Study of an Improved Comprehensive Magnetic Field Inversion Analysis for Swarm PM1, E2Eplus Study Work performed by Nils Olsen, Terence J. Sabaka, Luis R. Gaya-Pique, Lars Tøffner-Clausen, and Alexei Kuvshinov, Presented by: Nils Olsen

29. March 2006 | PM1 E2Eplus | page 2 Draft Agenda Swarm E2Eplus Progress Meeting 1, March , at DNSC Copenhagen 09:00 Welcome 09:05 Presentation of activities done so far (NIO) Fast Orbit Prediction (theory plus practical demonstration) Results of re-analysis of E2E CI Task 3 data using higher sampling rate Gradient Approach: first ideas and their implementation Plans for the near future General discussion 12:30 lunch 13:30 AOB 14:45 Adjourn

29. March 2006 | PM1 E2Eplus | page 3 Fast Orbit Prediction (FOP)

29. March 2006 | PM1 E2Eplus | page 4 Fast Orbit Prediction Approach used for E2E (Phase A): –Numerical integration of equations of motion, considering a lot of (tiny) effects –Some of the small effects are rather uncertain (e.g., air-drag), and therefore the position prediction error increases tremendously with time –Due to this uncertainty, a ”precise” orbit prediction (extrapolating several months/years in future) is not more precise than an approach that focuses on time- averaged effects (plus short-term effects due to change of air-drag) New approach –considering what is needed for the simulation : –circular near-polar orbits –realistic drift in local time –realistic altitude decay (solar activity effects …) –realistic maintenance of constellation

29. March 2006 | PM1 E2Eplus | page 5 Fast Orbit Prediction Circular orbit of radius a sma and inclination i in the orbit-fixed coordinate system Rotation by around z -axis to get orbit in ICRF: Rotation by - GAST around z -axis to get orbit in ITRF:

29. March 2006 | PM1 E2Eplus | page 6 Orbit Decay due to Air-Drag For a circular orbit, the decrease  a sma of the semi-major axis a sma per orbit is is the ballistic coefficient, and  is air density Since 1/T p with is the number of orbits per day, the decrease of the semi-major axis per day is Calculation of daily mean air density (MSIS) along orbit Linear distribution of  a sma over the day in consideration

29. March 2006 | PM1 E2Eplus | page 7 The Algorithm 1.Initial values ( a sma,,  ) for epoch t 0 2.Calculation of one day of positions r ITRF 3.Calculation of mean air density along orbit 4.Calculation of mean orbit decay,  a sma 5.Linear distribution of  a sma over the day, 6.New initial values ( a sma,,  ) for next day, i.e. epoch t=t 0 +1 day 7.Repeat steps 1 – 6 until end of mission (altitude < 200 km)

29. March 2006 | PM1 E2Eplus | page 8 Validation against CHAMP orbit observations Simulation of 5.5 years of CHAMP orbits Initial conditions, August 1, 2000, 00:00 UT –inclination i = ° –semi-major axis a sma = a km –mean anomaly = ° –RAAN  = ° Ballistic coefficient B = m/(A C D ) –m is satellite mass –C D is drag coefficient –A is effective satellite cross section (A x = 0.74 m 2, A y = 3.12 m 2, A z = 4.2 m 2 ) –5° misalignment between x-direction and actual flight direction: A = 1.01 m 2 –B = 230 kg/m 2 is a reasonable value, according to Hermann Lühr (compatible with A = 0.9 m 2, m = 500 kg, C D =2.4)

29. March 2006 | PM1 E2Eplus | page 9 Geomagnetic and Solar activity

29. March 2006 | PM1 E2Eplus | page 10 Observed vs. simulated altitude and LT

29. March 2006 | PM1 E2Eplus | page 11 Difference CHAMP observed - simulated

29. March 2006 | PM1 E2Eplus | page 12 Impact of higher sampling rate on lithospheric field recovery

29. March 2006 | PM1 E2Eplus | page 13 Comparison of Filter Method and CI, E2E Phase A: –CI superior at n<80, especially for terms m close to 0 –Gradient method is superior for n > 80 Gradient Method Sensitivity matrix CI

29. March 2006 | PM1 E2Eplus | page 14 Conceptual Example Orbit period: about 90 minutes, corresponding to 4°/min 1-min sampling rate: along-track structures smaller than 4° are not resolved Consider an orbit in the equatorial plane (inclination=0°) 1-min sampling rate: only spherical harmonic coefficients of order m 90 are unresolved Example: a) Equatorial orbit with spherical harmonic coefficients b) transformation to system with orbit inclination 86.8°

29. March 2006 | PM1 E2Eplus | page 15 Result:

29. March 2006 | PM1 E2Eplus | page 16 Assessment criteria Test quantities: Difference between recovered and original model –Power spectrum of the model SH coefficients and of the coefficients of the difference (original – recovered) –Degree correlation  n of coefficients –Sensitivity matrix –Global Maps (e.g., of B r ) of the model difference

29. March 2006 | PM1 E2Eplus | page 17 Assessment, lithospheric field, Phase A Combined solution: –CI result for n < 83 –Gradient method result for n ≥ 83

29. March 2006 | PM1 E2Eplus | page 18 Re-analysis of Constellation #2 data Phase A: 1 min sampling rate Now: 30 secs, respect. 15 secs sampling rate

29. March 2006 | PM1 E2Eplus | page 19 Re-analysis of Constellation #2 data

29. March 2006 | PM1 E2Eplus | page 20 The Gradient Method in the Comprehensive Inversion Approach

29. March 2006 | PM1 E2Eplus | page 21 On the Comprehensive Approach Comprehensive Approach: Modeling of all relevant contributions to Earth’s magnetic field Simultaneous (co-) estimation of all sources Presently: all data are sensitive to all parts of the model –Example 1: crustal field is obtained from all (also dayside) data insufficient description of day-side equatorial electrojet may lead to contamination of crustal field –Example 2: high- as well as low-order lithospheric field is determined from all data no explicit use of field gradient information

29. March 2006 | PM1 E2Eplus | page 22 “Selective Infinite Variance Weighting” Development of an approach that produces/identifies data subsets that are particularly sensitive to certain parameter subsets and applying appropriate weighting such that these data strongly influence the determination of such parameters –Example: high-order crustal field is resolved by gradient information (data difference) low-order field is resolved by data sum d 1, d 2, d 3 are data of Swarm 1,2,3 d s, d d, are sum and difference of Swarm 1,2 x is all model parameters but crustal field (sensed by all satellites ) y l is low-order crustal field (sensed by d s, d 3 ) y h is high-order crustal field (sensed by d d )

29. March 2006 | PM1 E2Eplus | page 23 Plans for the Near Future Implementation of selective weighting scheme in CI code Application to constellation # 3 data Results expected to be presented at Swarm workshop in Nantes (May 2006) Implementation of in-flight alignment (co-estimation of Euler angles) in CI code Application to constellation # 3 data Results expected to be presented at MTR (June 2006)

29. March 2006 | PM1 E2Eplus | page 24 Action Items AI-001 of KO meeting: ”Info on error characteristic of Optical Bench model in terms of Euler angles“ This information is requested needed at the beginning of May (Swarm workshop in Nantes), rather than MTR.

29. March 2006 | PM1 E2Eplus | page 25

29. March 2006 | PM1 E2Eplus | page 26 E2Eplus Study Logic

29. March 2006 | PM1 E2Eplus | page 27 Work Breakdown Structure

29. March 2006 | PM1 E2Eplus | page 28 Updated list of proposed Meetings and Deliverables Meeting PlaceParticipantsPlanned DateDeliverables Kick-Off Meeting (KO)DNSCAllJanuary 27, Progress Meeting 1DNSC3March 29, Midterm Review (MTR)ESTEC4June 19, 2006Draft report Progress Meeting 2DNSC3September 25, Final Presentation (FP)ESTEC4December 4, 2006 Final report