Sect_7.1.3

7.1.3 Introduction

LEAST SQUARES PROCEDURES APPROPRIATE FOR GPS SURVEYING

The functional models, or mathematical observation equations, appropriate for GPS survey adjustments are presented in section 7.2. The degree of model sophistication can vary considerably, though the basic models for GPS survey processing are now well defined.

In the following discussions, the basic range equation, in which the geometric range ( ) is a function of the satellite and receiver coordinates (see Figure below), will be considered:

(7.1-19)

where:

Xⁱ, Yⁱ, Zⁱ are the coordinates of satellite i, and

X_j, Y_j, Z_j are the coordinates of site j.

Figure 1. The geometric range: receiver to satellite.

In the case of GPS surveying, the coordinates of the satellites are assumed to be known, and hence enter the observation equations as apriori information. (Typically they are evaluated from the broadcast elements within the Navigation Message -- see Table in section 3.3.3 for an outline of the computational procedure.) It is necessary to distinguish between two classes of parameters:

Geodetic parameters, the station coordinates in eqn (7.1-19), parameterised as Cartesian coordinate components (in the same reference system as the satellite coordinates), and represented geometrically in figure above.
Non-geodetic parameters, which for GPS surveying consist of the receiver clock bias parameters, satellite clock bias parameters, and phase ambiguity parameters. These are generally considered to be "nuisance" parameters as they do not appear explicitly in the basic range equation (eqn (7.1-19)), but must be included in GPS model equations in order to account for the measurement biases present in the data. (When data is double-differenced, the clock terms are eliminated from the double-differenced observation equations.)

The data types processed in GPS survey adjustments are pseudo-ranges and integrated carrier phase (in either the differenced or undifferenced mode). Both types are measured by GPS surveying receivers, however only pseudo-range data are measured by GPS navigation receivers (LANGLEY, 1991a; LANGLEY, 1993). Because of the much higher measurement precision of carrier phase observations, the pseudo-range data plays only a minor role in GPS baseline processing (section 7.3.3).

The basic steps in a GPS adjustment therefore are:

Set up the solution: select the appropriate functional model for the data type being processed and for the solution application. Compute the design matrix A.
Select the observations: may require some pre-processing (to correct for some known bias -- section 6.2) or data manipulation (for example, data differencing -- section 6.3.2), and sufficient in quantity and quality to estimate the parameters reliably.
Obtain approximate values of the parameters: in particular the geodetic parameters (for example, from a previous solution or an approximate computation) .
Define the quality of the observations: this requires the specification of the VCV matrix Q_{l l}.
Define the datum in such a way as to overcome any rank defects in the system of equations A^TPA.
Iterate the solution until convergence is reached.

Calculation of the Partial Derivatives

As indicated above, the mathematical adjustment model must be linear, hence the basic GPS observation model has to be linearised. This requires that the parameter set consist of corrections to approximate values, rather than the parameters in the functional model themselves -- for example the coordinates of stations may be known to an accuracy of tens of metres, or even better. The elements of the design matrix in eqn (7.1-1) are the partial derivatives of the observables with respect to the geodetic and non-geodetic parameters in the functional model. (Note that the partial derivatives of the observables with respect to the clock terms need not be considered as they cancel in double-differenced and triple-differenced observations.)

The partial derivative of carrier beat phase with respect to the site coordinates is obtained from eqn (7.1-19):

(7.1-20)

where:

_bjⁱ is the phase observation from satellite i to receiver j (in units of cycles),

U are the site parameters, which will be defined for convenience in the Cartesian coordinate system (X, Y, Z),

f_o is the nominally constant carrier frequency,

c is the speed of EMR in a vacuum, and

_jⁱ is the geometric range between satellite i and receiver j (Figure 1).

From eqns (7.1-19) and (7.1-20) the partial derivative of phase with respect to site coordinates can be written as:

(7.1-21)

These are the partial derivatives for the one-way phase (or range) observation equation. Partials for the double-differenced and triple-differenced observation equations are obtained by the appropriate differencing (section 6.3 and section 7.2).

The partial derivative of the phase observable with respect to an ambiguity parameter is unity.

Least Squares Solutions in GPS Surveying

The following stages in GPS phase data processing require some form of Least Squares adjustment:

Point-position solutions using pseudo-range data to obtain preliminary WGS84 coordinates and approximate receiver clock error estimates (for the correction of observation time-tags at the microsecond level -- see section 6.3.8).

Triple-difference carrier phase solution using phase data obtained by differencing the double-differences between successive epochs. As ambiguity parameters are eliminated, such a solution can give good apriori coordinates, but is not recommended for precise GPS phase data processing.

Perhaps some form of (polynomial) curve-fitting to observation residuals to permit cycle slip detection and repair during phase data pre-processing.

Double-differenced phase data solutions estimating both the station coordinates and the ambiguity parameters (as real-valued quantities).

Solutions that combine all GPS baseline results into a single campaign adjustment (without processing all the raw double-differenced data, as above, in a single step solution).

Solutions to integrate the GPS results into a conventional geodetic network, involving the distortion of the minimally constrained GPS network to fit the surrounding control network.

Determination of the transformation parameters between the GPS datum and the local geodetic datum.

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Xⁱ, Yⁱ, Zⁱ	are the coordinates of satellite i, and
X_j, Y_j, Z_j	are the coordinates of site j.

_bjⁱ	is the phase observation from satellite i to receiver j (in units of cycles),
U	are the site parameters, which will be defined for convenience in the Cartesian coordinate system (X, Y, Z),
f_o	is the nominally constant carrier frequency,
c	is the speed of EMR in a vacuum, and
_jⁱ	is the geometric range between satellite i and receiver j (Figure 1).