1998 - Draper Laboratory

H R# δρ R,GPS -H A# δρ A,GPSThis deviation in position space can be mapped into pseudorangespace by premultiplying by H R .δρ R,GPS - H R H A# δρ A,GPS(13)Comparison of Eqs. (12) and (13) emphasizes the symmetrybetween the two errors.These results might suggest that if the survey error is expected tobe dominant, then position corrections should be transferred. Ifthe pseudorange biases are expected to be dominant, thenpseudorange biases should be transferred. Fikes (Ref. [1]) makesa numerical assessment of these errors. We now wish to extendthat assessment to cases in which the active receiver (at the target)only uses a subset of the satellites available at the referencereceiver.Numerical Analysis ofPseudorange Bias Transferfor Relative TargetingIn particular, we wish to assess the numerical error associatedwith the pseudorange bias format for relative targeting when largesurvey errors are present. Pseudorange biases are readily availablefrom reference receivers. If the reference receiver computesbiases for all Satellites in View (SVs), the active receiver can thenchoose any subset of these SVs for its solution. Of course, as theseparation between receivers grows, there will be fewer and fewerof these common SVs until finally there are less than fouravailable to the active receiver.The first-order error in this situation was given in Eq. (12). Byfirst order we mean that the error is proportional to the absolutesurvey error, not the difference in survey errors between referencereceiver and active receiver. Equation (12) will be rewritten toemphasize this. First define δδr survey to be the difference insurvey error at the target and reference receiver.δδr survey = δr T,survey - δr R,surveyThe error at the target can then be written:δr T,survey = δδr survey + δr R,surveySubstituting this definition for δr T,survey into Eq. (12) yields:δδr survey + δr R,survey - H A # H R δr R,survey =δδr survey + (I - H A # H R )δr R,survey )(14)If equivalent Eq. (14) is substituted for Eq. (12) in Eq. (11), theresult is:(15)ε = δδr survey + (I - H A # H R )δr R,survey + H A # δδρwhere δδρ GPS is defined to be δρ R,GPS - δρ A,GPS , the difference inpseudorange biases at the two locations. This term and itscompanion in position space, δδr survey , are differences in errors,whereas the middle term is a function of the absolute survey error.It is this term we wish to assess. It will cause the error on targetto be nonzero even if the relative survey error were to be zero andthe differential corrections were to be perfect.We will now depart from this linearized analysis and consider thecomplete nonlinear error in relative targeting due to the absolutesurvey error when pseudorange biases are chosen for the datatransfer format. This will be the only error considered. There willbe no pseudorange biases at either receiver and no differencebetween the survey error at reference receiver and active receiver.At the reference receiver, the computed pseudorange biases arenonzero only because the survey error is in the reference receiverposition. These biases are transmitted as corrections to the activereceiver at the target. At the active receiver, they are subtractedfrom the measured pseudoranges before computing the GPSposition solution. This solution is then compared with thesurveyed target/active receiver position. The difference is thetargeting error in question.The following expression simply defines the sign of the error(bias) in the following development.x msd = x true + x biasConsidering pseudorange biases: they are defined to be thedifference between the measured (by the receiver) pseudorangeand the “known,” “true” range determined by survey, for instance.ρ bias = ρ msd - ρ ”true”Note that the subscript is in quotes because in our specialsituation, the “true” range is in fact in error due to an imperfectsurvey, whereas the measured (by the receiver) range, ρ msd , hasbeen assumed perfect. Specifically, the two quantities are:ρ msd,R = |r SV - r R, true |ρ ”true” = |r SV - (r R, true + δr survey )|At the active receiver, these biases, ρ msd,R - ρ ”true” , aresubtracted from the measured ranges. (Again, the measuredranges are perfect.)ρ corr = ρ msd,A - ρ biaswhere ρ msd is the active receiver output (and is again perfect).ρ msd,A = |r SV - r A, true |The corrected ranges are used to compute the active receiver/target position.r A,corr = f(ρ corr )This position will be in error because a position error (the surveyerror) has been transmitted as pseudorange bias errors. Thequestion is, how much?Relative and Differential GPS Data Transfer and Error Analysis6