= AK

x

= AK

y

= AK

z

V

x

V

y

V

z

+X

(15)

+Y

(16)

+Z

(17)

= -(A'PA)

-1

(A'PX)

= -(A'PA)

-1

(A'PY)

= -(A'PA)

-1

(A'PZ)

K

x

K

y

K

z

where;

(18) (19) (20)

5

5t

5t^{2 }

5t

5t^{2 }

5t^{3 }

5t^{2 }

5t^{3 }

5t^{4 }

and the ' symbol represents the transpose matrix. For the moment consider the weight matrix P to be the Identity matrix I,

A'IA =

(21)

# A'IX =

x 1 x 1 t x 1 t 2

+

+x + x 2 t x 2 2 t 2

+x + x 3 t + x 3 3 t 2

+

+x + x 4 t x 4 4 t 2

+x + x 5 t + x 5 5 t 2

(22)

similar equations can be written for A'PY and A'PZ. VARIANCE-COVARIANCE WEIGHT MATRIX FOR GPS OBSERVATIONS

The variance-covariance matrix, P, is used to weight the contribution of each observation considering the span of time between the central observation point and the camera exposure

station.

The assumption is made that the five observations

are independent and, therefore,

observations

are

zero.

# Several

the co-variances between different choices for the

variances

were

considered.

The

first

choice

was

equal

weights.

This choice was not considered appropriate considering possible non-uniformity of the trajectory. A second choice was to compute the variances by giving more weight to the center

value of the interpolation, a central weight scheme.

The

justification

for

this

decision

is

based

on

the

increasing

difficulty to accurately between the central data

model point

a trajectory as the distance and its neighbors increases.

This fact was confirmed in Table 1. value is, therefore, greatest with

The weight for the decreasing weights

central for the

other data points as the central value increases.

time span from the data Several central weight

point to the systems were

tried including (Spilker 1980),

the use of the an estimate of

Geometric Dilution of Precision GPS relative accuracy, obtained

from the satellite geometry at the time of exposure.

# The

final

scheme

weights

the

data

points

as

a

binomial

expansion

technique.

The central variance was chosen to

The following formula (Sigma_{Lb}) follows:

for

the

variances

of

the

be 1.0 weight

cm^{2}. matrix

2^{2}*0.01 m^{2 }

0

0

2^{1}*0.01 m^{2 }

0

0

(23)

0

0

0

2^{0}*0.01 m^{2 }

0

0

0

2^{1}*0.01 m^{2 }

0

0

0

0

0

0