Come Together to Know The latest Technology & Computer System: Carrier phase tracking (surveying)

Utilizing the navigation message to measure pseudorange has been discussed. Another method that is used in GPS surveying applications is carrier phase tracking. The period of the carrier frequency times the speed of light gives the wave length, which is about 0.19 meters for the L1 carrier. With a 1% of wave length accuracy in detecting the leading edge, this component of pseudorange error might be as low as 2 millimeters. This compares to 3 meters for the C/A code and 0.3 meters for the P code.

However, this 2 millimeter accuracy requires measuring the total phase, that is the total number of wave lengths plus the fractional wavelength. This requires specially equipped receivers. This method has many applications in the field of surveying.

We now describe a method which could potentially be used to estimate the position of receiver 2 given the position of receiver 1 using triple differencing followed by numerical root finding, and a mathematical technique called least squares. A detailed discussion of the errors is omitted in order to avoid detracting from the description of the methodology. In this description differences are taken in the order of differencing between satellites, differencing between receivers, and differencing between epochs. This should not be construed to mean that this is the only order which can be used. Indeed other orders of taking differences are equally valid.

The satellite carrier total phase can be measured with ambiguity as to the number of cycles as described in CARRIER PHASE MEASUREMENT and CARRIER BEAT PHASE. Let $\ \phi(r_i, s_j, t_k)$ denote the phase of the carrier of satellite j measured by receiver i at time $\ t_k$ . This notation has been chosen so as to make it clear what the subscripts i, j, and k mean. In view of the fact that the receiver, satellite, and time come in alphabetical order as arguments of $\ \phi$ and to strike a balance between readability and conciseness, let $\ \phi_{i,j,k} = \phi(r_i, s_j, t_k)$ so as to have a concise abbreviation. Also we define three functions, : $\ \Delta^r, \Delta^s, \Delta^t$ which perform differences between receivers, satellites, and time points respectively. Each of these functions has a linear combination of variables with three subscripts as its argument. These three functions are defined below. If $\ \alpha_{i,j,k}$ is a function of the three integer arguments, i, j, and k then it is a valid argument for the functions, : $\ \Delta^r, \Delta^s, \Delta^t$ , with the values defined as

$\ \Delta^r(\alpha_{i,j,k}) = \alpha_{i+1,j,k} - \alpha_{i,j,k}$ ,

$\ \Delta^s(\alpha_{i,j,k}) = \alpha_{i,j+1,k} - \alpha_{i,j,k}$ , and

$\ \Delta^t(\alpha_{i,j,k}) = \alpha_{i,j,k+1} - \alpha_{i,j,k}$ .

Also if $\ \alpha_{i,j,k}\ and\ \beta_{l,m,n}$ are valid arguments for the three functions and a and b are constants then $\ ( a\ \alpha_{i,j,k} + b\ \beta_{l,m,n} )$ is a valid argument with values defined as

$\ \Delta^r(a\ \alpha_{i,j,k} + b\ \beta_{l,m,n}) = a \ \Delta^r(\alpha_{i,j,k}) + b \ \Delta^r(\beta_{l,m,n})$ ,

$\ \Delta^s(a\ \alpha_{i,j,k} + b\ \beta_{l,m,n} )= a \ \Delta^s(\alpha_{i,j,k}) + b \ \Delta^s(\beta_{l,m,n})$ , and

$\ \Delta^t(a\ \alpha_{i,j,k} + b\ \beta_{l,m,n} )= a \ \Delta^t(\alpha_{i,j,k}) + b \ \Delta^t(\beta_{l,m,n})$ ,

Receiver clock errors can be approximately eliminated by differencing the phases measured from satellite 1 with that from satellite 2 at the same epoch as shown in BETWEEN-SATELLITE DIFFERENCING. This difference is designated as $\ \Delta^s(\phi_{1,1,1}) = \phi_{1,2,1} - \phi_{1,1,1}$

Double differencing can be performed by taking the differences of the between satellite difference observed by receiver 1 with that observed by receiver 2. The satellite clock errors will be approximately eliminated by this between receiver differencing. This double difference is designated as $\ \Delta^r(\Delta^s(\phi_{1,1,1})) = \Delta^r(\phi_{1,2,1} - \phi_{1,1,1}) = \Delta^r(\phi_{1,2,1}) - \Delta^r(\phi_{1,1,1}) =$ $\ (\phi_{2,2,1} - \phi_{1,2,1}) - (\phi_{2,1,1} - \phi_{1,1,1})$ .

Triple differencing can be performed by taking the difference of double differencing performed at time $\ t_2$ with that performed at time $\ t_1$ . This will eliminate the ambiguity associated with the integral number of wave lengths in carrier phase provided this ambiguity does not change with time. Thus the triple difference result has eliminated all or practically all clock bias errors and the integer ambiguity. Also errors associated with atmospheric delay and satellite ephemeris have been significantly reduced. This triple difference is designated as $\ \Delta^t(\Delta^r(\Delta^s(\phi_{1,1,1})))$ .

Triple difference results can be used to estimate unknown variables. For example if the position of receiver 1 is known but the position of receiver 2 unknown, it may be possible to estimate the position of receiver 2 using numerical root finding and least squares. Triple difference results for three independent time pairs quite possibly will be sufficient to solve for the three components of position of receiver 2. This may require the use of a numerical procedure such as one of those found in the chapter on root finding and nonlinear sets of equations in Numerical Recipes ^[44]. Also see Preview of Root Finding. To use such a numerical method, an initial approximation of the position of receiver 2 is required. This initial value could probably be provided by a position approximation based on the navigation message and the intersection of sphere surfaces. Although multidimensional numerical root finding can have problems, this disadvantage may be overcome with this good initial estimate. This procedure using three time pairs and a fairly good initial value followed by iteration will result in one observed triple difference result for receiver 2 position. Greater accuracy may be obtained by processing triple difference results for additional sets of three independent time pairs. This will result in an over determined system with multiple solutons. To get estimates for an over determined system, least squares can be used. The least squares procedure determines the position of receiver 2 which best fits the observed triple difference results for receiver 2 positions under the criterion of minimizing the sum of the squares.

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Monday, July 20, 2009

Carrier phase tracking (surveying)

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