GNSS Dt Assimiltion nd GNSS Tomogrphy for Globl nd Regionl Models M. Bender, K. Stephn, A. Rhodin, R. Potthst Germn Wether Service (DWD) Interntionl Symposium on Dt Assimiltion, Munich 214 27. Februry 214 M. Bender,, K. Stephn,, A. Rhodin,, R. Potthst
GNSS Dt Assimiltion nd GNSS Tomogrphy Outline GNSS Observtion Opertor Atmospheric Phse Delys Inverse Problem Properties of the Opertor GNSS Tomogrphy Germn GNSS Network Reconstruction Techniques Results GNSS Dt Assimiltion DWD Wether Models GNSS Observtion Opertor Design Rytrcer Outlook nd Summry ISDA, Munich, Februry 214
Atmospheric Phse Dely Opticl pth length : L = S n(s) ds, n - refrction index Fermt s principle: The pth tken between two points by ry of light is the pth tht cn be trversed in the lest time. minimum of opticl pth length, L = L min Trvel time s compred to undisturbed signl propgtion in vcuum phse dely, slnt totl dely: L = n(s) ds dg = (n(s) 1) ds + (S G) = T k i S G S The Slnt Totl Dely (STD) is usully defined by neglecting the geometric dely (S-G): STD = 1 6 N ds with N = 1 6 (n 1) S ISDA, Munich, Februry 214
GPS Processing: Observtion Equtions GPS phse equtions: L k 1i = ϱ k i (t) + N k 1iλ 1 + T k i (t) + I k i (t) + c (δ i (t) δ k (t)) + ɛ for stellite k nd receiver i nd ϱ - distnce N - mbiguity T - troposphere correction δ i - receiver clock prmeter I - ionosphere correction δ k - stellite clock prmeter ɛ - residul Troposphere correction from the lest-squres djustment: Ti k = m ZTD i Ti k = m h ZHD i + m w [ZWD i + cot ε (G Ni cos φ + G Ei sin φ)] (+ δ) ISDA, Munich, Februry 214
Why Use Slnt Delys? ZTDs informtion bout horizontl structure but no verticl profiles Horizontl slnts (ε = ) verticl profiles but no horizontl structures 4 3 3 2 2 Rel STDs (5 ε ) 1 verticl profiles nd horizontl structures 1 informtion bout the 3D field 1 2 3 4 elevtion / deg. Number of entries / 2 deg. ISDA, Munich, Februry 214
Direct Problem, Observtion Opertor To compute relistic STD observtions 3D refrctivity filed is required. Numericl wether models provide 3D fields of temperture, pressure nd humidity. 1) Compute the 3D refrctivity field N(r) p N(r) = k d 1 T + k 2 e T + k 3 e, r is the position vector T 2 2) Estimte the signl pth S Solve Fermt s Principle for the given tmospheric stte nd given stllite-receiver geometry: Vritionl problem S N ds = min S = S(N)! 3) Compute the line integrl for given signl pth S Simulted STD STD = 1 6 S N ds S is prmeterized curve through the sclr field N. ISDA, Munich, Februry 214
Observtion Opertor The observtion opertor H mps the stte spce to the observtion spce nd covers ll observtions H(x) = y: H = {h i } i=1...r with h i = STD i N ds i, signl pth S i S i This is system of non-liner equtions which cn, e.g. be solved by minimizing the 3D-Vr cost function J: J(x) = 1 2 (x x) T B 1 (x x) + 1 2 (y H(x))T R 1 (y H(x)) Linerized nd discretized opertor H: H(x) = y = J c x = h j x i x=x x = H x = y System of liner equtions with mtrix H. ISDA, Munich, Februry 214
Ill-posed Inverse Problems Well-posed problems 1 A solution must exist. 2 There must be only one unique solution. 3 The solution must be stble. Tomogrphic problems violte t lest the two ltter conditions nd re therefore ill-posed. Conditioning of the mtrix An ill-posed problem leds to n ill-conditioned mtrix H. Ill-conditioned mtrices hve lrge condition numbers κ = σmx σ min σ i - singulr vlues. H 1 or (H T H) 1 re unstble nd unbounded The norml equtions (lest-squres djustment) led to meningless result: H T H x = H T m x = (H T H) 1 H T m ISDA, Munich, Februry 214
Number of Observtions - Avilble STD Dt number of slnts in 3 minutes 4 4 3 3 2 GPS GLONASS Glileo 6 7 8 9 1 11 12 13 14 15 16 17 18 time, UTC N sl 12 h N sl 3 min. N sl min. GPS 2594632 433 8642 GLONASS 26413 37281 74334 Glileo 2367267 46 899 Slnts for tomogrphy 121187 241635 GNSS slnt dely dt from 3 Germn GNSS sttions. ISDA, Munich, Februry 214
Impct of the GPS Stellite Constelltion 1:-1:15 UTC, 14 slnts ISDA, Munich, Februry 214 1:15-1:3 UTC, 11979 slnts
GNSS Tomogrphy I. GNSS Tomogrphy ISDA, Munich, Februry 214
Applictions of GNSS Tomogrphy Assimiltion of tomogrphiclly reconstructed humidity fields, retievl Smll, independent ppliction which cn run on PCs Only one vrible, e.g. N wet or bsolute humidity Rther smll number of grid nodes smll mtices H, stndrd liner lgebr (LAPACK) Provides 3D humidity fields without ny input from wether models Rel-time cpbilities ISDA, Munich, Februry 214
Design of GNSS Tomogrphy Systems GNSS STDs STDs Opertor H STD inversion humidity field Additionl observtions Synop dt, rdiosonde profiles, WVR or GPS IWF provide extr informtion nd stbility H = (H STD, H sy, H RS, H WVR,... ) Constrints Inter-voxel constrints or constrints on specific lyers Bckground x Fills gps in the observtions nd leds to more stble solutions: model fields or extrpolted surfce observtions Almost dt ssimiltion: J(x) = 1 2 (x x) T B 1 (x x) + 1 2 (y H(x))T R 1 (y H(x)) Solutions: Conjugte grdients, Tikhonov regulristion, Klmn filter, singulr vlue decomposition, ART,... ISDA, Munich, Februry 214
GPS-Tomogrphy - Germny 6 8 1 12 14 54 54 53 53 52 52 51 51 Grid resolution Grid dimension: 1 km3 Horizontl res. : 35 - km Verticl res. : 2 - m 3 min. temporl resolution 49 49 48 48 2 voxel in E-W direction 25 voxel in N-S direction 2-3 verticl voxels (up to 8-1 km) 6 8 1 12 1-1 Voxel 3 STDs 14 ISDA, Munich, Februry 214
Tomogrphic Reconstruction Slnt Totl Delys (STDs) Appliction of the Sstmoinen-Model Slnt Wet Delys (SWDs) Tomogrphic Reconstruction Estimtion of n initil field. Appliction of itertive Algebric Reconstruction Techniques MART: x k+1 j = x k j λa i j ( ) A i A m i,a i i,x k Field of the wet refrctivity N wet The bsolute humidity cn be obtined if the T-field is vilble. Tlk on dvnced tomogrphy: E. Altuntc, Kend workshop, 2 pm ISDA, Munich, Februry 214
25 4 35 25 Tomogrphic Reconstruction - horizontl Horizontl distribution of the wet refrctivity N wet t 19 m ASL. COSMO-DE: Tomogrphy: IWV: 54 53 52 51 49 48 54 53 52 51 49 48 4 3 35 4 4 54 53 52 51 49 48 3 35 35 25 3 3 3 35 3 35 35 35 35 3 35 25 35 25 35 35 3 3 3 35 35 5 6 7 8 9 1 11 12 13 14 15 N wet 5 6 7 8 9 1 11 12 13 14 15 N wet 5 6 7 8 9 1 11 12 13 14 15 IWV [kg m 2 ] 4 4 14 July 29, 16: UTC 24 26 28 3 32 34 36 38 4 42 Tomogrphic rekonstruction using 269 sttions nd 3 minutes of GPS dt. ISDA, Munich, Februry 214
Tomogrphic Reconstruction - verticl Verticl distribution of the wet refrctivity N wet up to m ASL. N wet N wet 1 2 3 4 1 2 3 4 3 4 3 2 1 47 48 49 51 52 53 54 ltitude 3 3 3 47 48 49 51 52 53 54 ltitude 3 3 4 3 2 1 ltitude ASL [m] ltitude ASL [m] COSMO DE Tomogrphy ISDA, Munich, Februry 214
Schleswig 27 8 9 h12 RSRwProfile.xy Fritzlr 27 8 9 h12 RSRwProfile.xy Greifswld 27 8 9 h12 RSRwProfile.xy Meiningen 27 8 9 h12 RSRwProfile.xy Emden 27 8 9 h12 RSRwProfile.xy Idr 27 8 9 h12 RSRwProfile.xy Bergen 27 8 9 h12 RSRwProfile.xy Stuttgrt 27 8 9 h12 RSRwProfile.xy Lindenberg 27 8 9 h12 RSRwProfile.xy Kuemmersbr 27 8 9 h12 RSRwProfile.xy Essen 27 8 9 h12 RSRwProfile.xy Muenchen 27 8 9 h12 RSRwProfile.xy Rdiosonde Vlidtion Sttion Schleswig 1 rdiosonde profile tomogrphy profile Sttion Greifswld 1 rdiosonde profile tomogrphy profile Sttion Emden 1 rdiosonde profile tomogrphy profile Sttion Bergen 1 rdiosonde profile tomogrphy profile Sttion Lindenberg 1 rdiosonde profile tomogrphy profile Sttion Essen 1 rdiosonde profile tomogrphy profile 4 4 4 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 Sttion Fritzlr 1 rdiosonde profile tomogrphy profile Sttion Meiningen 1 rdiosonde profile tomogrphy profile Sttion Idr 1 rdiosonde profile tomogrphy profile Sttion Stuttgrt 1 rdiosonde profile tomogrphy profile Sttion Kuemmersbr 1 rdiosonde profile tomogrphy profile Sttion Muenchen 1 rdiosonde profile tomogrphy profile 4 4 4 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 Rdiosonde profiles Tomogrphy profiles, interpolted to the RS sttion coordintes ISDA, Munich, Februry 214
Schleswig 27 8 28 h12 RSRwProfile.xy Fritzlr 27 8 28 h12 RSRwProfile.xy Greifswld 27 8 28 h12 RSRwProfile.xy Meiningen 27 8 28 h12 RSRwProfile.xy Emden 27 8 28 h12 RSRwProfile.xy Idr 27 8 28 h12 RSRwProfile.xy Bergen 27 8 28 h12 RSRwProfile.xy Stuttgrt 27 8 28 h12 RSRwProfile.xy Lindenberg 27 8 28 h12 RSRwProfile.xy Kuemmersbr 27 8 28 h12 RSRwProfile.xy Essen 27 8 28 h12 RSRwProfile.xy Muenchen 27 8 28 h12 RSRwProfile.xy Rdiosonde Vlidtion Sttion Schleswig 1 rdiosonde profile tomogrphy profile Sttion Greifswld 1 rdiosonde profile tomogrphy profile Sttion Emden 1 rdiosonde profile tomogrphy profile Sttion Bergen 1 rdiosonde profile tomogrphy profile Sttion Lindenberg 1 rdiosonde profile tomogrphy profile Sttion Essen 1 rdiosonde profile tomogrphy profile 4 4 4 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 Sttion Fritzlr 1 rdiosonde profile tomogrphy profile Sttion Meiningen 1 rdiosonde profile tomogrphy profile Sttion Idr 1 rdiosonde profile tomogrphy profile Sttion Stuttgrt 1 rdiosonde profile tomogrphy profile Sttion Kuemmersbr 1 rdiosonde profile tomogrphy profile Sttion Muenchen 1 rdiosonde profile tomogrphy profile 4 4 4 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 Rdiosonde profiles Tomogrphy profiles, interpolted to the RS sttion coordintes ISDA, Munich, Februry 214
Tomogrphy: GPS + GLONASS + Glileo Horizontl distribution of the wet refrctivity Nwet t 19 m ASL 8 1 9 1 11 5 12 1 35 4 95 1 588 slnts 14 July 29, 13:3 UTC 7 95 95 1 95 1 1 95 1 4 1 95 6 5 15 14 15 13 95 12 95 11 27912 slnts 1 13 Nwet Nwet 35 4 95 1 9 95 5 95 1 1 8 48 95 95 7 95 6 5 15 14 13 12 4 11 354 1 25 3 9 8 49 1 8 7 5 7 6 48 5 1 51 52 95 15 5 49 95 1 35 4 56 95 49 51 53 54 48 5 7 52 5 1 5 5 354 53 4 51 52 4 54 53 54 GPS+GLONASS+Glileo 95 GPS+GLONASS GPS 14 15 Nwet 35 4 95 1 776 slnts Resolution: 22 28 32 voxel 15 minutes of GNSS observtions ISDA, Munich, Februry 214
Tomogrphy? 54 N wet 3 km grid Slnts up to 5 km 53 52 4 blck GPS blue GLONASS white Glileo 1 11 12 13 14 ISDA, Munich, Februry 214
GNSS Dt Assimiltion II. GNSS Dt Assimiltion ISDA, Munich, Februry 214
Wether Models nd Assimiltion Systems Opertionl numericl wether models running t the DWD Globl Model: GME, ICON Resolution 2 km, 3D-Vr dt ssimiltion Europen Model: COSMO-EU Resolution 7 km, ssimiltion: nudging will be replced by ICON with locl grid refinement over EUROPE Germn Locl Model: COSMO-DE Resolution 2.8 km, ssimiltion: nudging, will soon be replced by locl trnsform ensemble Klmn filter (LETKF) ISDA, Munich, Februry 214
STD Assimiltion Opertor The STD ssimiltion opertor H STD consists of 3 prts: 1) Compute the 3D refrctivity field N(r) p N(r) = k d 1 T + k 2 e T + k 3 e, r is the position vector T 2 2) Estimte the bended signl pth S rytrcing Solve Fermt s Principle for the given tmospheric stte nd given stllite-receiver geometry: Vritionl problem S n(x, y(x), z(x)) 1 + y 2 + z 2 dx = min. 3) Compute the line integrl for given signl pth S STD = S n(x, y, z) 1 + y 2 + z 2 dx G S is prmeterized curve through the sclr field N or n, G is the geometric distnce between stellite nd receiver. ISDA, Munich, Februry 214
Fermt s Principle Vritionl Problem: n(x, y(x), z(x)) 1 + y 2 + z 2 dx = min. S The Euler-Lgrnge equtions led to set of differentil equtions: y = z = ( ny n n x n y ) (1 + y 2 + z 2 ) ( nz n n x n z ) (1 + y 2 + z 2 ) n - refrction index, n x = n x, n y = n y, n z = n z Derivtives re required in crtesin slnt coordintes but n is given in ellipsoidl coordintes: n = n(λ, β, h) ISDA, Munich, Februry 214
Fermt s Principle Difference Equtions Difference equtions with Lgrnge polynomils: j+1 L j,k(x j )y k n y (x j, y j, z j ) n(x j, y j, z j ) n x(x j, y j, z j ) n(x j, y j, z j ) k=j 1 j+1 k=j 1 L j,k(x j )z k 1 + j+1 k=j 1 L j,k(x j )y k + n z(x j, y j, z j ) n(x j, y j, z j ) n x(x j, y j, z j ) n(x j, y j, z j ) 1 + j+1 k=j 1 L j,k(x j )y k + 2 2 j+1 k=j 1 j+1 k=j 1 j+1 k=j 1 j+1 k=j 1 L j,k(x j )y k 2 L j,k(x j )z k = L j,k(x j )z k 2 L j,k(x j )z k = For given set of supporting points x i this leds to bnd mtrix of bnd width 7 which cn be solved by the Newton lgorithm. ISDA, Munich, Februry 214
Rytrcer Relistion Quntities re given in different frmes of reference: model (n) - ellipsoidl coordintes = crtesin ECEF coordintes = locl horizon system = slnt system Derivtives of the trnsformtions re required for n x, n y, n z The Newton lgorithm requires the Jcobin of the difference equtions: Exct derivtives re required s finite differences re much too slow. = second derivtives required Algorithm is very sensitive to errors in the derivtives but very stble nd fst for computing the bended signl pth: Only 1 or 2 Newton itertions re required down to elevtions of 5. ISDA, Munich, Februry 214
The Bended Signl Pth The rytrcer is strtet with predefined set of supporting points on the stellite-receiver xis ({x i }). The solution re the devitions y(x) nd z(x) from the stright line leding to the 3D signl pth S = (x, y(x), z(x)) z(x) [m] y(x) [m] 1 4 2,,,4,3,2,1, ε = 1 o ε = 5 o ε = 7 o ε = 1 o ε = 3 o 1 1 2 2 distnce to receiver, x [km] ISDA, Munich, Februry 214
Bended Signl Pth inside the Model 1 z(x) [m] 4 model top t 2 km model top t 4 km 2 y(x) [m],,,4,3,2,1, ε = 1 o ε = 5 o ε = 7 o ε = 1 o ε = 3 o 1 2 3 4 distnce to receiver, x [km] ISDA, Munich, Februry 214
Impct of the Rytrcer STD / m STD with rytrcing vs. stright line 4 3 2 1 STD rytrcing STD line 1 2 3 Elevtion ε 4 4 2 STD [mm] 2 12 1 4 2 STD / mm 4 3 2 1 1 mm error ε = 33 o STD = 1 mm 5 1 15 2 25 3 35 4 elevtion ε / deg. ISDA, Munich, Februry 214
Conclusions nd Outlook GNSS slnt delys provide vluble informtion bout sptil tmospheric structures. STDs nd ZTDs cn be ssimilted to wether models using observtion opertors but for elevtions below 35 rytrcer is needed to void systemtic overestimtion of the dely. Tomogrphiclly reconstructed humidity fields cn be ssimilted to void direct ssimiltion of lrge numbers of STDs. Tomogrphy s independent ppliction requires much more nd better observtions to reconstruct relible 3D fields from slnt delys. GLONASS, Compss nd Glileo observtions will be vilble in ner future, rel-time products within the next yers. To resolve the tmospheric wter vpour distributions more dense GNSS networks would be required. ISDA, Munich, Februry 214