GNSS Data Assimilation and GNSS Tomography for Global and Regional Models
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1 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 Februry 214 M. Bender,, K. Stephn,, A. Rhodin,, R. Potthst
2 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
3 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
4 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
5 Why Use Slnt Delys? ZTDs informtion bout horizontl structure but no verticl profiles Horizontl slnts (ε = ) verticl profiles but no horizontl structures Rel STDs (5 ε ) 1 verticl profiles nd horizontl structures 1 informtion bout the 3D field elevtion / deg. Number of entries / 2 deg. ISDA, Munich, Februry 214
6 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
7 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) (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
8 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
9 Number of Observtions - Avilble STD Dt number of slnts in 3 minutes GPS GLONASS Glileo time, UTC N sl 12 h N sl 3 min. N sl min. GPS GLONASS Glileo Slnts for tomogrphy GNSS slnt dely dt from 3 Germn GNSS sttions. ISDA, Munich, Februry 214
10 Impct of the GPS Stellite Constelltion 1:-1:15 UTC, 14 slnts ISDA, Munich, Februry 214 1:15-1:3 UTC, slnts
11 GNSS Tomogrphy I. GNSS Tomogrphy ISDA, Munich, Februry 214
12 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
13 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) (y H(x))T R 1 (y H(x)) Solutions: Conjugte grdients, Tikhonov regulristion, Klmn filter, singulr vlue decomposition, ART,... ISDA, Munich, Februry 214
14 GPS-Tomogrphy - Germny Grid resolution Grid dimension: 1 km3 Horizontl res. : 35 - km Verticl res. : 2 - m 3 min. temporl resolution voxel in E-W direction 25 voxel in N-S direction 2-3 verticl voxels (up to 8-1 km) Voxel 3 STDs 14 ISDA, Munich, Februry 214
15 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
16 Tomogrphic Reconstruction - horizontl Horizontl distribution of the wet refrctivity N wet t 19 m ASL. COSMO-DE: Tomogrphy: IWV: N wet N wet IWV [kg m 2 ] July 29, 16: UTC Tomogrphic rekonstruction using 269 sttions nd 3 minutes of GPS dt. ISDA, Munich, Februry 214
17 Tomogrphic Reconstruction - verticl Verticl distribution of the wet refrctivity N wet up to m ASL. N wet N wet ltitude ltitude ltitude ASL [m] ltitude ASL [m] COSMO DE Tomogrphy ISDA, Munich, Februry 214
18 Schleswig h12 RSRwProfile.xy Fritzlr h12 RSRwProfile.xy Greifswld h12 RSRwProfile.xy Meiningen h12 RSRwProfile.xy Emden h12 RSRwProfile.xy Idr h12 RSRwProfile.xy Bergen h12 RSRwProfile.xy Stuttgrt h12 RSRwProfile.xy Lindenberg h12 RSRwProfile.xy Kuemmersbr h12 RSRwProfile.xy Essen h12 RSRwProfile.xy Muenchen 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 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 Rdiosonde profiles Tomogrphy profiles, interpolted to the RS sttion coordintes ISDA, Munich, Februry 214
19 Schleswig h12 RSRwProfile.xy Fritzlr h12 RSRwProfile.xy Greifswld h12 RSRwProfile.xy Meiningen h12 RSRwProfile.xy Emden h12 RSRwProfile.xy Idr h12 RSRwProfile.xy Bergen h12 RSRwProfile.xy Stuttgrt h12 RSRwProfile.xy Lindenberg h12 RSRwProfile.xy Kuemmersbr h12 RSRwProfile.xy Essen h12 RSRwProfile.xy Muenchen 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 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 Rdiosonde profiles Tomogrphy profiles, interpolted to the RS sttion coordintes ISDA, Munich, Februry 214
20 Tomogrphy: GPS + GLONASS + Glileo Horizontl distribution of the wet refrctivity Nwet t 19 m ASL slnts 14 July 29, 13:3 UTC slnts 1 13 Nwet Nwet GPS+GLONASS+Glileo 95 GPS+GLONASS GPS Nwet slnts Resolution: voxel 15 minutes of GNSS observtions ISDA, Munich, Februry 214
21 Tomogrphy? 54 N wet 3 km grid Slnts up to 5 km blck GPS blue GLONASS white Glileo ISDA, Munich, Februry 214
22 GNSS Dt Assimiltion II. GNSS Dt Assimiltion ISDA, Munich, Februry 214
23 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
24 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
25 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
26 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 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
27 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
28 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 distnce to receiver, x [km] ISDA, Munich, Februry 214
29 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 distnce to receiver, x [km] ISDA, Munich, Februry 214
30 Impct of the Rytrcer STD / m STD with rytrcing vs. stright line STD rytrcing STD line Elevtion ε STD [mm] STD / mm mm error ε = 33 o STD = 1 mm elevtion ε / deg. ISDA, Munich, Februry 214
31 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
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