CET 318 10. Data Processing(2) From download data from receiver to Establishment of Data Processing Model and Solution 9. Data Processing(2) Book:p.203-276 Dr Guoding Zhou 3.1 Least Square Adjustment 1. Observation Equation 3. Adjustment, Filtering, and Smoothing-----Overview 2. Adjustment 3. Accuracy Analysis 3.2 Kalman filtering 3.3 Smoothing L Introduction 2. Prediction See explanation 3. Update 4.E
1 Dr. Guoqing Zhou 9. Data Processing (2) CET 318 Book: p. 203-276 10. Data Processing (2) From Download Data from Receiver to Establishment of Data Processing Model and Solution 3. Adjustment, Filtering, and Smoothing-----Overview 3.1 Least Square Adjustment 1. Observation Equation 2. Adjustment 3. Accuracy Analysis 3.2 Kalman Filtering 1. Introduction 2. Prediction 3. Update 4. Example 3.3 Smoothing See Explanation
4.1 Linearization of Observation Equation Approximate point position (Xo, Yo, Za) 4. Adjustment of Mathematical p()=√x(0)-xa)+(Y1()-Y2+(210)-z2 GPS Model ()=pl(n) p!(n)=pa(t)-a△x4-al2△y-a2△Z 4.2 Linearization model for point The satellite clock bias is assumed to be known. this Positioning with Code Ranges sumption makes sense because satellite clock correctors can be received from the navigation message The elementary model for point positioning with code ranges Four Satellit R()=p()+c6(t)-co1(t) =ax△X1+ay△Y+a2△Z-c6,(t) In the model, only the clocks are modeled. The ionosphere F2=a3△X+a3△Y4+a24Z·c6() troposphere, and other minor effects are neglected 13= R(1)-p()-c6l(t)=ax△X+ay△Y4+a2△Z1·c6,(t) Observation Equation =ax△X1+a△Y4+a2△Z1-c6,(t) l=Ax =?a=?ay=?a2=? 4.3 Linearization model for point Observation Equation Positioning with Carrier Phases Above fact reflects The procedure is the same as code range ineaized Equation: 1. Point positioning with phases in this form cannot be olved epoch by epoch 入()-P(0)-c6(0)=a△X+ayAY+a2,AZ,+N nt positioning with code ranges, the nu 0 knowns by a new clock term. of unknowns is now increased by the ambiguities 3. For two epochs there are eight eq and Four Satellites again unknowns(still an underdetermined problem ). For three epochs there are 12 equations and 10 unknowns, thus, slightly overdetermined problem. The coefficients of the coordinate increments are supplemented with the time parameter t. Obviously, the four equations do not I=Ax
2 4. Adjustment of Mathematical GPS Model 4.1 Linearization of Observation Equation Taylor series with respect to the approximate point ρ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) 2 i 2 i 2 i j i j j j t = + + f(X , Y , Z ) ρ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) i0 i0 i0 2 i0 2 i0 2 i0 j i0 = = + + j j j t (X , Y , Z ) Approximate point position i0 i0 i0 j i i i j j i i i j j i i i j Z p t Z t Z Y p t Y t Y X p t X t X t t ∆ − − ∆ − ∆ − − = − ( ) ( ) ( ) ( ) ( ) ( ) ρ ( ) ρ ( ) 0 0 0 0 0 j 0 i0 j i Xi i Yi i Zi Zi t = t − a ∆X − a ∆Y a ∆ j 1 1 1 i0 j i ρ ( ) ρ ( ) - 4.2 Linearization Model for Point Positioning with Code Ranges The elementary model for point positioning with code ranges is given by R ( ) p ( ) δ (t) - δ (t) i j j i j i t = t + c c In the model, only the clocks are modeled. The ionosphere, troposphere, and other minor effects are neglected. Lineaized Equation: R ( ) p ( ) δ (t) a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X j j i j i i i i t − t − c = ∆ + ∆ + ∆ c a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X 1 i i i l = ∆ + ∆ + ∆ c ? a ? a ? a ? 1 Z 1 Y 1 X 1 i i i l = = = = The satellite clock bias is assumed to be known. This assumption makes sense because satellite clock correctors can be received from the navigation message. Four Satellites: a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X 1 i i i l = ∆ + ∆ + ∆ c a X a Y a Z - δ (t) i i 2 i Z 2 i Y 2 X 2 i i i l = ∆ + ∆ + ∆ c !!!!! 3 l = !!!!! 4 l = Observation Equation: l = Ax 4.3 Linearization Model for Point Positioning with Carrier Phases λΦ ( ) ( ) δ (t) a X a Y a Z λN - δ (t) i j i i 1 i Z 1 i Y 1 X j 0 j i i i t p t c c j i − i − = ∆ + ∆ + ∆ + The procedure is the same as code range Compared to point positioning with code ranges, the number of unknowns is now increased by the ambiguities. Lineaized Equation: Four Satellites again: l = Ax The coefficients of the coordinate increments are supplemented with the time parameter t. Obviously, the four equations do not solve for the eight unknowns. Observation Equation: Above fact reflects 1. Point positioning with phases in this form cannot be solved epoch by epoch. 2. Each additional epoch increases the number of unknowns by a new clock term. 3. For two epochs there are eight equations and nine unknowns (still an underdetermined problem). For three epochs there are 12 equations and 10 unknowns, thus, a slightly overdetermined problem. l = Ax
4. 4 Linearization Model for Relative Positioning restricted to carrier phases, since it should be obvious how to change from the more expanded model of phases to a code 4a)「4,()吨)吗)40·0 6)-e)-a) M]4()呢))A0 The model for the double-difference u)-副)-e8") )吨()吗 A(r)=P1()+入Nk 2)吨(a)鸣(2)0A4 Where: PA(0=Pi(o)-pi(0-P(+pl(o 或)砖()哦)00x equation reflects the fact of four measurement Each of the four terms must be linearized Lineaized Equation YB+a2△B+)NA -o+x(Am△xn+y(=y△v Whe:l()=λ(1)-p2s()+p2()+p()-pl(n) a及 △XA+ ak=? Observation Equation E Ax 2(Q-2A△zA 5.1 Single baseline Solution Problem: 1. The adjustment principle requires observations are uncorrelated 5. Network Adjustment differences and triple-differences are correlated 3. The implementation of the double-difference correlation can be easily accomplished. Alternatively, decorrelated algorithm using a Gram-Schmidt orthogonalization. 4. The implementation of the correlation of the triple. differences is questionable since the noise of the triple-differences will always prevent to obtain a refined solution
3 4.4 Linearization Model for Relative Positioning For the case of relative positioning, the investigation is restricted to carrier phases, since it should be obvious how to change from the more expanded model of phases to a code model. The model for the double-difference jk AB jk λΦ (t) = p (t) + λN jk AB AB p (t) p (t) p (t) p (t) p (t) j A k A j B k B jk Where: AB = − − + Above equation reflects the fact of four measurement quantities for a double-difference. Each of the four terms must be linearized. Lineaized Equation: l = Ax jk X B Y B Z B AB X A Y A Z A jk AB a X a Y a Z λN δ (t) a X a Y a Z B B B A A A + ∆ + ∆ + ∆ + = ∆ + ∆ + ∆ jk jk jk jk jk jk ( ) λΦ ( ) ( ) ( ) ( ) ( ) 0 0 0 0 jk AB l t t p t p t p t p t j A k A j B k AB B jk = − + + − a ? a ? a ? a ? a ? a ? B B B A A A X Y Z X Y Z = = = = = = jk jk jk jk jk jk Where: Observation Equation: 5. Network Adjustment 5.1 Single Baseline Solution 1. The adjustment principle requires observations are uncorrelated. 2. The single-differences are uncorrelated, whereas doubledifferences and triple-differences are correlated. 3. The implementation of the double-difference correlation can be easily accomplished. Alternatively, decorrelated algorithm using a Gram-Schmidt orthogonalization. 4. The implementation of the correlation of the tripledifferences is more difficult. Furthermore, it is questionable since the noise of the triple-differences will always prevent to obtain a refined solution. Problem:
5.2 Multi-Point solutio 6. If ni denotes the number of observing sites, then ni(n-l) I. In contrast to the baseline by baseline solution, /2 baselines can be calculated. Note that only n -l of the multipoint solution considers at once all noints in the network. 7. The redundant baselines are either used for misclosure key difference compared to the single checks or for an additional adjustment of the baseline eline solution is that the correlations between the baselines are taken into account Disadvantage 3. The same theoretical aspects also apply to the extended case of a network correlation of the si eously observed baselines. 2. By solving baseline by baseline, this correlation is ignored Single-difference Example for A Network Covariance matrix Taking A as reference site, two baselines A-B, A-C, the two single-differences ①l()=①()-①(t) Cov(SD)=8CC= oc()=(1)-(n) A correlation of the single-differences of the two baselin 110 with a common point. 1D0-11000000 Double-difference Example for A Network 10-10-10 0000 C I Xn-1)independent double. m ●() For ni=3: n4, 6 independent 画A() double-difference 45(0)=()-()-4()+( 4()=46(0)-4()-()+中 Matrix expression ●AB(t 中()=(1)-4()-()+( 中()=5()-()-()+( 4()=(1)-()-4()+(0)C 中C()
4 5. For an observed network, the use of the single baseline method usually implies a baseline by baseline computation for all possible combinations. 6. If ni denotes the number of observing sites, then ni(ni-1) /2 baselines can be calculated. Note that only ni-1 of them are theoretically independent. 7. The redundant baselines are either used for misclosure checks or for an additional adjustment of the baseline vectors. 1. The simple single baseline solution from the theoretical point of view is that it is not correct because of the correlation of the simultaneously observed baselines. 2. By solving baseline by baseline, this correlation is ignored. Disadvantage: 5.2 Multi-Point Solution 1. In contrast to the baseline by baseline solution, the multipoint solution considers at once all points in the network. 2. The key difference compared to the single baseline solution is that the correlations between the baselines are taken into account. 3. The same theoretical aspects also apply to the extended case of a network. Single-difference Example for A Network Taking A as reference site, two baselines A-B, A-C, the two single-differences j A B C Epoch t Baseline Φ ( ) Φ ( ) Φ ( ) j A j t t t j AB = B − Φ ( ) Φ ( ) Φ ( ) j A j t t t j AC = C − = Φ ( ) Φ ( ) j j t t SD AC AB Φ = Φ ( ) Φ ( ) Φ ( ) j j j t t t C B A − − = 1 0 1 1 1 0 C Introducing = = 1 2 2 1 ( ) δ δ 2 T 2 Cov SD CC Covariance matrix A correlation of the single-differences of the two baselines with a common point. Double-difference Example for A Network j A B C Epoch t Baseline l m For ni points and nj satellites, (ni k -1)(nj-1) independent doubledifference For ni =3; nj=4, 6 independent double-difference Matrix expression Introducing
Covariance matrix 422211 5.3 Single Baseline vs Multi-point Solution 4212 Some Arguments Cor(DD)=8CC'=8 1. The correlation is not modeled correctly with the sing baseline solution because correlations between baseline symmetry are neglected 2. The computer program is, without doubt, much simpler 1. Solution without any correlation deviates from the 3. With modern software and hardware, the computational theoretically correct values by a greater amount time is not a real problen 2. It is estimated that the single baseline method deviates 4. Cycle slips are more easily detected and repaired in the from the multi-baseline(correlated) solution by a maximum of 28 5. It takes less effort in the single baseline mode to isolate bad measurements and possibly to eliminate them 6. The economic implementation of the full correlation for 5.4 Least Square Adjustment of Baselines I multipoint solution only works properly for networks with the same observation pattern at each receiver site In networks. the number of measured baselines will 7. Even in the case of the multipoint approach, it becomes usually exceed the minimum amount. questionable whether the correlations can be modeled properly Redundant information is available and the determination of the coordinates of the network points may be carried 8. For the dual frequen out by a least squares adjustment. is formed from LI and L2 and rocessed together with the li data of the single n、=X requency receivers. Thus, a correlation is because of the LI data. A proper model nr=Y-Y4·Y9 onosphere of the LI baseline, an effect The geometry of the visible satellites is an important factor in achieving high quality results especially for point positioning and kinematic surveying. The geometry changes with time due to the relative motion of 6. Dilution of precision A measure of the geometry is the Dilution of Precision(DOP) tes for point ning with code
5 = = 4 4 2 4 2 2 4 1 1 2 4 2 1 2 1 4 2 2 2 1 1 ( ) δ δ 2 T 2 Cov DD CC Covariance matrix 1. Solution without any correlation deviates from the theoretically correct values by a greater amount. 2. It is estimated that the single baseline method deviates from the multi-baseline (correlated) solution by a maximum of 2δ. symmetry 5.3 Single Baseline vs. Multi-point Solution Some Arguments: 1. The correlation is not modeled correctly with the single baseline solution because correlations between baselines are neglected. 2. The computer program is, without doubt, much simpler for the single baseline approach. 3. With modern software and hardware, the computational time is not a real problem. 4. Cycle slips are more easily detected and repaired in the multipoint mode. 5. It takes less effort in the single baseline mode to isolate bad measurements and possibly to eliminate them. 6. The economic implementation of the full correlation for a multipoint solution only works properly for networks with the same observation pattern at each receiver site. 7. Even in the case of the multipoint approach, it becomes questionable whether the correlations can be modeled properly. 8. For the dual frequency receivers, the ionosphere-free combination Lc is formed from L1 and L2 and processed together with the L1 data of the single frequency receivers. Thus, a correlation is introduced because of the L1 data. A proper modeling of the correlation biases the ionosphere-free Lc baseline by the ionosphere of the L1 baseline, an effect which is definitely undesirable. 5.4 Least Square Adjustment of Baselines j i In networks, the number of measured baselines will usually exceed the minimum amount. Redundant information is available and the determination of the coordinates of the network points may be carried out by a least squares adjustment. Xij X j i ij AB n = X − X - X Y j i ij AB n = Y − Y - Y Z j i ij AB n = Z − Z - Z 6. Dilution of Precision The geometry of the visible satellites is an important factor in achieving high quality results especially for point positioning and kinematic surveying. The geometry changes with time due to the relative motion of the satellites. − − − − = c c c c 4 Z 4 Y 4 X 3 Z 3 Y 3 X 2 Z 2 Y 2 X 1 Z 1 Y 1 X i i i i i i i i i i i i a a a a a a a a a a a a A A measure of the geometry is the Dilution of Precision (DOP) factor. 4 Satellites for point positioning with code ranges
DOP can be calculated from the co-variance matrix of the Qx=(AA) qxx qxy qx 4x Why 7. Accuracy Measures The diagonal elements are used for the following doP GDOP=axx+qrr+qn+qu Geometric dilution PDOP=√x+qy+qa Position dilution TDOP Time dilution y Why do we need accuracy measures? What have we learnt? How many accuracy measures are in use! Which parts are important? Chi-square Distribution uracy measures Two-dimensional accuracy measures Three-dimensional accuracy measures
6 T 1 X Q (A A) − = DOP can be calculated from the co-variance matrix of the solution. = tt ZZ Z YY YZ Y XX XY XZ X q q q q q q q q q q Q t t t X Symmetry The diagonal elements are used for the following DOP definitions: GDOP = q XX + q YY + q ZZ + qtt PDOP = q XX + q YY + q ZZ TDOP = qtt Geometric dilution Position dilution Time dilution Why 7. Accuracy Measures Why do we need accuracy measures ? How many accuracy measures are in use! • Qxx, and RMS • Chi-square Distribution • One-dimensional accuracy measures • Two-dimensional accuracy measures • Three-dimensional accuracy measures Summary What have we learnt? Which parts are important?