Detecting and monitoring the time-variable gravity field of Greenland

Transcription

1 Detecting and monitoring the time-variable gravity field of Greenland - using reprocessed GOCE gradients - V. Lieb, J. Bouman, M. Fuchs, M. Schmidt Deutsches Geodätisches Forschungsinstitut (DGFI) Centre of Geodetic Earth System Research (CGE) Munich, Germany lieb@dgfi.badw.de, schmidt@dgfi.badw.de 1

2 0. Motivation mass variations different measurement systems ice melting in Greenland GOCE: sufficient time span reprocessing -> improved accuracy time-variable gravity field of Greenland 2

3 1. Spectral sensitivity resolution levels split spectrum into frequency bands upper boundary corresponds to the maximum degree in a series expansion relation to the spatial resolution on the Earth s surface GOCE covers higher frequencies than GRACE MBW: 5 30 mhz , [level] [deg] [km] GRACE frequency [deg] GOCE

4 2. GOCE data set Measuring: Vzz GRF; Vzz Vrr Modifying: Vzz_mod = (Vzz-Vxx-Vyy)/2 40% noise reduction Filtering: MBW: 5 30 mhz low frequencies: GOCO03S Creating 2-month data sets: 2 months time span: 11/ / month overlap repeat cycle: ~ 61 days 4

5 3. Regional gravity field modeling j [level] l [deg] r [km] frequency [deg] GOCE level j =7 2 month highest level within MBW (5 30 mhz ~ d/o ) level j = 8 might be influenced by artefacts of non-equidistributed satellite tracks 1) Subtracting background model: Vzz = Vzz - Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( j = 7 ) (reduce static part completely) Vzz (approx.) Vback 5

6 3. Regional gravity field modeling background model: GOCO03S Dg [mgal], d/o 127 (j=7) 6

7 3. Regional gravity field modeling 1) Subtracting background model: Vzz = Vzz- Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( J = 7 ) (reduce static part completely) 2) Analysis: series expansion in terms of reproducing kernel krepro d/o 140 estimating unknown scaling coefficients d J avoiding omission errors Vzz (approx.) Vback 3) Synthesis: series expansion in scaling functions Blackman d/o 127 modeling approximation signal F j low-pass filter value j = 3 j = 4 level (scale) j = 5 degree l 7

8 3. Regional gravity field modeling 1) Subtracting background model: Vzz = Vzz- Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( J = 7 ) (reduce static part completely) 2) Analysis: series expansion in terms of reproducing kernel krepro d 140 estimating unknown scaling coefficients d 6 avoiding omission errors Vzz (approx.) Vback Blackman d/o 127 3) Synthesis: series expansion in scaling functions modeling approximation signal F j low-pass filter value j = 3 j = 4 level (scale) j = 5 degree l 8

9 3. Regional gravity field modeling 1) Subtracting background model: Vzz = Vzz- Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( J = 7 ) (reduce static part completely) 2) Analysis: series expansion in terms of reproducing kernel krepro d 140 estimating unknown scaling coefficients d 6 avoiding omission errors Vzz (approx.) Vback 3) Synthesis: series expansion in scaling functions Blackman d 127 modeling approximation signal F 7 low-pass filter 9

10 3. Regional gravity field modeling + full modeled signal: In: Goce Vzz Out: Dg [mgal], d 127 (j=7) background model: GOCO03S Dg [mgal], d/o 127 (j=7) approximation signal F J=7 : Dg [mgal], d 127 (j=7) 10

11 3. Regional gravity field modeling + full modeled signal: In: Goce Vzz Out: Dg [mgal], d 127 (j=7) background model: GOCO03S Dg [mgal], d/o 127 (j=7) approximation signal F J=7 : Dg [mgal], d 127 (j=7) 11

12 4. Time series i. Regional approach GOCE ii. Regional approach GRACE 2 months static 2 months static Dg [mgal], 7, months (11/09, 12/09) Dg [mgal], 7, 127 static (11/09 03/12) Dg [mgal], 7,

13 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Goce Vzz 2 months static Dg [mgal], 7, 127 diff_mean_dg [mgal]

14 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Regional approach - GOCE ( ) Goce Vzz 2 months static Dg [mgal], 7, 127 mean Vzz (approx.) Vback subtracted diff_mean_dg [mgal] variation: ~ 30 µgal seasonal variations max: Mar/Apr, Sep/Oct min: Dec/Jan, Jul/Aug no trend 2 months static 14

15 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Regional approach GRACE ( ) 1) Computing the potential differences, from GSM potential fields 2) Subtracting a background model:,,, dv (approx.) Vback [level] [deg] [km] GRACE frequency [deg]

16 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Regional approach GRACE ( ) Grace dv 2 months static Dg [mgal], 6, 63 mean dv (approx.) Vback subtracted diff_mean_dg [mgal] variation: ~ 5 µgal seasonal variations max: Mar/Apr min: Aug/Sep descending trend? 2 months static 16

17 4. Time series i. Regional approach GOCE ii. Regional approach GRACE iii. Comparison: regional vs. global Grace dv 2 months static Dg [mgal], 6, 63 mean dv (approx.) Vback subtracted diff_mean_dg [mgal] Grace GSM 1 month GOCO03S Dg [mgal], 6, 63 spherical harmonics diff_mean_dg [mgal] variation: 5 µgal seasonal variations (not so clear) max: Apr min: Oct descending trend:5 µgal -> parallels detectable 1month 17

18 5. Aim: combination regional: j=7 vs. j=6 diff_mean_dg [mgal] mean Vzz Vback subtracted Goce Vzz 7, 127 consistent maxima and minima j=6: regional vs. global diff_mean_dg [mgal] mean dv Vback subtracted Grace dv 6, 63 consistent trend diff_mean_dg [mgal] spherical harmonics Grace GSM 6, 63 18

19 5. Aim: combination Pyramidal algorithm J GOCE d J G J detail signal fine resolutio n level GRACE number of coefficients d J-1 d j-1 = H j-1 d j G J-1 detail signal H j-1 low-pass filter matrix structure s j d j G j detail signal coarse 19

20 Summary GOCE: originally planned to observe the Earth s static gravity field, but time variations visible?? 2-months solutions compared with static solution semi-seasonal variations detected with high sensitivity by GOCE good consistency to GRACE combination of GOCE + GRACE exploit highest degree of information from each measurement technique Open questions: - consistent data sets (2-months GOCE vs. 1-month GRACE-GSM)? - Vzz_mod: systematic errors caused by Vyy? - using full GOCE gradient tensor information? - combination by pyramidal algorithm (step-by-step introducing new observation techniques)? j=7 j=6 20

21 Appendix 21

22 3. Regional gravity field modeling With all these assumptions we obtain the observation equation for the modified GOCE gravity gradient with Δ, Δ Δ,,, (globally the condition 16, , has to be fulfilled). The modified scaling functions, are defined as, Φ,. With the 1 vectors,,,,,, and,,,,,, the general observation equation reads Δ. 22

23 3. Regional gravity field modeling Considering the prior information and for the expectation vector and the covariance matrix of the vector the linear model results, wherein and are unknown variance components, is the given positive weight matrix of the observations. Variance component estimation yields the estimation of the coefficient vector and its covariance matrix. Introducing the parameter / the solution can be rewritten as 23

24 3. Regional gravity field modeling i. General approach ii. MRR 1) Subtracting background model: Vzz = Vzz- Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( J = 7 ) (reduce static part completely) 2) Analysis: series expansion in terms of reproducing kernel krepro d 140 estimating unknown scaling coefficients d 6 avoiding omission errors Vzz (approx.) Vback 3) Synthesis: series expansion in scaling functions Blackman d 127 modeling approximation signal F 7 low-pass filter 4) MRR: series expansion in wavelet functions Vback: GOCO03S d/o 127 splitting approximation signal into detail signals G j=0,,6 using Blackman wavelet functions band-pass filter 24

25 3. Regional gravity field modeling i. General approach ii. MRR 1) Subtracting background model: Vzz = Vzz- Vback Vback: GOCO03S d/o 127 the same as used for filling up low frequencies according to modeling resolution ( j = 7 ) (reduce static part completely) 2) Analysis: series expansion in terms of reproducing kernel krepro d/o 140 estimating unknown scaling coefficients d j avoiding omission errors Vzz (approx.) Vback 3) Synthesis: series expansion in scaling functions Blackman d/o 127 modeling approximation signal F j low-pass filter 4) MRR: series expansion in wavelet functions Vback: GOCO03S d/o 127 splitting approximation signal into detail signals G j=0,,6 using Blackman wavelet functions band-pass filter 25

26 3. Regional gravity field modeling i. General approach ii. MRR up to 127 up to 63 up to 31 Multi-resolution representation (MRR)

27 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Regional approach GRACE ( ) 1) Computing the potential differences, from GSM potential fields 2) Subtracting a background model:,,, dv (approx.) Vback [level] [deg] [km] GRACE frequency [deg]

28 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Regional approach GRACE ( ) 1) Computing the potential differences, from GSM potential fields 2) Subtracting a background model:,,, dv (approx.) Vback 3) Analysis: series expansion in terms of reproducing kernel (up to degree 75 Δ,,,,,,,,, 28

29 4. Time series i. Regional approach GOCE ii. Regional approach GRACE Goce Vzz 2 months static Dg [mgal], j=7 (d/o127) mean Vzz (approx.) Vback subtracted diff_mean_dg [mgal] Goce Vzz 2 months static Dg [mgal], j=6 (d/o63) mean Vzz (approx.) Vback subtracted diff_mean_dg [mgal] Grace dv 2 months static Dg [mgal], j=6 (d/o63) mean dv (approx.) Vback subtracted diff_mean_dg [mgal]

30 5. Aim: combination Goce Vzz 2 months static Dg [mgal], j=7 (d/o127) mean Vzz (approx.) Vback subtracted diff_mean_dg [mgal] Grace GSM 1 month GOCO03S Dg [mgal], j=6 (d/o63) GOCE j=7 spherical harmonics variation: 75 µgal sensitive at high frequencies seasonal variations GRACE j=6 variation smaller clear trend diff_mean_dg [mgal] month 30