Faster, better, cheaper:News on seeking Gaia's Astrometric
Solution with AGIS
U. Lammers*, L. Lindegren**, A. Bombrun***, W. O'Mullane*, D. Hobbs*** European Space Astronomy Centre of ESA, Madrid, Spain, ** Lund Observatory, Lund, Sweden,*** Astronomisches Recheninstitut, University of Heidelberg, Germany

Abstract
Gaia is ESA's ambitious space astrometry mission with a foreseen launch date in early 2012. Its main objective is to perform a stellar census of the 1000 Million brightest objects in our galaxy(completeness to V=20 mag) from which an astrometric catalog of micro-arcsec level accuracy will be constructed. A key element in this endeavor is the Astrometric Global Iterative Solution(AGIS) - the mathematical and numerical framework for combining the ~80 available observations per star obtained during Gaia's 5yr lifetime into a single global astrometric solution. At last
year's ADASS XVIII we presented (O4.1) in detail the fundamental working principles of AGIS, its development status, and selected results obtained by running the system on processing
hardware at ESAC, Madrid with large-scale simulated data sets. We present here the latest developments around AGIS highlighting in particular a much improved algebraic solving method thathas recently been implemented. This Conjugate Gradient scheme improves the convergence behavior in significant ways and leads to a solution of much higher scientific quality.

AGIS input: Gaia’s astrometric measurements
RVS1

RVS3

RVS2

RP

AF9

BAM-NBAM-R WFS1

WFS2

BP

AF6

photometry radial velocity(dispersed (dispersed
images)images)
AF8

AF5

AF7

AF3

AF2

AF1

SM1

astrometricmeasurements

0.42 mSM2

detection
and FOV
discrimination

AF4

AGIS in a nutshell• AGIS is the mathematical and numerical scheme that shall be used to generate the core
astrometric mission products of Gaia: The five (six) standard astrometric parameters, position(2x), parallax (1x), proper motion (2x), radial velocity (1x) for all observed celestial objects
brighter than G=20mag with targeted accuracies (e.g. <10 uas [G<10mag], <25 uas [G=15mag],<300uas [G=20]).
• The best global match between all measurement data and an observational model involving
the unknown source, attitude, calibration, and global parameters is sought in a least-square
sense using normal equations and a block-iterative solution scheme• The system is developed in DPAC's CU3 by ESAC (Spain) and Lund Observatory (Sweden)• AGIS shall run every 6 months on an ever increasing raw data volume• The operational system is baselined to run on dedicated processing hardware at ESAC – a
10 TFLOP/s machine is envisaged – an alternative “Cloud” computing solution is being studied

AGIS input:
1012 observationsof millarcsec
accuracy

window of 6-18 samples
transmitted to ground0.93 m
BAM = basic angle monitor, WFS = wavefront sensor

3.2 Mio stars
CG referencecycle• Continuous large-scale validation tests aim at Demonstrating that the targeted mission
accuracies can be reached Study convergence behavior of the systemand devise acceleration schemes• About 40 iterations are currently (with SI
scheme, see below) needed to converge to asolution starting from conservative, noisy initialstarting values"Time of observation" for image centre relative to CCD
determined to ~200 μas precision (magnitude 15)
Some 700 such measurements per object in 5 years ⇒ 1012 observations“Simple Iterations” and “Conjugate Gradients”• Simple Iterations works but converges very slowly – after about 40 iterations
the spatial errors show still “zonal patterns”
Mean error about 20 muas max, so the system has reduced
the initial error by more than 3 powers of 10

Algebraic Details

Error pattern shows the eigenvector
corresponding to the largest eigenvalueof the iteration matrix (I – K–1N).
Solution is NOT converged!

• Assuming a centered Gaussian observation noise of standard deviation sL the astrometricproblem is equivalent to a weighted least-square minimization of()

sum over observations

statistical weight

Observed-Calculated (Residual)• So, the solution vector x=(si, aj, ck) gives the optimal agreement between the observationalmodel f and the actual observations• What is really solved in practice is a system of normal equations system for the updates Δx
to the parameters x:∂RL⎡ ∂RL ∂RL⎤⎢∑ ∂x ∂x′ WL ⎥ ⋅ Δx = −∑ ∂x RLWL
⎣L
⎦
L

⇔

attitude params

2wL
min J ( x) = ∑ t Lobs − f L ( x, aux)
= ∑ RL ( x, aux ) 2 WL
σ L2 + ε L2 Lx
LN ⋅ Δx = bx0“simple” iteration 2“simple” iteration 1
source params

N is a symmetric n×n matrix, b is an array of length n = number of parameters
Most of AGIS is about solving the system N·Δx = b
This needs to be done by iterations because of non-linearities and the very large
size of the Normal equation matrix N

sourceattitudes1s2s3 ...

a1

a2

a3

calibration...

Filledcone
5×5 matrix
per source

Sparsesource(5·108)

[S-A]

N =

attitude(4·107)

[A-C]
[C-A]

Zeroes[S-C]

K ⋅ Δx = b

one bandmatrix per
attitudeinterval• Setting the “disturbing” off-diagonal blocks [S-A],[S-C], [A-C], [C-A] to zero gives an approximation
to N which is the Gauss-Seidel Preconditioner
matrix K• Then what is solved is (easy)
K ⋅ Δx = b instead of (hard) N ⋅ Δx = b

calibration(~106)

the non-zeroeleaments here
connect thes, a, cparametersand make thesystem hard
to solve

set s, a, c to initial approximation

This is the”SimpleIteration” (SI)
schemeSource one
Blockstar at a time
Attitude Block
Calibration Blocks new = S ( obs | a old , c old )a new = A( obs | snew , cold )cnew = C ( obs | snew , a old )iterate untilconvergence

one matrix per cal. unit

Align s andadjust a

Subset of
QSOs
VLBI
positions

AGIS and Nano-JASMINE• Nano-Jasmine is a small Japanese near-infrared (z-band: 0.6-1 µm) astrometry
mission - foreseen launch: late 2010 - lifetime: 2 years• Aims at completeness to z=10 (~1 Mio stars) - expected accuracy: 2-3 mas(roughly “Hipparcos”)• Design and operating principles very similar to Gaia (2 telescopes separated bybasic angle, satellite is spinning about its main symmetry axis, one single CCD in
TDI mode)• Aim is to construct the astrometric catalog with AGIS
• No “special” AGIS version required – only needs nano-JASMINE version of Initial
Data Treatment (IDT) and the exchange of Gaia Parameter Data Base by NanoJasmine version• AGIS should process Nano-Jasmine data like coming from a “crippled” Gaia with
only 1 CCD strip and row• A problem with SI is that updates within a
block are constrained to be along the data
space axes and there is no direct “crosstalk” between the blocks• Conjugate Gradients (CG) is a standard
linear algebra method to solve large
systems of linear equations - CG does nothave this limitation – in theory it could
reach the minimum in a single step

Results
SI: Convergencein ~45 iterations

CG: Convergencein <15 iterations¾So far only results from AGISLab:CG works and can produce areally converged solution, free ofsystematic errors
Number of needed iterations is
reduced by a factor 2-3¾Implementation in AGIS completed– validation is ongoing – preliminary
results are encouraging¾Convergence acceleration perhaps
less spectacular than in AGISLabbecause of outliers in data (AGISLabuses ideal, noisefree data)