An improved treatment of the linearity correction of IR detectors

Massimo RobbertoJWST/NIRCam

STScI TIPS – Sep. 16, 2010

OUVERTURE

IR detectors are non linear

Linearity is assumed at the beginning of the ramp

linear fit to the first 20 samples

The “true” slope depends on the range of the assumed linear regime

In fact, the angular coefficient of the true slope is hard to find…

ACT 1CURRENT STATUS

How we do it now

In the case of NICMOS and WFC3, we apply the following correction

F are the measured counts Fc are the true counts. The calibration process assumes that they are known (fit to the first part of the ramp). Known both F’s, we derive the correction coefficients c2, c3 and c4 used for general linearity correction.

Problems with this approach

1) We do not really know what is the real slope of the calibration frame, and our estimate depends on the samples we use.

2) Physically, one has a linear true flux which is converted in a non-linear measured count rate by the detector. This is not what we model!

We modulate the observed data to get the real flux; instead, we should modulate the real flux to get the observed data.

A controlled experiment using simulated data

THIS IS THE WEIRD (NON POLYNOMIAL)NON-LINEARITY TERM

Let’s plot our baseline…

… and derive the correction “a’la HST”

I will assume that we know perfectly the true slope, i.e. problem 1 has been solved. I therefore get the best possible c coefficients.

THIS IS THE POLYNOMIALCORRECTION TERM

The result is:

Residuals

ACT 2A DIFFERENT APPROACH

Let’s look at the equationInstead of

We can try with the physically more correct expression:

i.e. we modulate the real flux Fc to get F, not viceversa

Fc × 1+ c2 × Fc + c3 × Fc

2 + c4 × Fc3( )=F

Fc =F× 1+ c2 × F+ c3 × F2 + c4 × F3( )

MethodIn Equation

the Fc and c2,c3,c4 values are unknown. I use IDL/curvefit.pro to derive them from the set of known ti and measured Fi:

having defined the function:

Fc × 1+ c2 × Fc + c3 × Fc

2 + c4 × Fc3( )=F

0.3% error on the slope!

Linearity correctionFrom the values of c2, c3, an c3 one can derive Fc by solving the equation:

Need to use an iterative method:

Fc =

F1+ c2 × Fc + c3 × Fc

2 + c4 × Fc3( )

Results

i=0

1

2

4

Check: different flux rate

Same “detector”, i.e. exponential non-linearity term

Correction: old vs. new method

Old

New

ConclusionThe current strategy we implement to correct for non-linearity seems less than ideal.1) Problems with the estimate of the coefficients, which depend

on the assumed “linearity” region of the detector2) Problems with the equation, which does not correctly

describes the non-linearity effect

The new method has two advantages3) Coefficients are estimated without any assumption on the

true, linear flux4) The correct equation, with an iterative solve, seems to

provide a much better estimate of the true linear flux.

Check on real data is in progress