Development of a Semi-Automated Approach for Regional Corrector Surface Modeling in GPS-Levelling

Transcription

1 Development of a Sem-Automated Approach for Regonal Corrector Surface Modelng n GPS-Levellng G. Fotopoulos, C. Kotsaks, M.G. Sders, and N. El-Shemy Presented at the Annual Canadan Geophyscal Unon Meetng Banff, Canada May 10-14, 003

2 Overvew Introducton to problem Background to prevous work Choosng the best model Assessng model performance Testng parameter sgnfcance Descrpton of data Swtzerland Canada Dscusson of results Conclusons CGU 003, Banff, Canada

3 Introducton (1/4) Standard practce: Use of a corrector surface to model the datum dscrepances and systematc effects when combnng GPS, geod and orthometrc heghts Theory: h H N 0 N GPS / levellng N Practce: h H N l N GPS / levellng N Model: l h H N T a x v resduals parametrc corrector model/surface CGU 003, Banff, Canada

4 Introducton (/4) Profound reasonng for choosng a specfc model s mssng Spatal modellng and analyss of the adjusted resdual values over a network of GPS/levellng benchmarks are useful for a varety of applcatons: External accuracy evaluaton of sphercal harmonc models of the Earth s gravty feld and regonal gravmetrc geod solutons Refnement of regonal geod solutons by elmnatng long wavelength errors through tes to GPS/levellng benchmarks Check and mprove the accuracy of vertcal datums through combnng geod, GPS and levellng data CGU 003, Banff, Canada

5 Introducton (3/4) Development of corrector surface models to be used wth GPS and gravmetrc geod models for GPS-Levellng H p h p N p T a p xˆ known heght data corrector surface Data orthometrc heght at new pont GPS: h, Δh j Orthometrc heghts: H, H Geod model : N, j N j Predcton surface am s to derve a surface from data whch s to be appled to new data CGU 003, Banff, Canada

6 Introducton (4/4) Objectve: To elmnate some of the arbtrarness n both choosng the model type and assessng ts performance General Pontwse Case: h H N T a x v where, x a v vector of unknown parameters vector of known coeffcents (depend on horzontal coords) resduals CGU 003, Banff, Canada

7 Classc Emprcal Approach polynomal (order)? Δh Data h,h,n j,δh,δn j j h H N 0 h Corrector Models H N a T x 0 base functons (trg.)? physcal meanng of terms network geometry pre-corrected pre-adjusted over constraned adjustment Fnal model selecton Least-squares adjustment xˆ k Statstcs of adjusted resduals vˆ h H N a T xˆ CGU 003, Banff, Canada

8 Corrector Surface Model Selecton Corrector Models h H N a T x 0 Selecton of analytcal model suffers from a degree of arbtrarness (Why?) type of model (.e. polynomal) type of base functons (.e. trgonometrc) number of coeffcents Need statstcal tools to assess choces made compare dfferent models Factors for model selecton/analyss may vary f nested models orthogonal vs. non-orthogonal models No straghtforward answer, data dependent (geometry) CGU 003, Banff, Canada

9 CGU 003, Banff, Canada Assessng the Goodness of Ft n 1 n 1 ) ( ) v ( 1 R ˆ 1) (n ) ( m) (n ) v ( 1 R n 1 n 1 ˆ n # of observatons N H h m # of parameters Statstcs of adjusted resduals x a N H h v T ˆ ˆ Coeffcent of determnaton R Adjusted coeffcent of determnaton R orgnal after ft

10 Cross-Valdaton Addtonal Emprcal Approach Use a subset of all ponts to compute the model parameters xˆ Predct the resdual values at a new pont and compare the predcted value wth the known heght value Δvˆ p h p H p N p a T p xˆ Repeat for each pont and compute the average rms, n μ 1 σ p Cross-valdaton (emprcal approach) CGU 003, Banff, Canada

11 Testng Parameter Sgnfcance Reasons for reducng the number of model parameters Smplcty, computatonal effcency Over-parameterzaton (.e. hgh-degree trend models) unrealstc extrema n data vods where control ponts are mssng Unnecessary terms may bas other parameters n model hnders capablty to assess model performance Parameter Sgnfcance Need for automated selecton process CGU 003, Banff, Canada

12 Stepwse Procedures Backward Elmnaton Procedure Start wth hghest order model Elmnate less-sgnfcant terms one-by-one (or several at once) Crtera for determnng parameter deleton Partal F-test Level of sgnfcance, Problem: correlaton between parameters nested models only Forward Selecton Procedure Start wth smple model Add parameter wth the hghest coeffcent of determnaton (or partal F-value) Stepwse Procedure Combnaton of backward elmnaton and forward selecton procedures Starts wth no parameters and selects parameters one-by-one (or several) After ncluson, examne every parameter for sgnfcance (partal F-test) CGU 003, Banff, Canada

13 Testng Parameter Sgnfcance Statstcal tests are more powerful n pontng out napproprate models rather than establshng model valdty Test f a set of parameters n the model s sgnfcant or not: x x ( I) x I I set of parameters tested (I) remanng parameters (complement) hypothess H0 : xi 0 vs H a :xi 0 test statstc crtera F xˆ Q kσˆ 1 xˆ I xˆ ~ I I ~ F F k, f k number of tested terms Q ˆx I H 0 accepted... submatrx of Q = N -1 CGU 003, Banff, Canada

14 Testng Parameter Sgnfcance Test statstc (regardless of form) s a functon of observatons F xˆ Q kσˆ 1 xˆ I xˆ ~ I I ~ F partal ( v) ˆ ( v) ˆ full ( v) ˆ n m full k No need to repeat combned least-squares adjustment (frst case) Problems No unque answer (depends on ntal selecton, ) Hgh parameter correlaton may skew results Hghly correlated parameters should be deleted (detecton) CGU 003, Banff, Canada

15 Stepwse Procedure Enter parameter Perform regresson yes Termnate no Compute partal F-values, choose the hghest one ~ F F n Re-compute partal F-values for each model parameter Start Select regresson model Least-squares adjustment ~ F F out Delete parameter Backward elmnaton CGU 003, Banff, Canada

16 Lattude Descrpton of Data 111 statons n Swtzerland 343 km 1 km regon Form resduals : h H N Statstcs of resduals before ft mn -4.9 cm max 19 cm mean 1.1 cm std 3.8 cm rms 3.9 cm 30' 48 N 30' 47 N 30' 46 N 30' 45 N 6 E 7 E 8 E 9 E 10 E 11 E Longtude GPS on Benchmarks (and resduals) CGU 003, Banff, Canada

17 Lattude Descrpton of Data 63 statons n Southern Brtsh Columba & Alberta 495 km 334 km regon Form resduals : h H N Stats of resduals before ft mn cm max 5. cm mean 4.5 cm std 8.1 cm rms 9.3 cm 5 N 51 N 50 N 49 N 14 W 1 W 10 W 118 W 116 W Longtude GPS on Benchmarks (and resduals) CGU 003, Banff, Canada

18 Nested blnear polynomal seres Analytcal Models 1 d dλ d dλ d dλ d dλ d dλ d dλ d d d dλ d dλ d dλ 4 Classc trgonometrc-based polynomal fts 1 cos cos cos sn sn 1 cos cos cos sn sn sn Dfferental smlarty transformaton cos cos cos sn sn sn cos sn W sn cos cos W 1 f sn W sn W where, W 1 e sn CGU 003, Banff, Canada

19 Analytcal Models (polynomals) 4th order 3rd order nd order 1st order CGU 003, Banff, Canada

20 Other Analytcal Models Classc 4-parameter Classc 5-parameter 7-parameter dfferental smlarty Notes all values shown n m GPS BMs n Swtzerland used Full models shown (no parameters omtted) CGU 003, Banff, Canada

21 Example - Coeffcent of Determnaton R Swtzerland 0 R R R Canada model performance R -0.1 A B C D E F G A 1 st order polynomal B Classc 4-parameter C Classc 5-parameter D nd order polynomal E Dfferental Smlarty F 3rd order polynomal G 4 th order polynomal CGU 003, Banff, Canada

22 RMS (cm) Emprcal Testng Conclusons Resduals after ft 4th order polynomal 14 1 predcton Predcton (external test) Any model except 4th order polynomal Not enough of a dfference between models to justfy statstcal parameter sgnfcance testng use lowest order model resduals after ft predcton resduals after ft A B C D E F G Swtzerland Canada CGU 003, Banff, Canada

23 Results - Southern BC/AB Dfferental Smlarty Ft (7-parameters) cos cos cos sn sn sn cos sn W sn cos cos W 1 f sn W sn W Selecton crtera R R vˆ T vˆ 53 cm condton number rms after ft rms (predcton) 6.7 cm 7.9 cm cm CGU 003, Banff, Canada

24 Results - Swtzerland Classc 4-parameter ft 1 cos cos cos sn sn Selecton crtera R R vˆ T vˆ 4.5 cm condton number rms after ft rms (predcton).4 cm.4 cm cm CGU 003, Banff, Canada

25 Sem-automated procedure for comparng corrector surface models and assessng model performance was presented Sem no unque straghtforward soluton some user nterventon requred In most cases, the best test s cross-valdaton (predcton) ndependent external test depends on qualty of data Conclusons When model parameters are hghly correlated (as s the case wth polynomal regresson), statstcal testng may not be conclusve Use orthogonal polynomals to elmnate problems wth hgh correlaton between parameters (.e. Fourer Seres) Procedure should nclude a combnaton of emprcal and statstcal testng CGU 003, Banff, Canada