1998 - Draper Laboratory

This expression will be evaluated first for transfer of positioncorrections, δr c , then for the transfer of pseudorange biases, δρ c .We will now proceed to develop an expression for error on targetusing position corrections. The guidance constraint, that thecorrected solution is driven to the surveyed target coordinates, isexpressed below.r A,c = r T,survey(8)r(ρ A )- δr c = r T,surveyThe pseudoranges at the active receiver, ρ A , and the surveyedtarget position may both be expanded into true values pluserrors.ρ A = ρ A,true + δρ A,GPSr T,survey = r T,true + δr T,surveySubstituting these expressions for ρ A and r T,surveyed into Eq. (8)yields:r(ρ A,true + δρ A,GPS ) - δr c = r T,true + δr T,surveyA first-order expansion of the function on the left side of theequation yields:r A,true + H A# δρA,GPS - δr c = r T,true + δr T,surveyRearranging terms yields:r A,true - r T,true = δr T,survey - H A# δρA,GPS - δr cThe two terms on the left side of the equation are recognized asthe error-on-target and the expression for δr c , Eq. (6), can besubstituted on the right side of this equation. The result is:ε = δr T,survey -H A # δρ A,GPS + H R# δρ A,GPS - δr R,surveyAgain, substituting these expressions for ρ A and r T,survey into Eq.(10) yields:r(ρ A,true + δρ A,GPS - δρ c ) = r T,true + δr T,surveyA first-order expansion of the function on the left side of theequation yields:r A,true + H A # (δρ A,GPS - δρ c ) = r T,true + δr T,surveyRearranging terms yields:r A,true - r T,true = δr T,survey - H A # (δρ A,GPS - δρ c )The two terms on the left side of the equation are recognized asthe error-on-target. The expression for δρ c , Eq. (7), can besubstituted on the right side of this equation. The result is:ε = δr T,survey -H A # (δρ A,GPS - δρ R,GPS +H R# δr R,survey )Multiplying through and rearranging terms yields:ε = δr T,survey -H A # H R δr R,survey + H A# (δρ R,GPS - δρ A,GPS )(11)So the error is seen to consist of a relative surveying error and adifference in GPS bias errors.For convenience, the expressions for error-on-target for the threedata transfer formats: transfer of target coordinates, transfer oftarget position corrections, and transfer of pseudorangecorrection, are repeated below.ε = δr T,survey - δr R,survey + H R# δρ R,GPS -H A # δρ A,GPS(5)Simply reordering terms yields the expression for error-on-targetwhen using position corrections.ε = δr T,survey - δr R,survey + H R# δρ R,GPS -H A # δρ A,GPS (9)Finally, a similar analysis for the use of pseudorange biases will begiven. Starting from the same point, the guidance constraint is:r A,c = r T,survey(10)r(ρ A - δρ c ) = r T,surveywhere the function r(ρ A - δρ c ) represents the corrected solution atthe active receiver.The pseudoranges at the active receiver, ρ A , and the survey targetposition may both be expanded into true values plus errors.ρ A = ρ A,true + δρ A,GPSr T,survey = r T,true + δr T,surveyε = δr T,survey - δr R,survey + H R# δρ R,GPS -H A # δρ A,GPS(9)ε = δr T,survey -H A # H R δr R,survey + H A# (δρ R,GPS - δρ A,GPS )(11)The errors in transfer of target coordinates and transfer of targetposition corrections are identical, as would be expected. Notethat for the transfer of pseudorange corrections, Eq. (11), therelative survey error is sensitive to the separation between thereference receiver and the target/active receiver. The term is:δr T,survey - H A # H R δr R,survey(12)The product H A # H R would be identity if the reference receiverwere colocated with the active receiver. It is nonidentical fornonzero receiver separation or if a different set of satellites is usedat the two receivers. For the transfer of position corrections, Eq.(9), the term that is sensitive to receiver separation is thedifferential pseudorange bias term. That term is:Relative and Differential GPS Data Transfer and Error Analysis5