Systematic Error Propagation into Photometric Zero Points

Nick Mostek
Stuart Mufson

Nick and I have used a toy model to compute model parameters and a covariance matrix for the determination the photometric zero points.  It is our attempt at the solution to a  problem that Alex Kim/Ramon Miquel and the simulation group have requested from the calibration group so that they can make more realistic simulations of the precision to which SNAP can determine the cosmological parameters. 

I. Zero Point Computation

As we understand it, the problem is to determine how systematic errors propagate into photometric color measurements in the SNAP filter system.  Since there has been a good deal of confusion over this in the calibration group, we must emphasize here that these computations do NOT represent the solution to the photometric calibration problem.  Rather they are aimed at the determination of model parameters and the construction of a covariance matrix, the combination of which propagates systematic errors.  And the construction of this covariance matrix requires the solution of a minimization problem.  Here we describe our initial toy model simulation of this problem.

For our propagation of systematic errors we use the standard astronomical technique for the transformation of one photometric system to another, the method of "zero point" determination.  In our case, photometric measurements of stellar fluxes with systematic errors represent one phototmetric system, and the photometric system yielding "true" fluxes represent the second photometric system.  By "true" fluxes we mean the theoretical flux determined from an assumed a priori knowledge of the stellar spectrum and filter response.  So far we have included systematic errors resulting from (1) uncertainties in the flux scale set by the fundamental calibrator and (2) uncertainties in the characterization of the SNAP filters.   We expect to include additional systematic errors in the future (e.g., flat fielding errors) if our approach proves to be acceptable (and we have labored long and hard to figure out an acceptable approach).

We base our computation of the zero points on the method used to transform the magnitudes measured by WFPC2 broad band filter set to standard UBVRI magnitudes (J.A. Holtzman et al.  1995, PubAstr.Soc.Pacific, 107, 1065).    The zero point equations used here for the SNAP problem were modified from the WFPC2 equations as follows:  (1) The transformation equations we use describe a flux-based calibration, rather than a magnitude-based calibration; (2) the flux ratios in our transformation equations (or color terms in a magnitude-based system) incorporate local as well as information over longer wavelength baselines; and (3) we do not use higher order color terms.  Assumptions (2) and (3) reflect our own biases and could be changed quite simply to accomodate different approaches to the problem.  We make assumption (2) to propagate local SED slope information into the transformation equations; we make assumption (3) to reduce the fit parameters in the problem.

Our transformation equations are:

    B = a_0*b

    F_1 = a_1*f_1 + a_2*(B/F_1) 

    F_2 = a_3*f_2 + a_4*(B/F_2)  + a_5*(F_1/F_2)

    F_3 = a_6*f_3 + a_7*(B/F_3)  + a_8*(F_2/F_3)
    .
    .
    .
    F_7 = a_18*f_7 + a_19*(B/F_7)  + a_20*(F_6/F_7)

    F_8  = a_21*f_8 + a_22*(B/F_8)  +a_23*(F_7/F_8)

where B is the "true" stellar flux through an analytic approximation to the Bessel B filter, F_1 is the known stellar flux through the first redshifted Bessel B filter, and so forth; lower case b, f_1, ... are the fluxes with systematic errors through this same filter set; and the 24 fit parameters in this model are labelled a_0, a_1, ..., a_24. 

The minimization problem consists of forming the chi**2 statistic for the set of parameters a_1, ..., a_25 that give the smallest deviations between the "true" fluxes and the fluxes with systematic errors for a set of primary calibration standard stars

    chi**2 = SUM_i { [B - a_0*b ]_i **2/(sigma_p)**2

                  + [F_1 - (a_1*f_1 + a_2*(B/F_1))]_i **2/(sigma_p)**2

                  + [F_2 - (a_3*f_2 + a_4*(B/F_2) +  a_5*(F_1/F_2))]_i **2/(sigma_p)**2

                   .
                   .
                   .

                   + [F_8 - (a_21*f_8 + a_22*(B/F_8) + a_23*(F_7/F_8)]_i **2/(sigma_p)**2 }

where i runs over the set of primary calibration standard stars and sigma_p represents the statistical error associated with the photometric measurements.  For these computations, we assumed that sigma_p = 0.003; bright stars observed with a 2m telescope in space should have very small photometric errors.


  A.  Primary Calibration Standard Stars

The primary calibration standard stars were modelled as blackbodies for simplicity.  The set of blackbody primary calibration standard stars we studied was suggested by Michael Richmond who has graciously agreed to assume all blame if these calculations are found to be faulty in any way whatsoever:

     "The Richmond 8": T = 3,000K; 4,000K 5,000K; 6,000K; 8,000K; 10,000K; 15,000K; 20,000


  B. SNAP Filter Set

The filters used in these studies were an analytic form for the Bessel B filter generated by Chuck Bower. 

    lambda <= 360 nm
        T = 0.

   360 nm < lambda <= 420 nm
         T = 1/{1 + exp[-0.17*(lambda-390)]}  + 0.006*(lambda-390)/30

    420 nm < lambda <= 560 nm
         T = {cos[pi/2*(lambda-420)]/140}**2.4

     lambda > 560 nm
          T = 0.
 
The SNAP filter set was generated by redshifting this Bessel B filter according to

    1 + z = (1.16)**n,   where n = 0-8


  C.  Systematic Uncertainties in the Flux Scale

The measured fluxes for the primary calibration stars were assumed to have systematic errors that derive from the transfer of calibration from the fundamental calibration standard star.  These systematic errors were taken from Ralph Bohlin's flux models for the fundamental white dwarf standard G191 B2B.  As in the calibration transfer computations discussed previously (fundamental calibration errors), the systematic error as a function of wavelength was assumed to be of the form

     error(lambda)  = 2*(nlte - lte)/(nlte + lte),

where nlte and lte are the non-LTE and the LTE flux models for G191 B2B which are assumed to bracket the true flux. 

  D.  Systematic Uncertainties in the SNAP Filters

Both Mufson/Mostek and Lampton have investigated on-orbit calibration of the filters.  In the Mufson/Mostek studies, the filters were characterized by 3 parameters: central wavelength, bandpass, and integrated transmission (peak transmission * bandpass). 





We investigated a problem somewhat different than the one investigated by Mike Lampton.  In our studies, we assumed that the filter parameters had shifted from their measured laboratory values for some reason during launch/on-orbit, and we tested the precision to which measurements of a set of standard stars with a wide range in color could track these changes.  The Mufson/Mostek computations use the same Richmond8 set of primary calibration standard stars for the filter characterizations.  To find the shifted filter parameters we minimized a chi**2 statistic for each individual filter; for the B filter, this chi**2 statistic was written

    chi**2 = SUM_i [B - B'(a_1,a_2,a_3)]**2 _i/sigma_p**2

where B is the flux measured through the shifted filter, and B'(a_1,a_2,a_3) is the flux measured through a filter where the 3 filter parameters, a_1, a_2, a_3, were allowed to vary.  The sum runs over the 8 calibration stars, and sigma_p = 0.003 as before. 

For primary calibration stars without systematic errors in their flux scale, this scheme recovers the shifted filter parameters to high precision.  For instance, for the case in which the central wavelength has shifted by 0.25 nm (delta_lambda = 0.25), the bandpass has increased by 1% (stretch = 0.01), and the integrated transmission, norm, increases by 2% (1% change in peak transmission, 1% increase in bandpass) we find the following results for the fits to the 9 filters:

 

No sys error in blackbody flux
norm = 1.02, delta_lambda = 0.25, stretch = 0.01
************************************************

Filter Parameters
filter  norm    delta_lambda    stretch
-----------------------------------------
0       1.0200  0.2500          0.0100
1       1.0200  0.2500          0.0100
2       1.0200  0.2500          0.0100
3       1.0200  0.2500          0.0100
4       1.0200  0.2500          0.0100
5       1.0200  0.2500          0.0100
6       1.0200  0.2500          0.0100
7       1.0200  0.2500          0.0100
8       1.0200  0.2500          0.0100

On the other hand, when systematic errors in the primary standards' flux scale are included, the shifted filter parameters are not well tracked.  For the same shifted filter as above

 
Systematic error in blackbody flux
norm = 1.02, delta_lambda = 0.25, stretch = 0.01
************************************************

filter  norm    delta_lambda    stretch
-----------------------------------------
0       0.9970  0.0407          0.0002
1       0.9992  0.0311          0.0002
2       1.0011  0.0283          0.0002
3       1.0031  0.0348          0.0004
4       1.0054  0.0357          0.0003
5       1.0078  0.0389          0.0004
6       1.0104  0.0401          0.0004
7       1.0132  0.0469          0.0005
8       1.0159  0.0325          0.0007

II.  Computations

For each set of blackbodies, the computations proceeded as follows.  For each blackbody in the set, the broad band fluxes without systematic errors, B, F_1, F_2, ..., F_8, and the broad band fluxes with systematic errors, b, f_1, f_2, f_8  were calculated.  The chi**2 statistic described above was then used in both the ROOT MINUIT fitter and the NAG fitter to find the best-fit set of parameters a_0 - a_24.  (NOTE: For these computations, the fluxes were normalized by 10**(-16).)  Checks show that both fitters give similar results.

  A.  Systematic Errors in the Fundamental Calibration Standard Flux Scale

For the set of 8 blackbodies with the systematic errors flux in the flux scale derived from Ralph Bohlin's non-LTE and LTE models for G191 B2B, the results of the computations are found in the file zero points - 8 blackbodies.  In this file, first the temperatures for the 8 blackbodies used in the computations are listed, then the best fit parameters, then the 24x24 covariance matrix. 

  B.  Systematic Errors in the Flux Scale + Systematic Errors in the Filter Parameters

Systematic errors in the filter parameters were included in the zero point computations as follows.  In the chi**2 statistic for the zero point determinations, the "true" fluxes (B, F_1, ..., F_8) were computed with the shifted filter parameters (e.g., norm = 1.02, delta_lambda = 0.25, stretch = 0.01) and the blackbody fluxes at temperature T without systematic errors in the flux scale; the "measured" fluxes (b, f_1, ..., f_8) were computed with the fitted filter parameters (e.g., table above - "Systematic error in blackbody flux") and blackbody fluxes that include systematic errors in the flux scale.  For the same set of 8 blackbodies with the toy shifted filter above, the results of the computations are found in the file zero points - 8 blackbodies + filter errs.  In this file, the temperatures for the 8 blackbodies, the best fit parameters, and the 24x24 covariance matrix are given.