriate for gravity observations at the leveling points. For gravity observations at isolated points within
the same area, the heights cannot be determined and
H = H 0 must be set to obtain ∆H = 0 and
g b − γ 0 = (t − t 0 )a g H + c Tg gradT + (t − t 0 )c Tg gradT + v g .
Repeated gravity observations can be used to obtain the
"velocity" of gravity g , or of gravity anomalies ∆g , which
(95)
can be integrated to obtain velocities ζ of the geoid undulations ζ . This procedure corresponds to a solution of a
geodynamic boundary value problem, presented by Sanso
and Dermanis (1981) and Heck (1981).
For leveling the observation equation is eq. (76):
∆h b − ∆h 0 = a hP ∆H P + (t − t 0 )a hP H P +
Apart from the above classical approach to the combination
of independent leveling and gravimetric work, an integrated
approach is also possible. A first step is the use of gravity
data in the leveling area for the prediction of gravity values
necessary for the conversion of primarily observed "height
differences" ∆h into the geopotential differences
∆W = g∆h , which will enter in the adjustment of the leveling network. However in high precision leveling aiming to
the study of crustal motion it is standard practice to make
gravity observations along the leveling route, instead of relying on predicted gravity, or even worst, in theoretical
normal gravity values γ in place of g . Further information
on the above approach can be found in Heck and Mälzer
(1983, 1986).
+ a hQ ∆H Q + (t − t 0 )a hQ H Q +
+ c ThP gradTP + (t − t 0 )c ThP gradTP +
+ c ThQ gradTQ + (t − t 0 )c ThQ gradTQ + v h .
(96)
The parameters of the above equations can be distinguished
to deterministic parameters x , displacement signals d ,
gravity signals g , and gravity variation signals q :
 
 H 
 
 P
 ∆H 
 
P





x=
, d =  H Q  ,


 
 ∆H Q 


 
 H 


 


A purely integrated approach will be outlined here, based on
the precise models of leveling and gravity observations,
which make use of geometric heights H above the reference ellipsoid as parameters. The observation equation for
gravity is eq. (72a):
g b − g 0 = a Tg (x − x 0 ) + (t − t 0 )a Tg x (x 0 ) +
+ c Tg gradT (x 0 ) + (t − t 0 )c Tg gradT (x 0 ) + v g
x = mH
(91)
12
 
gradT 
P

 


g = gradTQ  ,
 


 gradT 
 


 
gradT 
P

 


q = gradTQ 
 


 gradT 
 


(97)
We can differentiate between observations at leveling points
(subscript 1) and observations in isolated points (subscript
2)


 H 
 P
d1 =   ,
 
H Q 
 




gradT 
P

g1 =   ,


gradTQ 
 




gradT 
P

q1 =   ,


gradTQ 
 


 
g 2 = gradT  ,
 

d 2 =  H  ,
 

 c T
hP



1
T
G g =  c gP


 0


(98)
 
q 2 = gradT  ,
 
to obtain two sets of observation equations
b 1 = A 1 x + G 1d d 1 + G 1g g 1 + G 1q q 1 + v 1 =
b2 =
G d2 d 2
+ G 2g g 2
(100)
+ G q2 q 2
+ v 2 = G 2s 2 + v 2
G 2g
(101)



0
 ∆h b − ∆h 0 







 b
0 
b 1 =  g P − γ P  A 1 = 0




 b
0 
0
 gQ −γ Q 






a hP
a hQ
a gP
0
0
a gQ
c TgQ


=  (t g − t 0 )c Tg



=  c Tg

(t h − t 0 )c ThQ
0
(t gQ − t 0 )c TgQ

 (104)







 ,







.


(105)

0


0 ,


0


Appendix A: Linearization of basic geodetic
observables
In this appendix the coefficients a , c of the linearized observation equations (35)-(40) will be given adopted from
Dermanis (1985b) and Dermanis and Rossikopoulos (1988).
For any observable y = y (x, , g (x) ) , the linearized observation equations for the observed value y b have the form
 
b 2 =  g b − γ 0 
 

G d2 =  (t g − t 0 )a Tg

0
G 2g
The sequential adjustment algorithm described in section
(5.4) can now be applied.
where

 (t − t )a
0
h
hP



G 1d =  (t gP − t 0 )a gP


0








 ,






 (t − t )c T
0 hP
h



G 1q =  (t gP − t 0 )c TgP


0



(99)
= A 1 x + G 1s 1 + v 1
c ThQ
y b − y (x 0 , , (x 0 ) ) = a Ty (x − x 0 ) + + c Ty gradT (x 0 ) .
(102)
(A1)
(t h − t 0 )a hQ
0
(t gQ − t 0 )a gQ

,













The coefficients rows a Ty , c Ty , for the various observables
take the following values:
c αT
ξ
=  22
d
−
sin φξ 2 − cos φξ 3
 
− sin φξ 1
0 
0 
cos φξ 1

ξ1
d2
sin λ

 γ cos φ

1
×  sin φ cos λ
γ
 cos φ cos λ


(103)
ξ
a αT P =  22
d
13
−
ξ1
d
2
−
cos λ
γ cos φ
1
sin φ sin λ
γ
− cos φ sin λ
0
−ξ3
ξ2



1
− cos φ 

γ
− sin φ 

0
0

0 ×
0 0
0
∂ 0

0 (−R 0 ) + c αT
(x P ) ,
∂x
0
(A2)
R0 = R
(
λ (x 0P ), φ (x 0P )
ξ
a αT Q =  22
d
 cT =  1 23
 ds
−
ξ1
d
2
−
ds 2
 sin λ
 γ cos φ

sin φ cos λ
×

γ
 cos φ cos λ


 a TQ =  1 23
 ds
c TΛ
 sin λ
=
 γ cos φ
2 3
ds
cos λ

0 
γ cos φ

sin φ sin λ
cos φ 
−
γ
γ 
− cos φ sin λ − sin φ 

 0
−
cos λ
−
γ cos φ
 sin φ cos λ
c TΦ = 
γ

a TΦ = c TΦ

0
0
= [− cos φ cos λ
a Tg
=
∆g = g − γ 0 = a Tg m
(A6)
∂ 0
(x )
∂x
+ c Tg gradT (x 0 ) =
γ (x 0 )n T0 (x 0 )m
T (x 0 ) − n T0 (x 0 )gradT (x 0 )
1
1 + n TP n Q
(n P + n Q ) T (x Q − x P ) =
1
1 + n TP n Q
(n P + n Q ) T (x QE + H Q m Q − x EP − H P m P )
is carried out by fixing x EP (b P , λ P ) , x QE (bQ , λ Q ) and
(A12)
m P (b P , λ P ) , m Q (bQ , λ Q ) , i.e. by assuming that the horizontal position of leveling points is known. The linearization follows the scheme
(A13)
 ∂∆h
∆h − ∆h = 
 ∂H P
0
A different type of linearization for g utilizes knowledge of
W (x) and the telluroid mapping x → x 0 ( H → H 0 ) defined by W (x) = U (x 0 ) . The term (x − x 0 ) in the linearized single point observation (38) for g is related to the
disturbing potential T , by noting that
cos b 0 cos λ 0 


x − x 0 = ( H − H 0 )  cos b 0 sin λ 0  = ( H − H 0 )m
 sin b 0



γ (x 0 )n T0 (x 0 )m
(B1)
(A11)
− sin φ ]0 =
m.
The linearization of the leveling model (58) which in terms
of gravity has the form
=
− cos φ sin λ
γ (x 0 )
Appendix B: Linearization of leveling observations
(A10)
−n T0
T (x 0 )
n T0 (x 0 )M (x 0 )m
∆h PQ =
cos φ 

γ 0
T (x 0 )
(A17)
(A8)
−
γ (x 0 )n T0 (x 0 )m
m≈
With this value of (x − x 0 ) the linearized relation for g
becomes
(A5)
(A9)
sin φ sin λ
γ
T (x 0 )
(A16)
(A7)
∂ 0
(x )
∂x
c Tg
x − x 0 = ( H − H 0 )m =
=−
d 
 R0
s2 0
(A15)
which in view of W (x) = U (x 0 ) and = −γn 0 ( n 0 being
the unit vector in the normal vertical direction), leads to the
relation
−
∂ 0
d 
(−R 0 ) + c T
(x P )
2
∂x
s 0
∂U 0
(x )(x − x 0 ) + T (x 0 ) =
∂x
= U ( x 0 ) + ( H − H 0 ) T ( x 0 )m + T ( x 0 )
0 0
− 3 0 ×
2 0 0
∂ 0
(x )
∂x
a TΛ = c TΛ
c Tg
2
W ( x) = U ( x) + T ( x) ≈ U ( x 0 ) +
(A4)
sin 3 2 − cos 3 3
d 
− sin 31
s 2  0 

cos 31
−
2
2 3
ds
)

0 R 0 ,
0
2 3
 a TP =  1 23
 ds
(A3)
*
 ∂∆ h

 ( H P − H P0 ) + 
 ∂H Q
0

*

 ( H Q − H Q0 ) +

0
 ∂∆ h 
 ∂∆h 
 gradT (x 0Q ) .
 gradT (x 0P ) + 
+ 


∂
∂
g
g
 P 0
 Q 0
(B2)
The derivatives with respect to H P and H Q are implicit
ones and must be calculated from the explicit ones taking
into account the dependence n = n(g ) = − 1
g g , and
(A14)
g = g (H ) . The explicit derivatives are easily found to be
where m 0 = m is supposed to be known, and
14
∂∆h
1
=
(n P + n Q ) T m P ,
∂H P 1 + n TP n Q
∆h 0 =
1
T
1 + n 0P n Q0
(n 0P + n Q0 ) T (x Q0 − x 0P )
(B11)
(B3)
∂∆h
1
=−
(n P + n Q ) T m Q
T
∂H Q
1+ n PnQ
∂n Q
∂n P
1
=
(n P n TP − I ) ,
∂g P
gP
∂g Q
=
n 0P = n 0 (x 0P ) ,
1
(n Q n TQ − I )
gQ
c ThP =
(B4)
c ThQ =
(B5)
∂H Q
=
∂g Q ∂x Q
∂x Q ∂H Q
=
∂g Q
∂x Q
1
(B12)
T
T
γ (x 0P )(1 + n 0P n Q0 )
(x Q0 − x 0P − ∆h 0 n Q0 ) T (n 0P n 0P − I )
(B13)
∂g P
∂g ∂x P
∂g
= P
= P mP ,
∂H P ∂x P ∂H P ∂x P
∂g Q
n 0Q = n 0 (x Q0 )
1
T
T
γ (x Q0 )(1 + n 0P n Q0 )
0
0
− ∆h 0 n 0P ) T (n Q0 n Q
− I)
(x 0P − x Q
(B14)
mQ
a hP =
(n 0P + n Q0 ) T m P
and the implicit ones are
T
1 + n 0P n 0Q
+ c ThP M (x 0P )m P ,
(B15)
 ∂∆h

 ∂H P
T
∂
∂  ∂U 
=
M=
 ,

∂x ∂x  ∂x 
*

∂∆h ∂∆h ∂n P ∂g P
 =
+
,
∂H P ∂n P ∂g P ∂H P

M ij =
∂ U
∂x i ∂x j
2
(B6)
 ∂∆h

 ∂H Q

*

∂n Q ∂g Q
 = ∂∆h + ∂∆h

∂H Q ∂n Q ∂g Q ∂H Q

a hQ = −
The derivatives with respect to g can be obtained using the
chain rule
∂∆h ∂∆h ∂n P
=
,
∂g P ∂n P ∂g P
∂∆h ∂∆h ∂n Q
=
∂g Q ∂n Q ∂g Q
(n 0P + n Q0 ) T m Q
T
1 + n 0P n Q0
+ c ThQ M (x Q0 )m Q .
Appendix C: Adjustment algorithms for the
mixed linear model.
(B7)
Given the mixed linear model of the form
b = A x+G s+ v
n×1
where
∂∆h
1
=−
(x Q − x P ) T (n P + n Q )n TQ +
2
T
∂n P
g P (1 + n P n Q )
+
=
1
g P (1 + n TP n Q )
1
g P (1 + n TP n Q )
n× q q×1
n×1
(C1)
s ~ (v s , C s ) ,
(C2)
C sv = E{(s − v s ) v T } = 0
Cs = σ 2Qs ,
(B8)
C v = σ 2Q v
(C3)
the adjustment for the estimation of the deterministic parameters x and the prediction of the random parameters s
and v can be performed using alternative algorithms which
of course give identical optimal results. The choice of algorithm in each specific application depends on numerical
considerations. One tries in general to implement the inversion of matrices with smaller dimensions.
with a similar expression for Q . It follows after some rather
lengthy computations that
∆h − ∆h 0 = a hP ( H P − H P0 ) + a hQ ( H Q − H Q0 ) +
+ c ThP gradT (x 0P ) + c ThQ gradT (x Q0 )
n× m m×1
v ~ (0, C v ) ,
(x Q − x P ) T =
( x Q − x P − ∆h n Q ) T
(B16)
Algorithm A:
(B9)
M = GQ s G T + Q v
(C4)
Q xˆ = ( A T M −1 A) −1
(C5)
xˆ = Q xˆ A T M −1b
(C6)
where:
x 0P = x EP + H P0 m P ,
x Q0 = x QE + H Q0 m Q ,
(B10)
15
sˆ = Q s G T M −1 (b − Axˆ )
(C7)
vˆ = Q v M −1 (b − Axˆ )
ˆ , C
ˆ , C
ˆ of the covariance matrices C ,
Estimates C
xˆ
xˆ
xˆ sˆ
sˆ
C xˆ sˆ , C sˆ , can be obtained by multiplying the "cofactor"
(C8)
matrices Q xˆ , Q xˆ sˆ , Q sˆ , respectively, with the estimate
Q xˆ sˆ = −Q xˆ ( A T M −1G )Q s
(C9)
σˆ 2 =
−1
Q sˆ = Q s − Q s (G M G )Q s +
T
+ Q s (G T M −1 A)Q xˆ ( A T M −1G )Q s
Q vˆ = Q v M −1Q v − Q v M −1 AQ xˆ A T M −1Q v
vˆ T Q −v1 vˆ + sˆ T Q s−1sˆ
.
n−m
(C27)
(C10)
When the covariance matrices C s , C v , are assumed to be
(C11)
completely known (no common unknown factor σ 2 ) the
above algorithms still hold by replacing all cofactor matrices Q with the corresponding covariance matrices C .
However the estimate σ̂ 2 should still be obtained and tested
N G = G T Q −v1G + Q s−1
(C12)
−1 T
Q xˆ = [ A T Q −v1 A − G T Q −v1 AN G
A Q −v1G ] −1
(C13)
−1 T
xˆ = Q xˆ A T Q −v1 (b − GN G
G Q −v1b)
(C14)
statistically for σ 2 = 1 , in order to evaluate the overall validity of the model.
Prediction of new signals s ′ with E{s ′} = 0 not contained
in the model can be performed when the cross-covariance
matrix C s′s = E{s ′s T } is known. It is also assumed that the
new signals are uncorrelated to the observational errors, i.e.,
C s′v = E{s ′v T } = 0 . The prediction is given by
−1 T
sˆ = N G
G Q −v1 (b − Axˆ )
(C15)
sˆ ′ = C s′s C s−1sˆ
vˆ = b − Axˆ − Gsˆ
(C16)
The prediction errors e = sˆ ′ − s ′ have covariance matrix
−1
Q xˆ sˆ = −Q xˆ A T Q −v1GN G
(C17)
C e = C s′ − C s′s G T C −v1Q vˆ C −v1GC Ts′s
−1
−1 T
−1
+ NG
Q sˆ = N G
G Q −v1 AQ xˆ A T Q −v1GN G
(C18)
where C s′ is the covariance matrix of s ′ .
Algorithm B:
−1 T
Q vˆ = [I − ( A − Q xˆ + GQ sˆxˆ A T Q −v1 ][Q v − GN G
G ]×
× [I − ( A − Q xˆ + GQ sˆxˆ A
T
Q −v1 ]T
(C28)
(C29)
References
(C19)
Bencini, P., A. Dermanis, E. Livieratos and D. Rossikopoulos (1982): Crustal Deformation at the Friuli Area
from Discrete and Continuous Geodetic Prediction
Techniques. Bollettino di Geodesia e Scienze Affini,
XLI, 2, 137-148.
Bruns, H. (1978): Die Figur der Erde. Publ. Preuss. Geod.
Inst., Berlin.
Dermanis, A. (1976): Probabilistic and Deterministic Aspects of Linear Estimation in Geodesy. Report No.
244, Department of Geodetic Science, The Ohio
State University.
Dermanis, A. (1978): Adjustment of Geodetic Networks in
the Presence of Signals. Proceedings, International
School of Advanced Geodesy, 2nd Course: "Spacetime Geodesy, Differential Geodesy and Geodesy in
the Large". 'Ettore Majorana' Centre for Scientific
Culture, Erice, Sicily, May 1978. Bollettino di Geodesia e Scienze Affini, 38, 4, pp. 513-539.
Dermanis, A. (1981): Geodetic Estimability of Crustal Deformation Parameters. Quaterniones Geodaesiae,
vol. 1, no. 2, 159-169.
Dermanis, A. (1983): Theory and Applications of Collocation in Surveying. In: M. Unguendoli (ed.): "Techniche moderne di analisi dei dati geodetici con particolare riguardo alla collocazione", pp. 37-67, Editice
CLUEB, Bologna.
Algorithm C: Same as above with the following alternative
relations
Q sˆ = [G T Q −v1G − G T Q −v1 A( A T Q −v1 A) −1 A T Q −v1G + Q s−1 ] −1
(C20)
sˆ = Q sˆ [G T Q −v1b − G T Q −v1 A( A T Q −v1 A) −1 A T Q −v1b] (C21)
xˆ = ( A T Q −v1 A) −1 A T Q −v1 (b − Gsˆ )
(C22)
vˆ = b − Axˆ − Gsˆ
(C23)
Q xˆ sˆ = −( A T Q −v1 A) −1 A T Q −v1GQ sˆ
(C24)
Q xˆ = ( A T Q −v1 A) −1 +
+ ( A T Q −v1 A) −1 A T Q −v1GQ sˆ G T Q −v1 A( A T Q −v1 A) −1
(C25)
Q vˆ = R − RGQ sˆ G T Q −v1 R ,
(C26)
R = [I − A( A T Q −v1 A) −1 A T Q −v1 ]
16
Grafarend, E.W. (1978): The definition of the telluroid.
Bulletin Géodésique, 52, 25-37.
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Acknowledgement
This work was prepared during the author's stay at the Department of Earth Sciences, Ibaraki University. Part of the paper has
been contained in lectures presented to the Department of Earth Sciences, Ibaraki University and a lecture at the Department of
Geophysics, University of Kyoto. The author would like to thank his host Prof. Yoichiro Fujii for motivating discussions and
his valuable comments. Financial support of the author's stay in Japan by the Japanisches-Deutsches Zentrum Berlin is
gratefully acknowledged.
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