CIDA/STARDUST Example of Data Calibration                                     
                                                                              
Jochen Kissel and Johan Silen                                                 
                                                                              
We present a step by step tutorial of how to                                  
calibrate CIDA/STARDUST data using a laboratory                               
spectrum. We convert the numerical units of the                               
instrument readout into meaningful physical                                   
values. A guide for the interpretation of  this                               
time-of-flight signal as a mass spectrum is                                   
outlined.                                                                     
                                                                              
                                                                              
1) The Calibration Principle                                                  
============================                                                  
                                                                              
The CIDA instrument is described in the document                              
"Cometary and Interstellar Dust Analyzer for                                  
comet Wild 2" by Kissel, J., et.al.,                                          
J.Geophys.Res., 108, (E10), 8114, doi:                                        
10.1029/2003 JE002091, 2003 [KISSELETAL2003]. Be                              
sure to have understood this document before                                  
proceeding.                                                                   
                                                                              
The term calibration is used here for the                                     
conversion of the digital output of the                                       
instrument into physical values. Calibration can                              
be done to                                                                    
                                                                              
  - the housekeeping data                                                     
  - the target-channel                                                        
  - the spectrum data                                                         
                                                                              
Among them, the last needs a more sophisticated                               
procedure, which is outlined here and                                         
demonstrated with laboratory and in-flight                                    
examples.                                                                     
                                                                              
The data as provided by the instrument consist                                
of housekeeping information and science data in                               
one or several redundant hardware channels. The                               
housekeeping data serve as technical support                                  
information and are not normally needed for data                              
analysis. The data to be analysed therefore                                   
consist of four 8192 samples-long vectors,                                    
representing two high- and two low-sensitivity                                
channels. Two of the vectors may contain zeros                                
to indicate missing data. For details of the                                  
experiment, see the paper we refer to in the                                  
beginning of the section; the data format is                                  
described in FMT files in the DATA directory.  A                              
non-zero channel is used for two purposes. The                                
first half (samples 1 to 4096) carries the                                    
output of the charge sensitive amplifier                                      
connected to the target, the second half is                                   
redundant to the previous nonzero data channel,                               
i.e. serves as an additional measurement of the                               
same signal, but with slightly different gain                                 
properties. At the end of each data vector, a                                 
calibration sequence is recorded that is the                                  
result of injecting four distinct current pulses                              
into the four instrument channels. The sequence                               
usually produces an unwanted transient at its                                 
start.  The calibration pulses are the response                               
to the four injected currents with the values                                 
shown in Table 1. Note, that the starting                                     
positions of calibration signals vary and must                                
be determined independently for each spectrum.                                
                                                                              
                                                                              
Table 1: Values of injected calibration signal                                
into a TOF-spectrum for Channel 1 as an example                               
                                                                              
              Injected    Start                                               
  Name        Signal, uA  Position                                            
  ==========  ==========  ========                                            
  Cal-1       53.2        7351                                                
  Cal-2       167         7501                                                
  Cal-3       532         7651                                                
  Cal-4       1670        7810                                                
  Background  0           8101                                                
                                                                              
                                                                              
A sample laboratory spectrum is shown in Figure                               
1. The y-axis indicates instrument digital                                    
numbers; the x-axis shows instrument 25 ns bin                                
number.  The left upper corner shows the primary                              
signal (the charge released from or introduced                                
to the target) recorded in the low-sensitivity                                
delayed channel (blue line). The middle bottom                                
part of the figure shows the spectra recorded in                              
redundant channels: high-sensitivity direct                                   
(black) and delayed (red) channels and                                        
low-sensitivity direct (green) channel. The                                   
right lower corner shows the calibration signals                              
for the four channels.                                                        
                                                                              
                            
    
                                                                              
Figure 1:  Example spectrum used to demonstrate                               
the calibration process.                                                      
                                                                              
                                                                              
The relation between the current I (in uA) and                                
the numerical reading N is described by a                                     
theoretical equation:                                                         
                                                                              
  N = C1 + C2 I + C3 log( 1 + I / C4 )   [Eq. 1]                              
                                                                              
A useful inverse solution to reconstitute the                                 
current I from the digital number N is given by                               
                                                                              
                    (C2 ( N - N0))                                            
  I = C1 ( N - N0) e                     [Eq. 2]                              
                                                                              
We prefer to use eq. (2) for any practical work.                              
Using the values from Table 1 it is possible to                               
solve it in a least square sense to get the                                   
values for C1 and C2. Here N and N0 are                                       
respectively the numerical reading and the                                    
spectrum numerical background level of the                                    
recorded, compressed raw signal.                                              
                                                                              
The relation between time-of-flight and ion mass                              
is:                                                                           
                                                                              
  t = a sqrt(m) + b                      [Eq. 3]                              
                                                                              
This relation holds for all ions, which are                                   
formed simultaneously, are stable during their                                
flight time and are accelerated to the same                                   
kinetic energy. The instrument's reflector                                    
compensates for initial energies the ions might                               
have. Ions with different distribution functions                              
for their initial energy may be represented by                                
different peak shapes and thus provoke a                                      
slightly different set of a-, b-values                                        
determining the peak mass. Note that there is a                               
small ~1% variation of a, depending on whether                                
the impact is central or peripheral due to a                                  
small difference in the ion trajectories from                                 
the target to the detector.  Also, there may                                  
occasionally be some spurious peaks present                                   
because of ion chemistry processes or collisions                              
with the walls.                                                               
                                                                              
Knowing positions of two or more mass peaks in a                              
given spectrum, it is possible to find the                                    
constants a and b defining the relation between                               
mass and sample time.                                                         
                                                                              
                                                                              
2) Application to Laboratory Spectrum                                         
=====================================                                         
                                                                              
The example presented is chosen to demonstrate                                
how to produce a time of flight spectrum from                                 
the recorded raw signal. The steps involved are                               
the following:                                                                
                                                                              
i) Compute the mean background of the spectrum:                               
This is conveniently done by taking the very                                  
last samples in the spectrum, positions starting                              
at, e.g., 8101. The result is slightly different                              
for the different channels. Make sure that the                                
region used for the background estimation does                                
not contain any real signal.                                                  
                                                                              
ii) Compute the mean of say 50 samples from each                              
of the channels present starting from the                                     
positions given in Table 1. The difference                                    
indicates directly the difference in gain.  Make                              
sure that there is no contamination by real                                   
signal.                                                                       
                                                                              
iii) Knowing the injected currents (Table 1) and                              
the levels (step 2), the constants Ci and N0                                  
can be calculated from formula 2.                                             
                                                                              
iv) Having the constants Ci and N, one can build                              
a lookup table for I = I ( Ci, N0, N ).  Knowing                              
this completes the amplitude calibration. The                                 
spectrum amplitude I is given in uA sampled at                                
40 MHz (samples are 25 ns apart).                                             
                                                                              
v) Looking at the TOF-spectrum and knowing (or                                
guessing) that the first major peak seen is                                   
Hydrogen and that the large double peak at end                                
of the spectrum is Silver, it is possible to                                  
compute an estimate for a and b (Equation 3)                                  
connecting the mass and the time of flight.                                   
                                                                              
vi) It is important to note that the relative                                 
yields of ions are strongly dependent on the                                  
detailed chemical properties of the incident                                  
dust grain and of the local target composition                                
and structure.  Therefore it is incorrect to                                  
infer relative elemental abundances from the                                  
above procedure directly. Filters in the                                      
detector amplifier eliminate the difference in                                
amplitude due  to the smaller flight time of the                              
light ions where the peak width would be below                                
200 ns.                                                                       
                                                                              
                                                                              
3) Numerical Values for the Example Spectrum                                  
============================================                                  
                                                                              
This section provides values for the example                                  
spectrum (Figure 1).                                                          
                                                                              
                                                                              
3.1) Background and Calibration Injection Levels                              
------------------------------------------------                              
                                                                              
The background and the calibration levels                                     
estimated from 50 samples starting at the                                     
positions indicated in Table 1 are shown in                                   
Table 2.                                                                      
                                                                              
                                                                              
Table 2:  Background and calibration injection                                
levels computed for the example spectrum                                      
                                                                              
       Background  Cal-1  Cal-2  Cal-3   Cal-4                                
       ==========  =====  =====  ======  ======                               
  Ch1  18.58       35.84  62.32   90.46  126.86                               
  Ch2  17.16       33.78  59.92   88.76  124.18                               
  Ch3  16.64       41.12  68.24  101.58  143.00                               
  Ch4  15.24       38.78  64.34   96.84  134.36                               
                                                                              
                                                                              
3.2) Computation of Constants                                                 
-----------------------------                                                 
                                                                              
The computation of the numerical constants has                                
to be made numerically and iteratively using                                  
some standard technique. The result is shown in                               
Table 3, which contains the results obtained                                  
based on Equations 1 and 2. The solution based                                
on Eq. 2 is more stable and sufficiently                                      
accurate. C1 in Eq. 1 is identical to N0 in Eq.                               
2.                                                                            
                                                                              
                                                                              
Table 3:  Coefficients connecting the numerical                               
raw value and the physical current as calculated                              
from the example spectrum using the values in                                 
Table 1.                                                                      
                                                                              
       C1    C2                                                               
       ====  ======                                                           
  Ch1  2.01  0.0183                                                           
  Ch2  2.10  0.0181                                                           
  Ch3  1.35  0.0179                                                           
  Ch4  1.38  0.0192                                                           
                                                                              
                                                                              
For calibration, one can either use a lookup                                  
table built from Eq.(1) or calculate the current                              
directly from inverse Eq.(2).  The first                                      
approach may be numerically, unstable but when it                             
works gives somewhat more accurate results. The                               
second approach is stable and simple to use, but                              
for amplitudes larger than the highest injected                               
calibration impulse, can be somewhat inaccurate.                              
However, the errors are smaller than the                                      
uncertainties in the physics of ion production                                
rates.                                                                        
                                                                              
                                                                              
3.3) Transforming Time-of-Flight Spectrum into a                              
Mass-Spectrum                                                                 
-------------------------------------------------                             
-------------                                                                 
                                                                              
Identifying the Hydrogen peak at t = 2944,                                    
[107]Ag at t = 4614 and [109]Ag at t=4632 gives                               
the three equations:                                                          
                                                                              
  2944 = a + b                                                                
  4614 = a sqrt(107) + b             [Eqs. 4]                                 
  4632 = a sqrt(109) + b                                                      
                                                                              
The solution is a = 178.8 and b = 2765. In an                                 
analog way, one can identify a larger number of                               
masses and solve the corresponding linear                                     
equations in a least square manner. The mass                                  
expressed in sample coordinates would then be:                                
                                                                              
                                                                              
                2                 2                                           
       ( t - b )        (t - 2765)                                            
  m =  ---------    =   -------------    [Eq. 5]                              
            2                       4                                         
           a             3.1969 x 10                                          
                                                                              
                                                                              
Equations (2) and (5) together with data from                                 
Table (3) and the values for a and b completes                                
the calibration procedure. The resulting impact                               
mass spectrum is shown in Figure 2. The                                       
interpretation of it requires a solid knowledge                               
of mass spectrometry and is beyond the scope of                               
this paper.                                                                   
                                                                              
                           
     
                                                                              
Figure 2:  Calibrated spectrum. The mass is                                   
fixed using equation (5) and the amplitude using                              
equation (2) and the data values in Table 3.                                  
                                                                              
                                                                              
4) Calibration of Negative Mode Flight Spectrum                               
===============================================                               
                                                                              
We selected the event NEG31 as an in-flight data                              
example. Its spectrum is shown in Figure 3. The                               
negative mode is indicated by the information in                              
the housekeeping header data.                                                 
                                                                              
                            
    
                                                                              
Figure 3:  Flight spectrum, event NEG31 used as                               
an example illustrating the calibration                                       
procedure. The amplitude calibration data shown                               
at the end of the spectrum represent the data                                 
given in Table 4.                                                             
                                                                              
                                                                              
4.1) Background and Calibration Injection Levels                              
-------------------------------------------------                             
                                                                              
The background and the calibration levels are                                 
estimated from 120 samples starting at the                                    
positions indicated in Table 4.                                               
                                                                              
                                                                              
Table 4: Values of injected calibration signal                                
into a TOF-spectrum, sample position and average                              
value of signal for channel 1 as an example.                                  
                                                                              
              Injected    Starting                                            
  Name        Signal, uA  Position                                            
  ==========  ==========  ========                                            
  Cal-1       53.2        7310                                                
  Cal-2       167         7460                                                
  Cal-3       532         7600                                                
  Cal-4       1670        7740                                                
  Background  0           8000                                                
                                                                              
                                                                              
4.2) Computation of Constants                                                 
-----------------------------                                                 
                                                                              
We use equation (2) to perform the amplitude                                  
calibration. The logarithm of this equation gives                             
                                                                              
  log(I) = log(C1) + log(N-N0) + c2 (N-N0)  [Eq.                              
6]                                                                            
                                                                              
This equation has to be solved for C1 and C2,                                 
which can be made in a least-square sense using                               
standard techniques, see e.g. Menke(1984),                                    
Geophysical Data Analysis, p 57-59. Here we use                               
a constrained least-square fit requiring that                                 
the multiplier for the term log(N-N0) equals 1.                               
Table 5 contains the values representing the                                  
average calibration injection levels computed                                 
from the injected signal levels.                                              
                                                                              
                                                                              
Table 5:  Background and calibration injection                                
levels computed for the NEG31 spectrum                                        
                                                                              
       Background  Cal-1  Cal-2  Cal-3  Cal-4                                 
       ==========  =====  =====  =====  ======                                
  Ch1  15.88       29.24  48.31  71.06   99.29                                
  Ch2  15.01       27.75  46.94  68.18   95.58                                
  Ch3  15.35       33.30  53.24  77.42  108.18                                
  Ch4  15.09       32.02  51.67  75.95  105.26                                
                                                                              
                                                                              
The resulting values for C1 and C2 are shown in                               
Table 6.  No lookup table is required since Eq.                               
(2) provides an analytic function, which                                      
describes the relation between physical current                               
expressed in uA and the numeric value of the                                  
data.  The constrained least-square fit provides                              
values for the constants which are given in                                   
Table 6.                                                                      
                                                                              
                                                                              
Table 6:  Constants C1 and C2 computed for the                                
NEG31 spectrum using a constrained least square                               
solution to equation (6), expressing currents in                              
uA in equation (2)                                                            
                                                                              
       C1     C2                                                              
       =====  ======                                                          
  Ch1  0.978  0.0237                                                          
  Ch2  1.01   0.0245                                                          
  Ch3  0.612  0.0245                                                          
  Ch4  0.681  0.0246                                                          
                                                                              
                                                                              
4.1) Transforming Time-of-Flight Spectrum into a                              
Mass-Spectrum                                                                 
-------------------------------------------------                             
-------------                                                                 
                                                                              
Determining the mass scale requires identifying                               
some peaks in the spectrum. This can most                                     
conveniently be made by identifying at least two                              
peaks. Using these as an initial guess providing                              
an estimate for a and b in Eq. 3, gives a simple                              
approximate scale, which can be refined by                                    
identifying additional peaks. This is an                                      
iterative process that does require general                                   
knowledge about the interpretation of mass                                    
spectra.                                                                      
                                                                              
In general, negative mode spectra do not show a                               
pronounced step in the target signal. Instead                                 
they usually contain a large electron peak                                    
indicating the impact time (mass zero). Often                                 
also M=1 is present as is M=13 (CH-). For                                     
this example an inspection of the spectrum shows                              
that there are peaks at positions shown in Table                              
7.                                                                            
                                                                              
                                                                              
Table 7:  Peaks identified from the data for                                  
channel 1. The mass is computed together with                                 
the values for "a" and "b$ using equations (4)                                
and (5). The values for the constants are                                     
calculated to be a = 174.05 and b = 4104. The                                 
resulting masses are shown.                                                   
                                                                              
  Peak      Identification  Calculated                                        
  Position  as mass         Mass                                              
  ========  =======         =======                                           
  4100      0                5.9E-4                                           
  4283      1                1.05                                             
  4707      12              12.00                                             
  4731      13              12.97                                             
  4800      16              15.98                                             
  4827      17              17.25                                             
  4951      24              23.67                                             
  4976      25              25.09                                             
                                                                              
The mathematics are similar to those in Section                               
3.3, equations (4) and (5). The results are                                   
indicated as calculated mass in Table 7. The                                  
calibrated mass spectrum (amplitude and mass                                  
both calibrated) is shown in Figure 4.                                        
                                                                              
                           
     
                                                                              
Figure 4:  Enlarged portion of the calibrated                                 
mass spectrum. Note the well-behaving channels 1                              
and 2 superimposed to well within 1-bit                                       
accuracy. The low sensitivity channel is not                                  
accurate at this low 2-bit amplitude level.