###  JPL D-26533

**Earth Observing System (EOS)**

###### Aura Spacecraft

#### **TROPOSPHERIC**

**EMISSION SPECTROMETER**

**(TES)**

### 

### 

###  Instrument Calibration Report

***[Volume 1]{.underline}***

##### Preliminary Version

##### September 26, 2003

![](media/image1.wmf)

![](media/image2.png){width="0.84375in" height="0.2916666666666667in"}

##### Jet Propulsion Laboratory

**California Institute of Technology**

**Pasadena, CA 91109**

### JPL D-26533

### TROPOSPHERIC EMISSION

### SPECTROMETER (TES)

### 

###  Instrument Calibration Report

> Reinhard Beer Jet Propulsion Laboratory
>
> Kevin W. Bowman Jet Propulsion Laboratory
>
> Brendan M. Fisher Jet Propulsion Laboratory
>
> Michael R. Gunson Jet Propulsion Laboratory
>
> Ron G. Holm Jet Propulsion Laboratory
>
> Mingzhao Luo Jet Propulsion Laboratory
>
> David M. Rider Jet Propulsion Laboratory
>
> Edwin Sarkissian Jet Propulsion Laboratory
>
> Susan Sund Kulawik Jet Propulsion Laboratory
>
> Denis A. Tremblay Raytheon
>
> Helen M. Worden Jet Propulsion Laboratory

######### Approved by: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

#######  Reinhard Beer, TES Principal Investigator

**\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_**

> **Thomas A. Glavich, TES Project Magnager**
>
> ![](media/image2.png){width="0.7229166666666667in"
> height="0.25069444444444444in"}

######## **Jet Propulsion Laboratory**

### Preliminary Version California Institute of Technology

### September 26, 2003 Pasadena, CA 91109 

######### TABLE OF CONTENTS

***[Volume 1]{.underline}***

1.  **Relevant Documents
    ....................................................... 4**

2.  **Calibration requirements Summary ............................. 5**

3.  **SYSTEM TESTING BACKGROUND
    .......................................\... 5**

4.  **TEST CONFIGURATIONS
    ....................................................... 6**

5.  **KEY DATA SUMMARY
    .......................................................... 10**

6.  **CALIBRATION RESULTS SUMMARY ......................................
    12**

    1.  **Channel Shifts
        ............................................................
        13**

    2.  **FFT Size / Filter Spectral Range
        ..................................... 15**

    3.  **Shear / Co-Boresight
        ..................................................... 17**

    4.  **Time Trend of Signal Level, Ice Buildup
        ............................. 17**

    5.  **Channeling
        ................................................................
        19**

    6.  **Electrical Filter
        ........................................................... 20**

    7.  **Gain
        .........................................................................
        21**

    8.  **Spectral Spikes
        ........................................................... 22**

    9.  **Pointing Angles for Calibration Sources
        ............................ 23**

    10. **PCS Calibration
        ......................................................... 25**

    11. **Non-Linearity
        ............................................................
        30**

    12. **Radiometric Calibration
        ............................................... 31**

    13. **NESR
        .........................................................................
        36**

    14. **Frequency Calibration
        ................................................. 40**

    15. **Instrument Line Shape (ILS)
        .......................................... 42**

    16. **Field of View (FOV) Response
        ........................................ 48**

7.  **CONCLUSIONS -- IMPACTS ON ALGORITHM REQUIREMENTS .... 52**

***[Volume 2]{.underline}***

1.  **RUN LOGS FOR SYSTEM TESTS**

2.  **TES DESIGN FILE MEMORANDA (DFMs)**

**1. RELEVANT DOCUMENTS**

*Tropospheric Emission Spectrometer: Science Objectives & Approach,
Goals & Requirements*. JPL D-11294, Version 6.0, April 1999.

*Tropospheric Emission Spectrometer: Instrument Pre-Flight Calibration
Requirements.* JPL D-20040, Version 1.2, February 9, 2003.

*Tropospheric Emission Spectrometer: Spectrometer Instrument Calibration
Plan.* JPL D-13432,1996.

*Tropospheric Emission Spectrometer:Integration and Test Plan.* JPL
D-15389,2000.

*Tropospheric Emission Spectrometer: Level 1B Algorithm Theoretical
Bases Document.* JPL D-16479, Version 1.1, October 1999.

*Tropospheric Emission Spectrometer: Level 2 Algorithm Theoretical Bases
Document.* JPL D-16474, Version 1.1, October 1999.

**2. Calibration requirements Summary**

Detailed calibration requirements are given in *Tropospheric Emission
Spectrometer: Instrument Pre-Flight Calibration Requirements.* JPL
D-20040, Version 1.2, February 9, 2003. We summarize them below, under
the categories: radiometric, spectral and spatial.

# ***Radiometric Calibration***

Measurements taken for radiometric calibration must be sufficient to
determine instrument response linearity, radiometric accuracy, specified
in equivalent brightness temperature and precision, specified as the
noise equivalent spectral radiance (NESR) over the expected range of
source temperatures, with calibration sources that are traceable to NIST
radiance standards. There is also a requirement to calibrate relative
gain settings in the signal chain; however, this has very low priority
since we do not plan to exercise the different settings at this time.

# ***Spectral Calibration***

Spectral calibration results must characterize spectral response and
stability, frequency positions of spectral lines with respect to known
line positions (e.g., the HITRAN database) and the instrument line shape
(ILS).

# ***Spatial Calibration***

Spatial calibration measurements must provide the field of view (FOV)
response function for each detector at each filter setting, along with
the relative co-boresight for the four focal plane arrays at a spatial
resolution sufficient for accurate radiance models in the limb view. In
addition, co-boresight measurements from the on-board spatial
calibration source (OBSCS) must be verified using the external spatial
calibration source (target projector). Finally, the pointing control
system (PCS) must be calibrated with respect to the TES and Aura
spacecraft alignment reference cubes.

**3. SYSTEM TESTING BACKGROUND**

**4. TEST CONFIGURATIONS**

The characterization measurements of the TES instrument are performed in
the same thermal environment that is expected on orbit. Eight
temperature control zones provide the means for simulating the on-orbit
thermal environment. Seven of these zones supply gaseous or liquid
nitrogen to thermal panels and test equipment. The eighth thermal zone
is supplied with helium. Six thermal shrouds are provided within the
system; each is independently supported from the chamber through thermal
insulators and is independent of the vibration-isolated optical bench.
The operational temperature range of the chamber is from 220 to 280 K.
It operates with an active load of 350 W and a maximum temperature
gradient of 16 K. Figure 4.1 presents the layout, within the thermal
vacuum chamber, of the TES instrument and its calibration equipment.

![](media/image3.wmf){width="5.998611111111111in"
height="5.243055555555555in"}

Figure 4.1. Thermal Vacuum Chamber Layout

### *Radiometric Test Sources* 

The radiometric calibration of the instrument is performed using two
thermally controlled, high-emissivity blackbody sources. Each blackbody
source is mounted inside the chamber in direct view of the instrument.
There are no additional optics placed between the instrument and the
calibration sources. One blackbody source is cooled with LN~2~ to
approximately 80 K to simulate the view of cold space. The second
blackbody is a variable-temperature source (Figure 4.2) that operates
between approximately 90 and 350 K. The effective temperature of the
sources is controlled using heater elements and LN~2~ cooling. The
temperature is monitored using readings from calibrated PRT temperature
sensors placed on the source.

![](media/image4.wmf)

Figure 4.2. Variable Temperature Blackbody Layout

### *Spatial Calibration Source* 

Field-of-view (FOV) and spatial characterization tests are performed
using a vacuum-qualified target projector. This projector (Figure 4.3)
consists of a high-temperature infrared source (uncalibrated), off-axis
collimator, chopper (operating at 80 Hz), and a set of interchangeable
apertures on a computer-controlled X**-**Y stage. The IR-illuminated
aperture of the target projector is positioned at the focus of an
off-axis paraboloid and imaged by the TES instrument onto its detectors.
A remotely controlled translational stage scans the illuminated aperture
across the FOV of the instrument.

The instrument views the target projector via the TES PCS mirror.
Mechanical stability is provided by mounting the target projector on a
vibration-isolated optical table.

![](media/image5.wmf){width="5.9944444444444445in"
height="4.060416666666667in"}

## Figure 4.3. TES Target Projector

***Spectral Calibration Sources***

The spectral calibration of the instrument is measured by comparing
recorded spectra of a gas cell (Figure 4.4) filled with a known gas, at
a known temperature and pressure, with the spectral line widths of
reference test gases. To avoid the strong pressure-broadened water vapor
lines that occur in the atmosphere, all spectral measurements are taken
in the evacuated chamber with a pressure of \<10^-5^ torr. The gas cell
is a cylinder with a 40-cm path length and a 13.25-cm IR transmitting
window. A stable large-area (20-by-20-cm) blackbody source operating at
340 K provides the input radiance used to illuminate the gas cell.
Temperature sensors placed on the external surface of the cell provide
the temperature monitoring required to accurately predict the line
widths of the gas under test. The gas cell has two input lines that
interface with chamber feedthroughs and connect to a manifold outside
the chamber. The manifold provides the input port and purge line for the
test gases and performs the monitoring and control of the pressure
within the cell.

A second spectral measurement is performed using a stable monochromatic
source. This measurement provides a single-frequency measurement of the
instrument line shape and provides a validation of the gas-cell
measurements. Spectra are recorded while viewing the monochromatic
source, which illuminates the instrument through an IR-transmitting
window placed in one of the chamber viewing ports.

![](media/image6.wmf){width="5.15in" height="2.872916666666667in"}

## Figure 4.4. TES Spectral Calibration Source

**5. KEY DATA SUMMARY**

***Radiometric Calibration and NESR***

**[Primary dataset:]{.underline}** ST-5 runs 504-515 (testIDs 416-427)
with external variable blackbody at 120K to 340K, 20K increments. This
dataset used the same gain settings for data taken from the external
variable blackbody, the OBRCS and the external cold (90K) blackbody.

**[Secondary dataset:]{.underline}** ST-5 runs 447-449,453-460, (testIDs
358-360,364-371) with external variable blackbody at 120K to 340K, 20K
increments. This dataset also used different gain settings for the
external blackbody as temperature was varied. Data from the OBRCS view
were always taken with a low gain setting and data from the external
cold (90K) blackbody were always taken with the highest gain settings.

# ***Spectral Calibration***

**[CO~2~ Laser:]{.underline}** ST-5 run 498 (testID 410). TES was
illuminated with scattered light from a CO~2~ laser. Such a device emits
many lines but the ones used in this study were the two strongest: P20
at 1046.8543 cm^-1^ and R24 at 1081.0877 cm^-1^, both from the
00^0^1-02^0^0 band. TES filters 1B1 and 1B2 cover the CO~2~ lines.

**[Gas Cell:]{.underline}** ST-5A runs 461-473 (testIDs 372-384) and
ST-5B runs 516-529 (testIDs 428-441). The gas cells include CO2 at 150
and 10 Torr respectively, N2O at 150 and 5 Torr respectively, and CO at
5 Torr. The gas cell temperatures were at room temperature, about 290 K.
The blackbody background temperatures for the gas cell measurements were
all set to 340 K. For a given gas cell measurement, two empty cell
measurements were taken before and after, with 340 K and 320 K blackbody
background temperatures respectively. The OBRCS and Cold BB measurements
are also taken.

***Spatial Calibration***

The spatial calibration data was taken for FOV response and
co-boresight. For each testid/runid, there were 200 scan steps, with one
scan per step.

  ------------- ------------- ------------- ----------------- -------------
  Set Number    TestIDs^1^    RunIDs^1^     Filters           Notes

  1             354, 355, 356 443,444,445   1A1,1B1,2A1,2B1   Missed
                                                              detectors

  2             385,386,387   474,475,467   1A1,1B1,2A1,2B1   Filter set 1

  3             392           481           1A1,1B1,2A1,2B1   Out of field,
                                                              above

  4             393           482           1A1,1B1,2A1,2B1   Out of field,
                                                              below

  5             395,396,398   484,485,487   1A2,1B2,2A2,2B1   Filter set 2

  6             400,401,402   489,490,491   1A3,1B1,2A3,2B1   Filter set 3

  7             404,405,406   493,494,495   1A4,1B1,2A4,2B1   Filter set 4

  8             442,442,444   530,531,532   1A1,1B1,2A1,2B1   Filter set 1

  9             447           535           1A1,1B1,2A1,2B1   Out of Field,
                                                              above

  10            448           536           1A1,1B1,2A1,2B1   Out of Field,
                                                              below

  11            450,451,452   538,539,540   1A2,1B2,2A2,2B1   Filter set 2

  12            454,455,456   542,547,548   1A3,1B1,2A3,2B1   Filter set 3

  13            458,459,460   546,547,548   1A4,1B1,2A4,2B1   Filter set 4

  14            527,528,529   634,635,636   1A1,1B1,2A1,2B1   Filter set 1

  15            541,542,543   648,649,650   1A1,1B1,2A1,2B1   Filter set 1
  ------------- ------------- ------------- ----------------- -------------

^1^ ID numbers are for along track, cross track top, and cross track
bottom sets

The FOV data used for the level 2 OSP were set 2 for filters 1A1, 1B1,
2A1, 2B1, set 5 for filters 1A2,1B2,2A2, set 6 for filters 1A3 and 2A3,
and set 7 for filters 1A4 and 2A4. The co-boresight fot the OSP was
derived from set 15. Set 13 is contaminated with interferogram spikes.

**6. CALIBRATION RESULTS SUMMARY**

The following 16 sections briefly summarize the analysis results on the
TES calibration data. More detailed analyses can be found from the JPL
internal memorandums in Volume 2 of this report.

**6.1 Channel Shifts**

# ***Background***

In ST-1, incorrect assignments of pixel were first observed in data from
the target projector. The data was processed using the known
channel-to-pixel map from the hardware specifications. Since the target
projector scans across the detector arrays, if the channel assignment
has been shifted during the data collection, the incorrect pixel order
is obvious. Since then, other methods to detect channel shifts have been
discovered that do not require target projector data and channel shifts
in the data have been observed throughout the system testing with
relatively high occurrence, on average, around 1 shift per run.

If there is any mis-timing in the real-time interrupts sent to the
signal chain MUX (multiplexer), the channel being read can jump to
another channel. Following a flight software fix after ST-1, this only
happens at the beginning of scans. Since channel number is inferred by
the order of the interferogram samples, with no other ancillary
identification, the apparent channel order is shifted when this occurs.
To arrive at the correct assignment of pixel, the interferogram array
must be shifted back to the correct order, removing the possible junk in
the first interferogram sample from each channel before the channel
shift occurred. The mapping from channel to pixel can then be applied.

Hardware channel maps:

1A, 2A arrays 1B, 2B arrays

  -------- ---------------- ---------------------- ---------------- --------
           pix 15           -ch 00                 pix 15           -ch 08

  ch 08-   pix 14           ch 00-                 pix 14           

           pix 13           -ch 01                 pix 13           -ch 09

  ch 09-   pix 12           ch 01-                 pix 12           

           pix 11           -ch 02                 pix 11           -ch 10

  ch 10-   pix 10           ch 02-                 pix 10           

           pix 09           -ch 03                 pix 09           -ch 11

  ch 11-   pix 08           ch 03-                 pix 08           

           pix 07           -ch 04                 pix 07           -ch 12

  ch 12-   pix 06           ch 04-                 pix 06           

           pix 05           -ch 05                 pix 05           -ch 13

  ch 13-   pix 04           ch 05-                 pix 04           

           pix 03           -ch 06                 pix 03           -ch 14

  ch 14-   pix 02           ch 06-                 pix 02           

           pix 01           -ch 07                 pix 01           -ch 15

  ch 15-   pix 00           ch 07-                 pix 00           
  -------- ---------------- ---------------------- ---------------- --------

# ***Channel shift detection and correction***

# The two methods that have been used for channel shift detection are summarized as

1)  Interferogram mean value. This method makes use of a fortuitous
    artifact of the signal chain. Due to differences in biasing
    resistors, all the detectors have slightly different interferogram
    mean values after A-D conversion. These mean values provide a
    16-element array for each detector array that can be used for
    correlation with the mean interferogram values from each scan. A
    channel shift is observed as a shift in channel position for the
    peak correlation.

2)  Non-linear behavior in slope of zpd offset vs. pixel. If pixels were
    not correctly assigned, the pattern of zero-path-difference (zpd)
    offsets, which should fall on a straight line as a function of pixel
    becomes obviously distorted. This method cannot be used if the
    spectral phase is indeterminate, i.e., when the signal is too low.

The correct assignment of pixel is critical for both L1B and L2
processing. Therefore, the channel shift correction requires a robust
method that can detect a channel shift from any scan, regardless of the
target source. The current processing approach applies the first method,
using interferogram mean values to detect channel shifts and apply the
corrections in L1A. The correction of the channel shift is verified in
L1B for OBRCS (340K) calibration spectra when the interferometer shear
is computed, which uses the slope of zpd offsets and checks for
linearity vs. pixel.

[Channel Shift DFMs:]{.underline}

TES A&S DFM \# 435, "Absolute channel mean value map", R. Beer,
4/23/2003.

TES A&S DFM \# 443, "Phase analysis approach to shear estimation", K.
Bowman, E. Sarkissian, 5/1/2003.

TES A&S DFM \# 445, "Validation of channel shift detection using
interferogram mean value map", H. Worden, K. Fry, 5/6/2003.

TES A&S DFM \# 493, "Correcting interferogram sample timing to account
for channel shifts", E. Sarkissian, H. Worden, 8/10/2003.

**6.2 FFT Size / Filter Spectral Range**

***FFT Size Determination***

Raw interferograms are truncated to a prime-factor FFT size that allows
efficient computing. The FFT size determines the spectral resolution
used for processing the calibration data taken during the System Test 5
(ST5). It will be used for the operational level 1B calibration
processing.

The interferogram Zero Path Difference (ZPD) corresponds to the maximum
deviation from the mean. Taking a symmetric number of data points about
the ZPD index and truncating it to the next smaller prime-factor number
gives a good FFT size. The determination of the operational FFT size is
based on the statistical distribution of two sided interferogram size
with a centered ZPD index.

A total of 89,079 interferograms taken during ST5 were processed. For
example, Figure 6.2.1 shows the half size of the two sided interferogram
for the clock divisor value 9 and for low resolution scans. For this
case, the FFT size half is found to be 16200. The long scan FFT size
will be 4 times larger. Table 6.2.1 shows the FFT size results.

Table 6.2.1. FFT Size of Short and Long Scans.

  ------------------------ -------------------------- --------------------
  **Divisor Clock**        **Short Scan**             **Long Scan**

  8                        18200                      72800

  9                        16200                      64800

  10                       14580                      58320

  11                       13312                      53248

  12                       12150                      48600
  ------------------------ -------------------------- --------------------

It was found that the smaller side of a full interferogram for a forward
scan direction occurs after the ZPD index. It occurs before the ZPD
index for the reverse scan direction. Referring to Figure 3, DFM
456-6.1, the ZPD index distribution is somewhat uniform outside the main
cluster. For ZPD indices outside the main cluster, no correlation was
found with any physical attributes (Filter, scan resolution, scan
direction, scan view) suggesting it is caused by the TES instrument
start scanning mechanism.

For more details, see the TES A&S DFM 456-6.1, "FFT Size Determination
From ST5 Calibration Data".

Figure 6.2.1. Histogram of interferogram ZPD indices for clock divisor
value 9, low resolution scans. For this case, the FFT size half is found
to be 16200.

![](media/image7.wmf){width="5.998611111111111in"
height="4.284722222222222in"}

***Filter Range***

To the nearest 10 cm^-1^, the filter half-power point frequency ranges
observed for 340K OBRCS system test spectra are given in the following
table.

+-------------+-----------------------------+--------------------------+
| TES Filter  | Frequency Range             | Wavelength Range         |
|             |                             |                          |
|             | (cm^-1^)                    | \(m\)                    |
+-------------+-----------------------------+--------------------------+
| 2B1         | 650 - 930                   | 15.4 - 10.8              |
+-------------+-----------------------------+--------------------------+
| 1B1         | 810 - 1060                  | 12.3 - 9.4               |
+-------------+-----------------------------+--------------------------+
| 1B2         | 920 - 1160                  | 10.9 - 8.6               |
+-------------+-----------------------------+--------------------------+
| 2A1         | 1090 - 1350                 | 9.2 - 7.4                |
+-------------+-----------------------------+--------------------------+
| 2A2         | 1280 - 1580                 | 7.8 - 6.3                |
+-------------+-----------------------------+--------------------------+
| 2A3         | 1480 - 1780                 | 6.8 - 5.6                |
+-------------+-----------------------------+--------------------------+
| 2A4         | 1670 - 1980                 | 6.0 - 5.1                |
+-------------+-----------------------------+--------------------------+
| 1A1         | 1890 - 2260                 | 5.3 - 4.4                |
+-------------+-----------------------------+--------------------------+
| 1A2         | 2190 - 2470                 | 4.6 - 4.0                |
+-------------+-----------------------------+--------------------------+
| 1A3         | 2410 - 2690                 | 4.1 - 3.7                |
+-------------+-----------------------------+--------------------------+
| 1A3         | 2570 - 2850                 | 3.9 - 3.5                |
+-------------+-----------------------------+--------------------------+
| 1A5         | 2750 - 3050                 | 3.6 - 3.3                |
+-------------+-----------------------------+--------------------------+

Reference with figures: TES DFM #1367.

**6.3 Shear / Co-Boresight**

**6.4 Time Trend of Signal Level, Ice Buildup**

During the TES system tests, the measured signals from the on-board
radiometric calibration source (OBRCS) were used to monitor the
radiometric stability of the instrument over time. The OBRCS was set to
its operating temperature 340 K and the PCS track/x-track angles were
kept the same through all five system tests.

The spectral magnitudes as a function of frequency derived from fixed
number of interferogram points per filter/pixel are used to examine the
spectral signatures, the pixel dependencies, and the time trends of the
signal level. The signal levels are mainly affected by two factors, the
ice/solid CO~2~ buildups on the detectors and the instrument alignment.

The ice / solid CO~2~ buildups on the detected had been expected. This
effect dominated the downward time trends of the signal levels of filter
2B1 and 1B1 (TES DFM #1362 and DFM A&S 491). Periodically warming up of
the detectors is therefore necessary though less ice is expected in
space. Figures 6.4.1 and 6.4.2 below show the dominant effect of ice
buildups on integrated spectral magnitudes of 2B1 and its spectral
features respectively.

![](media/image8.png){width="3.1104166666666666in"
height="4.034027777777778in"}![](media/image9.png){width="4.091666666666667in"
height="3.1743055555555557in"}

> Figure 6.4.1. Integrated spectral magnitudes Figure 6.4.2. Spectral
> magnitudes of best
>
> for 2B1 as functions of time (day) for ST2, (no-ice) and worst (a lot
> of ice and poor
>
> ST3, ST4, and ST5. The beam splitter alignment) measurements taken in
> ST5.
>
> temperature time trend plot is also provided.

The signal level changes with the change in the instrument alignment.
This effect dominates the time trends of the signal levels of filters
2A4 and 1A1 (TES DFM A&S 491). Figure 6.4.3 and 6.4.4 below show the
dominant effect of alignment on 2A4 integrated spectral magnitudes and
its spectral features respectively. The signal level time trends for
other filters showed the mixed effects of ice buildup and alignment.
Great deals have been learned about the instrument through examining the
relationships among the analyses of the shears, the co-boresights, and
the time trends of the signal levels.

Keeping track of the integrated spectral magnitude is a simple way of
monitoring the instrument radiometric performance. This method should be
applied for on-orbit calibration data.

> ![](media/image10.png){width="2.8875in"
> height="3.7284722222222224in"}![](media/image11.png){width="3.8444444444444446in"
> height="2.963888888888889in"}
>
> Figure 6.4.3. Integrated spectral magnitudes Figure 6.4.4. Spectral
> magnitudes of best
>
> for 2A4 as functions of time (day) for ST2, (alignment), typical
> (alignment) and worst
>
> ST3, ST4, and ST5. The beam splitter (alignment) measurements taken in
> ST5.
>
> temperature time trend plot is also provided.

5.  **Channeling**

Channeling is a common artifact in FTS spectra that appears as a
near-sinusoidal modulation in the spectral continuum. Although wedging
of optical elements can significantly minimize the effect, it is hard to
eliminate altogether. However, if the channeling spectral features are
stable enough, they are completely removed in the process of radiometric
calibration since they are common to both target and calibration
spectra. The issues that concern us for characterizing the channeling
are therefore the stability of the amplitudes, frequency positions and
periods of the spectral features.

Since channeling features have appeared in all system test data, it was
possible to perform an analysis before ST-5. Using ST-4 data, the
channeling amplitude was measured as less than 1%, as expected, and the
period was greater than 1 cm^-1^ as required. An investigation of the
stability of the features showed that over a 5 day period, the features
are cancelled in spectral ratios. This meets the requirements for global
survey calibration where calibration spectra are applied to target scans
acquired within about 36 hours of the calibration scans. Channeling
spectral features also had to be tested for cancellation when lower
resolution (4 sec.) calibration spectra are interpolated and used to
calibrate higher resolution (16 sec.) scans. In all filters, there were
no residual channeling features in the differences between interpolated
short scans and measured long scans.

**6.6 Electrical Filter**

From science test (ST) 1, it became immediately obvious that spectra
measured from the on-board radiometric source (OBRCS) contained
unexpected features in both the magnitude and phase when the electrical
filters were turned on. In ST 4, measurements were made of the OBRCS
with both the electrical filters on and off. The difference in the
magnitude spectrum of the electrical filter-on case relative to the
magnitude spectrum of the electrical filter-off case is over 10% at
certain frequencies. Furthermore, the phase of the OBRCS measurement is
non-linear with respect to frequency near the optical filter edge. As a
result, the electrical filters were turned off for all subsequent ST
experiments. The loss of the electrical filters resulted in a roughly
factor 3 increase in signal-to-noise ratio.

The spectral distortion seen in the spectra appears to result from a
subtle relationship between the electrical filters and the velocity of
the interferometer relative to the analog-to-digital (A/D) samplers. The
non-linear phase response of the electrical filters induced a phase
dispersion in the interferogram. The non-uniform time sampling of the
phase dispersion in the interferogram, which is due to the nonlinear
velocity characteristics of the interferometer arm, resulted in a
distorted signal. A simulation study was conducted based on this TES
sampling model that produced spectral characteristics similar to those
observed in ST4. A method for correcting the nonuniform velocity when
the electrical filters are on was proposed in TES A&S DFM 521 given a
sufficiently accurate knowledge of the velocity. However, the
preliminary results have not been conclusive. Additional investigation
is required.

**6.7 Gain**

# ***Requirement***

# 

Each of the 64 TES signal chains has four switchable gains, nominally
x0.5, x2, x8 and x32 for the "A" arrays and x0.5, x1, x2 and x8 for the
"B" arrays. Since it may occur that, for example, OBCRS 340K
calibrations are taken at a different gain setting from "cold space"
calibrations, it is essential that the *relative* gains be known to
better than 0.5%. Note that the *absolute* gains are irrelevant (because
they are subsumed in the radiometric calibration process), so the lowest
gain setting is arbitrarily assigned an exact value of 0.5 and the other
gains determined with respect to this value.

# ***Method***

TES detectors have a "closed" filter wheel setting so that the detectors
see nothing but a cavity at a temperature of about 80K(?). Measuring the
gain then reduces to a measurement of the mean noise power at each
setting. The exact approach is fully described in A&S DFM 436 (see
Volume II).

The method determines the relative gains to a precision of 0.3% or
better, so the requirement has been satisfied.

# ***Comment***

It is highly probable that we will operate TES at a constant 0.5 gain
setting, thus rendering this calibration moot. Nevertheless, it will be
repeated on orbit in case different gains should be re-introduced.

**6.8 Spectral Spikes**

The spikes appeared in nearly all processed TES spectra. These spikes
were found reproducible over the system tests, *e.g.,* occurred in the
same spectral frequency points for a given type of measurement (*e.g.,*
global survey). The analysis of the frequencies and amplitudes points
directly to electromagnetic pickup of the analog-to-digital
converter-sampling clock by the signal chains.

DFM A&S 447 describes the general characteristics of the spikes, the
algorithm used to identify the spikes, and the spike lists generated
from global survey data taken in ST5 Part B. The products of this
analysis will be used in initial Level 1B data processing for
de-weighting the spectral points with spikes. Figure 6.8.1 and 6.8.2
below show two examples of the spiky magnitude spectra, 2B1 and 1B2.

> ![](media/image12.png){width="3.275in"
> height="2.5305555555555554in"}![](media/image13.png){width="3.3409722222222222in"
> height="2.5881944444444445in"}

Figure 6.8.1. Illustration of spikes in Figure 6.8.2. Illustration of
spikes in

magnitude spectra, 2B1, short scan. magnitude spectra, 1B2, short scan.

**6.9 Pointing Angles for Calibration Sources**

Since an absolute calibration of the PCS (pointing control system) was
not available, optimal pointing angles were determined for the on-board
and external calibration sources using target search procedures designed
to produce maps of source signal levels over a grid of pointing angles.

***Spectral and radiometric sources***

The same methods for determining the optimal pointing angles were
applied for the external gas cell, external cold (90K) source, external
variable blackbody and the on-board radiometric calibration source
(OBRCS), *i.e*., the internal variable blackbody.

The measurements for these "target searches" were obtained by stepping
through a 9 x 9 grid of track and x-track PCS angles with 2 scans at
each grid point. The grids were centered on the best known angles for
each source before the target search.

The data were processed by converting interferograms to magnitude
spectra and performing a spectral average, after removing any spikes. It
was also critical to account for channel shifts since the pixels were
processed separately. Using the spectral averaged magnitude, a single
values could be assigned for each pixel, each PCS angle. Contour maps of
magnitude values vs. track and x-track angles showed that an adequate
range of target signal levels were subtended by the grid of PCS angles.

To find the angles that produce the maximum signal levels (or minimum in
the case of the external cold blackbody), we compute the angular
intersection of two signal-weighted curves. For example, the values in
the x-track curve are the signal-weighted x-track angles computed from
each along-track cross-section. See Fig. 6.9.1 for an example. The
optimal angles were reported for each focal plane, using the average
over the detectors. Final PCS angle values used in ST-5 command macros
were the focal plane averages.

![](media/image14.png){width="4.529166666666667in"
height="3.4743055555555555in"}

Figure 6.9.1. Example of intersecting curves used to determine pointing
angles for the external calibration sources and OBRCS. The
signal-weighted curves are overplotted on a contour map of signal levels
for the external variable blackbody.

Reference: TES DFM \# 1366, "ST-4 Target Search Results", H. Worden,
3/6/2003.

***Spatial Calibration Source***

The on-board spatial calibration source (SCS) was used to help validate
the PCS pointing and TES absolute boresight. Additionally, it was used,
and will be used on orbit, to evaluate co-boresight in the along track
direction. The SCS was also used for evaluation of the jitter in the PCS
system.

**6.10 PCS Calibration**

The TES pointing and control system (PCS) operates a two-axis gimbaled
mirror, M1, that points the line of sight (LOS) of the instrument over
abroad range of angles with respect to the Aura spacecraft. The
objective of the PCS calibration was to determine the angular
relationship between the TES line of sight and the Aura spacecraft base
coordinate system for any given set of PCS gimbal angles. The final
result of the calibration together with spacecraft position and attitude
is used to locate TES measurements on the surface of the earth. The
calibration required a number of measurements of the alignment of
various elements of TES and the spacecraft. These measurements were
brought together to determine the overall relationship between the LOS,
the PCS gimbal encoder angles and the Aura spacecraft. A detailed
description of the data and analysis is reported in DFM 1437. The
material presented here provides an overview of the approach, a summary
of the data and the final result.

The calibration utilized TES radiometric measurements of the TES onboard
spatial calibration source (SCS) and theodolite measurements of the SCS
slit and of optical alignment reference cubes, commonly referred to
simply as alignment cubes, mounted on the SCS, on the instrument base
structure and on the spacecraft. The radiometric measurement provided
the relationship between the LOS and PCS gimbal angles. Theodolite
measurements of the SCS alignment cube, the instrument alignment cube
and of an alignment cube mounted on the spacecraft provided the
relationship between the SCS and the Aura spacecraft base coordinate
system.

# ***LOS to SCS Slit Measurement***

The SCS is a collimated slit source mounted on the interior of the TES
instrument. In flight it will be used to measure the relative alignment
of the four test detector arrays. However, its absolute position with
respect to the instrument alignment cube is known with enough accuracy
that it can also serve as a pointing reference with respect to the
instrument structure. During system test 5 a series of interferograms of
the SCS were recorded over a range of PCS angles. The prescribed angles
were chosen to lie on lines parallel to the long axis of the detector
arrays. The interferograms were converted to spectra and integrated to
provide the response of the 16 pixels of the 2A detector array as a
function of PCS angles. Figures 6.10.1 illustrated the PCS angles used
for the measurement. Figure 6.10.2 is an example of the integrated
response data.

The PCS pointing reference for the field-of-view of the instrument is
defined as the midpoint of the line defined by the junction between
pixels 8 and 9. The PCS angles for this point with respect to the SCS
slit was determined from a two-dimensional fit to the response curves.
The details of the fit are given DFM 1406. The final results are
summarized in Table 6.10.1.

  -----------------------------------------------------------------------
  ![](media/image15.wmf)

  Figure 6.10.1. The PCS scan lines used to determine the instrument
  boresight with respect to the SCS slit. The rotated grid is represents
  the nominal location of a detector array. The numbers in parentheses in
  the legend are the run counter setting of the recorded data.
  -----------------------------------------------------------------------

  -----------------------------------------------------------------------
  ![](media/image16.wmf)

  Figure 6.10.2. Example of the integrated spectra from a scan of the
  PCS. This is from the 2A1 detector array/filter.
  -----------------------------------------------------------------------

  ----------------------------------- -----------------------------------
  Table 6.10.1. PCS angles that place 
  the center the TES line of sight on 
  center of the SCS slit.             

  PCS Gimbal                          PCS Gimbal Angle (deg.)

  Pitch                               -44.958 ± 0.002

  Roll                                -221.952 ± 0.002
  ----------------------------------- -----------------------------------

# 

# ***Alignment Cube Measurements***

Of the four sets of thodolite measurements used to relate the boresight
to the spacecraft coordinate system, three of the sets, the relationship
of the SCS slit to the SCS alignment cube, the relationship of the SCS
alignment cube to the instrument alignment cube (IAC) were made at JPL.
The results, reproduced in Tables 6.10.2, 6.10.3 and 6.10.4, are
reported as Euler direction cosines (unit vectors) in the instrument
base coordinate system. The relationship between the instrument and the
spacecraft was measured at by NGST, using the instrument alignment cube
as a reference. The Euler direction cosines are given in Table 6.10.5.

  ----------------- ----------------- ----------------- -----------------
  Table 6.10.2. SCS                                     
  slit direction                                        
  cosines in the                                        
  instrument                                            
  coordinate                                            
  system.                                               

                    Instrument X      Instrument Y      Instrument Z

  SCS Slit X        0.000107294       0.642648041       -0.766161526

  SCS Slit Y        -0.647484997      0.583919641       -0.489694835

  SCS Slit Z        -0.762078189      -0.496025552      0.416167616
  ----------------- ----------------- ----------------- -----------------

  ----------------- ----------------- ----------------- -----------------
  Table 6.10.3. SCS                                     
  alignment cube                                        
  direction cosines                                     
  in the instrument                                     
  base coordinate                                       
  system.                                               

                    Instrument X      Instrument Y      Instrument Z

  SCS Cube X Face   0.999997213       -0.000309776      -0.000689598

  SCS Cube Y Face   0.000314787       0.996536593       0.083123097

  SCS Cube Z Face   0.000660201       -0.083124884      0.996540199
  ----------------- ----------------- ----------------- -----------------

  ----------------- ----------------- ----------------- -----------------
  Table 6.10.4. IAC                                     
  direction cosines                                     
  in the instrument                                     
  base coordinate                                       
  system.                                               

                    Instrument X      Instrument Y      Instrument Z

  IAC X Face        0.553999774       -0.832516887      -0.000056208

  IAC Y Face        0.832516871       0.553999258       0.000715970

  IAC Z Face        -0.000564919      -0.000443429      0.999999714
  ----------------- ----------------- ----------------- -----------------

  ----------------- ------------------- ------------------- -------------------
  Table 6.10.5.                                             
  Spacecraft body                                           
  fixed direction                                           
  cosines in the                                            
  TES alignment                                             
  cube coordinate                                           
  system. This                                              
  matrix is                                                 
  referred to below                                         
  as S.                                                     

                    IAC X               IAC Y               IAC Z

  S/C X             0.55428212442495    0.83232884714490    -0.00012943492856

  S/C Y             -0.83232852135924   0.55428203376661    0.00081214253476

  S/C Z             0.00074771311512    -0.00034242370680   0.99999966183549
  ----------------- ------------------- ------------------- -------------------

# ***Transformation of the Line of Sight in PCS angles to Spacecraft Coordinates***

The direction of the line of sight after reflection by the M1 mirror is
give by the rotation matrix

$\text{LOS}_{\text{PCS}} = \begin{bmatrix}
\text{cos}2q & \text{sin}2q\text{sin}f & - \text{sin}2q\text{cos}f \\
\text{sin}2q\text{sin}f & - \text{cos}2q\text{sin}^{2}f + \text{cos}^{2}f & \left( \text{cos}2q + 1 \right)\text{cos}f\text{sin}f \\
 - \text{sin}2q\text{cos}f & \left( \text{cos}2q + 1 \right)\text{cos}f\text{sin}f & - \text{cos}2q\text{cos}^{2}f + \text{sin}^{2}f
\end{bmatrix}$,

where $q$ is PCS pitch encoder angle and $f$ is the PCS roll encoder
angle. The rotation matrix brings the line of sight calculated from
$\text{LOS}_{\text{PCS}}$ parallel with the measured SCS slit-PCS angles
was calculated. This rotation matrix, C, is given in Table 6.10.6.

  ----------------- ----------------- ----------------- -----------------
  Table 6.10.6. The                                     
  direction cosine                                      
  matrix, C, that                                       
  brings the                                            
  LOS~PCS~                                              
  coordinate system                                     
  parallel to the                                       
  instrument base                                       
  coordinate                                            
  system.                                               

                    Instrument X      Instrument Y      Instrument Z

  LOS~PCS~ X        0.999412569       0.026375448       -0.021882692

  LOS~PCS~ Y        -0.026974415      0.999256588       -0.027543658

  LOS~PCS~ Z        0.021139948       0.028117751       0.999381056
  ----------------- ----------------- ----------------- -----------------

The LOS in the spacecraft body fixed coordinate
system,$\text{LOS}_{\text{SC}}$~C~, is given by the matrix product

$\text{LOS}_{\text{SC}} = S^{T}\text{IAC}^{T}\text{CLOS}_{\text{PCS}}$.

The superscript T indicates the standard matrix transpose. The matrix
product *S^T^IAC^T^C* that transforms $\text{LOS}_{\text{PCS}}$ to
$\text{LOS}_{\text{SC}}$ is given in Table 6.10.7.

  ----------------- ----------------- ----------------- -----------------
  Table 6.10.7. The                                     
  direction cosine                                      
  matrix to                                             
  transform                                             
  LOS~PCS~ to                                           
  LOS~SC~                                               

                    SC X              SC Y              SC Z

  LOS~PCS~ X        0.999420057       0.026738156       -0.02108889

  LOS~PCS~ Y        -0.027335715      0.999217281       -0.028593972

  LOS~PCS~ Z        0.020307821       0.029153871       0.999368595
  ----------------- ----------------- ----------------- -----------------

**6.11 Non-Linearity**

# ***Requirement***

Any non-linearity in TES detectors and signal chains must be known to
within 1% or better.

# ***Method***

One of the tests performed during calibration was to illuminate TES with
scattered light from a 9.5 m CO~2~ laser. Despite the scatter plate, the
laser shows a peak brightness temperature in excess of 4000K, a vastly
stronger signal than TES will ever see on orbit.

Examination of these data show that there is, indeed, a small
non-linearity in the system. However, the magnitude is below 0.01% and
will have no measurable impact on TES retrievals. Thus the requirement
is automatically satisfied and the issue can be safely ignored.

A full description of the method can be found in A&S DFM 441 (see Volume
II).

**6.12 Radiometric Calibration**

Applying the complex calibration method, The calibrated target radiance,
*R~tgt~*, is given by:

![](media/image17.wmf)

Where C indicates the phase-aligned complex spectrum for target (*tgt*),
and phase-aligned averaged spectra for the 90 K cold blackbody and
OBRCS. We can assume the Planck function for the cold blackbody,
*B(,T~cold~)* is negligible. We also approximate the OBRCS emissivity,
~OBRCS~ to be 1.0, as discussed in TES A&S DFM \# 504.

# ***Phase Alignment***

In order to average complex calibration spectra over different scans,
(same detector, same scan direction) we have to perform the initial step
of phase alignment to remove scan-to-scan integer sampling phase
differences. These sampling phase differences are the result of
variations in the starting points for interferogram sampling, giving
optical path differences between scans. The difference in optical path
between two scans is an integer multiple of the sampling distance
defined by the laser fringe count. A difference in optical path between
interferograms is equivalent to a linear phase difference between
spectra. The spectral phases of different scans are aligned by looping
through integer samples *k*, adjusting the phase by exp(-*i2k/l*) where
is the spectral frequency and *l* is the laser frequency, and checking
for a minimum phase difference from a reference scan. Figure 6.12.1
shows an example of single scan phases before and after phase alignment.

Once calibration spectra are phase aligned and averaged, the target
spectrum, cold blackbody average and OBRCS average have to be phase
aligned. This is normally done using a double loop with integer samples
*j* and *k* for adjusting the cold blackbody and target spectrum
sampling phases relative to the OBRCS. Phases are considered aligned
when the RMS of the imaginary term in the complex calibration equation
above is minimized or when the phase difference between numerator and
denominator in the complex calibration equation is minimized.

# ***ST-5 Calibration Data***

The data taken to assess radiometric accuracy and precision were sets of
runs with views of the external blackbody set at temperature increments
of 20K, the OBRCS set to 340K and the cold external (90K) blackbody.
This analysis was performed mostly using ST-5 part B data, runs 509 and
513, corresponding to target temperatures of 220K and 300K. This data
set was chosen, despite poor signal levels due to shear, because it was
the only data taken with consistent gain settings for all views, which
allows for radiometric calibration without added errors from gain
uncertainties. Consistent gain settings are most important at the lower
frequencies (2B1, 1B1 and 1B2 filters) where the instrument offset
radiance contributes significantly to the measured signal and gain
calibration errors would be non-negligible for the cold blackbody view
set to a different gain from the OBRCS.

# ***Radiometric calibration results***

Brightness temperature precision and accuracy for the calibration
results are summarized in Table 6.12.1 for two temperatures of the
external variable blackbody, which was used as the target source with
the cold (90K) blackbody and OBRCS (340K) used for calibration spectra.
Radiometric calibration precision is given by the NET (Noise equivalent
delta temperature) for the average of the calibrated scans. NET is
computed by adding the NESR (Noise equivalent spectral radiance) to the
Planck function for the source temperature and converting the summed
radiance back to a brightness temperature. Systematic errors in the
radiometric calibration are estimated using differences of the best fit
to the calibrated brightness temperatures with the temperatures taken of
the variable external blackbody during data collection. The numbers
given in the Table 6.12.1 are spectral averages over frequencies inside
the optical filter band passes. Figure 6.12.2 shows an example of
averaged calibrated radiances and corresponding brightness temperature
for a 220 K source.

Table 6.12.1. Estimated radiometric calibration precision (NET) and
systematic error for ST5 data.

+-----+---------+------------+------------+------------+-------------+
| Fil | Number  | 220 K      |            | 300 K      |             |
| ter | of      | source     |            | source     |             |
|     | scans   | (run 509)  |            | (run 513)  |             |
|     |         |            |            |            |             |
|     | A       |            |            |            |             |
|     | veraged |            |            |            |             |
+-----+---------+------------+------------+------------+-------------+
|     |         | NET (K)    | Sys. Err   | NET (K)    | Sys. Err    |
|     |         |            | (K)        |            | (K)         |
|     |         |            |            |            |             |
|     |         |            | T(ext      |            | T(ex        |
|     |         |            | BB)-T(best |            | tBB)-T(best |
|     |         |            | fit)       |            | fit)        |
+-----+---------+------------+------------+------------+-------------+
| 2B1 | 36      | 1.27       | 0.84       | 0.58       | 0.30        |
+-----+---------+------------+------------+------------+-------------+
| 1B1 | 20      | 0.44       | 0.62       | 0.23       | 0.19        |
+-----+---------+------------+------------+------------+-------------+
| 1B2 | 20      | 0.49       | 0.83       | 0.24       | 0.23        |
+-----+---------+------------+------------+------------+-------------+
| 2A1 | 10      | 1.21       | -0.13      | 0.46       | -0.24       |
+-----+---------+------------+------------+------------+-------------+
| 2A4 | 10      |            |            | 2.0        | -0.63       |
+-----+---------+------------+------------+------------+-------------+
| 1A1 | 20      |            |            | 2.6        | -0.56       |
+-----+---------+------------+------------+------------+-------------+

***Radiometric Accuracy Requirement***

# 

# The requirement for radiometric accuracy is 0.5 K. For a 300 K source we meet this requirement for the 2B1, 1B1, 1B2, 2A1 filters, based on the limited cases that have been processed. For 2A4 and 1A1, our determination of accuracy may be limited by the number of scans averaged in the case reported here and we need to run more calibration cases for these source temperatures. This report will be updated as more calibration scans are processed, with the results for the remaining target source temperatures appended.

References:

TES L1B ATBD (Algorithm Theoretical Basis Document), JPL D-16479, V1.1,
10/1999.

TES DFM \# 1427, "Initial assessment of radiometric accuracy", H.
Worden, 6/16/2003.

TES DFM \# 1438, "Comparisons of external blackbody to OBRCS with ST-5
data", H. Worden, 8/7/2003.

TES A&S DFM \# 504, "Relatively small errors in calibrated brightness
temperatures when OBRCS emissivity is neglected", H. Worden, 8/26/2003.

![](media/image18.png){width="7.25in" height="3.720833333333333in"}

![](media/image19.png){width="7.25in" height="3.582638888888889in"}

> Frequency (cm^-1^)

Figure 6.12.1. Example of phase alignment. Top shows the phases of
individual scans before sampling phase alignment, bottom shows the
phases after alignment.

![](media/image20.jpeg){width="6.620833333333334in"
height="5.697916666666667in"}Figure 6.12.2. Example of calibrated scan
average for 220 K source. The top panel shows the averaged calibrated
radiance (W/cm^2^/sr/cm^-1^) for 20 spectra, 1B1 filter. The second
panel shows the average residual radiance for the complex imaginary
terms with the red line indicating zero bias. The third panel shows the
calibrated radiance converted to brightness temperature and compared to
the red line indicating the expected temperature for the external
blackbody. The blue-green line indicates the best fit for the calibrated
brightness temperature and the difference between this and the red line
is the measure of systematic error for radiometric calibration. The
bottom panel shows the NESR for the scan average (20 scans in this
case).

**6.13 NESR**

***Methods***

Two different methods are used to estimate NESR (noise equivalent
spectral radiance) from the ST-5 data. The first relies on having
several spectra for the same radiometric target while the second method
is used to estimate NESR for each spectrum as a part of standard L1B
calibration processing.

Method 1: If several measurements of the same target (e.g., the
blackbody sources from the TES system tests) are available, the NESR, by
definition, is the standard deviation of the measurements at each
spectral point.

Method 2: To estimate the NESR for a single radiance spectrum during
complex calibration, we use the rms of the magnitude of an out-of-band
spectral region:

![](media/image21.wmf)

where *C~tgt~* is the complex target spectrum, *~out_of_band~* is the
frequency range for extracting the uncalibrated noise in data numbers
(DN), *r()* is the instrument response function to convert uncalibrated
DN to radiance units (W/cm2/sr/cm-1) and is computed from averaged
complex calibration spectra *C~OBRCS~* and *C~cold~* and their
corresponding Planck functions *B(,T~OBRCS~)* and *B(,T~cold~)*. In our
case, *B(,T~cold~)* is negligible and we assume the OBRCS emissivity is
1.0.

Figure 6.13.1 shows a comparison of NESR estimates using both methods.
It is important to demonstrate here that the NESR estimates using single
scans agree with the statistical standard deviation of several scans of
the same target since multiple scans of a stable external target, other
than cold space, will not be available in orbit. Both NESR calculations
are performed in the process of system test radiometric calibration, and
they have consistently agreed. The in-band RMS NESR values for the two
methods agree to within 2% for the cases checked, satisfying the 5%
accuracy requirement for NESR measurements.

***ST-5 data used for NESR estimates***

Most of the NESR estimates reported here were performed using the same
data set from ST-5 part B used for radiometric calibration, with
significantly lower signal levels due to interferometer shear. Lower
signal levels in the OBRCS data result in higher NESR values, which are
less optimistic than what we expect on-orbit. However, as explained in
the section on radiometric calibration, the dataset used was the only
set where gain calibration errors did not have to be considered. For
filter ranges at higher frequencies, exact correction of the relative
gains used for data from the OBRCS view and the cold external blackbody
is not as critical. Using the secondary dataset for radiometric
calibration from ST-5 part A, some NESR estimates can be made for higher
frequency filters (2A1-2A4, 1A1) for cases where the gain setting used
for data from the external variable blackbody was the same as the gain
setting used for data from the OBRCS.

![](media/image22.jpeg){width="6.247222222222222in"
height="4.696527777777778in"}

Figure 6.13.1. Comparison of NESR estimate using standard deviation of
20 scans (red) vs. estimate from calibration of out-of-band noise
magnitude (black).

***ST-5 NESR results***

The tables below give frequency band average NESR values for the filters
used in the TES global survey. Table 6.13.1 lists results from ST-5 part
B runs 509 and 513, while Table 6.13.2 gives a comparison of ST-5 part A
to ST-5 part B, 2A1 filter only. ST-5 part A data had significantly
greater signal levels in the OBRCS magnitude spectra (about 25% larger
for the 2A1 filter) compared to the part B data.

***Model NESR results***

The radiometric model used to predict the system signal-to-noise is
described in Appendix C of the TES SOAGR (Scientific Objectives &
Approach, Goals & Requirements), JPL D-11294, V6.0, 1999. The inputs to
the model were adjusted so that model predictions of NESR agreed
reasonably well with the ST-5 external cold (90K) measurements. Figure
6.13.2 shows the model results of NESR for limb and nadir spectral
resolution at average expected source temperatures. Except for 2B1, the
model NESR predictions are better than the measured ST-5 values since
they assume no ice buildup or shear. For 2B1, which is detector noise
limited rather than background noise limited, the NESR value is worse
because the model accounts for the average detector noise while the
measurement used only detector 10, which is one of the better detectors.
Limb NESR values are a factor of 2 higher than nadir due to 4 times
longer integration time, however, they also have 4 times better spectral
resolution.

References:

TES A&S DFM #172, "Estimating L1B NESR from single spectra", H. Worden,
11/27/2000.

TES DFM \# 1329, "ST-2 NESR estimates", H. Worden, 11/13/2002.

TES A&S DFM \# 492, "Measured and Predicted TES NESR for ST-5 data", H.
Worden, R. Beer, 8/14/2003.

Table 6.13.1. NESR estimates from ST-5 Part B data for 4 sec scan
resolution (i.e., nadir resolution). Measurements are for detector 10.
NESR values are in-band averages in nW/cm2/sr/cm-1.

+-------------------+-----------+-----------+-----------+-----------+
| Filter            | 90 K      | 220 K     | 300 K     | 340 K     |
|                   | Source    | Source    | Source    | Source    |
|                   |           |           |           |           |
|                   | (ext.     | (ext.     | (ext.     | (OBRCS)   |
|                   | cold BB)  | var. BB)  | var. BB)  |           |
+-------------------+-----------+-----------+-----------+-----------+
| 2B1 (650-930      | 590       | 590       | 640       | 660       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+
| 1B1 (810-1060     | 98        | 110       | 170       | 200       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+
| 1B2 (920-1160     | 100       | 110       | 170       | 210       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+
| 2A1 (1090-1350    | 100       | 110       | 180       | 230       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+
| 2A4\* (1670-1980  | 170       | Not meas. | 240       | 330       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+
| 1A1\* (1890-2260  | 140       | Not meas. | 250       | 380       |
| cm^-1^)           |           |           |           |           |
+-------------------+-----------+-----------+-----------+-----------+

\* 2A4 and 1A1 average NESR values are \~ factor of 2 worse than they
would be for more optimal alignment.

Table 6.13.2. Comparison of ST-5 NESR estimates for 2A1 (pixel averages)
for different shear conditions (ST-5 part A vs. part B). NESR values are
in-band averages in nW/cm2/sr/cm-1.

+-----------------+-----------------+-----------------+-----------------+
| Run number      | 90 K Source     | 300 K Source    | 340 K Source    |
|                 |                 |                 |                 |
|                 | (ext. cold BB)  | (ext. var. BB)  | (OBRCS)         |
+-----------------+-----------------+-----------------+-----------------+
| ST-5 part A run | 83              | 142             | 188             |
| 458             |                 |                 |                 |
+-----------------+-----------------+-----------------+-----------------+
| ST-B part B run | 97              | 191             | 254             |
| 513             |                 |                 |                 |
+-----------------+-----------------+-----------------+-----------------+

![](media/image23.jpeg){width="6.502083333333333in"
height="4.773611111111111in"}

Figure 6.13.2. Model predicted NESR for optimal conditions on orbit
e.g., minimal interferometer shear and no ice-buildup.

**6.14 Frequency Calibration**

# ***Introduction***

The SOAGR (JPL D-11204 ver. 6) lists 6 ways by which frequency errors
can accumulate in TES spectra. They are:

1\) All pixels are off-axis;

2\) All pixels have a finite field-of-view;

3\) Aberrations and diffraction modify ray paths;

4\) Drifts of the control laser;

5\) Interferogram sampling errors;

6\) Doppler shift.

This section treats only two of them: off-axis effects and laser drifts
(which are believed to be due to temperature variations of the Nd:YAG
crystal). Finite field-of view (which subsumes aberrations and
diffraction) appears to have only a minor effect on frequency,
interferogram sampling errors are negligible and Doppler shift is
analytically correctible given knowledge of the orbital velocity,
pointing angle and Earth rotation.

# ***Laser Drift***

The temperature coefficient of the Nd:YAG laser is said to be --3.1
GHz/K

= -0.103 cm^-1^/K (higher temperature reduces the frequency). Given a
temperature control of 0.1 K, the output frequency can change by as much
as 0.01 cm^-1^. Though it appears that this drift is reasonably slow, it
is large enough that regular on-orbit frequency calibration will be
necessary. Fortunately, most prominent atmospheric lines have
frequencies known to a few parts in 10^-6^ cm^-1^, which will permit
more-than-adequate determination of the laser frequency.

For example, the P20 CO~2~ laser line (see section 6.15.1) is located at
1046.8543 cm^-1^. The analysis used there assumed that the Nd:YAG laser
frequency was 9394.3482 cm^-1^ but the apparent CO~2~ frequency was
1046.8832 cm^-1^, from which we may directly infer that the true Nd:YAG
frequency was 9394.0889 cm^-1^. It is also worth noting that the
illustration (Fig. 6.15.1.1) is a 20-scan average which took almost 6
minutes to acquire but there is no evidence (at the .001 cm^-1^ level)
that there were any frequency drifts during that period.

***Off-axis Effects***

As was discussed in A&S DFM 467 (see V2), the predicted frequency shift
across a detector array is parabolic:

Observations of CO~2~ laser lines (see Fig. 6.14.1) and gas cells
confirm this as well as providing a means of determining the location of
the optic axis in the in-track direction.

# ***Conclusion***

Thus although most frequency shifts are predictable, enough uncertainty
remains that frequent recalibration of the frequency scale will be
necessary on orbit.

![](media/image24.wmf)

# **Instrument Line Shape (ILS)**

# **6.15.1 CO~2~ Laser**

# 

# ***Introduction***

During ST5, observations were made of 2 CO~2~ laser lines (*via* a
scatter plate to reduce the intensity to a manageable level). The lines
were 00^0^1 -- 02^0^0 P20 at 1046.8543 cm ^--1^ and 00^0^1 -- 02^0^0 R24
at 1081.0877 cm ^--1^. Both lines were strong enough to provide an SNR
\~1000.

Observing a monochromatic line is the most direct way of determining the
instrumental line shape (ILS). ILS is a fundamental determinant of
spectral quality and purity and it is essential that it be known to high
accuracy in order for the retrieval process to proceed. Unfortunately,
it is not a general solution because there are too few lines to cover
the full TES spectral range, so the approach should be considered as a
*validation* of the alternative approach which requires deconvolution of
gas cell data (see Section 6.15.2).

However, the laser data are of sufficient quality that they have allowed
a direct comparison of observed and predicted ILS, where the prediction
is based on numerical integration of monochromatic interferograms across
a field-of-view model that was, itself, based on the deconvolved IFOV
data (see Section 6.16). The match is almost perfect, from which we draw
the following conclusions:

1\) Self-apodization can probably be ignored for all short scans;

2\) Correction for self-apodization for long scans can probably be
accomplished by employing a low-order polynomial amplitude function
developed from the model for a central frequency for each filter
together with a linear amplitude correction to the resultant spectra.

3\) At the longest wavelengths (certainly 2B1 and possibly 1B1 & 1B2),
no correction of any kind will be needed.

# ***Methods***

The method for producing laser spectra was straightforward -- the
interferograms were simply subjected to classical FSV phase correction.
The result is shown as the red curve in Fig. 6.15.1.1 where the *only*
manipulation done on the result was normalization to unity. In
particular, note that the ILS "sits" firmly on the zero line.

The method for producing the ILS model (black curve in Fig. 6.15.1.1) is
somewhat more complex. First, the mean in-track and cross-track
half-widths of the deconvolved target projector FOV data were
determined. From these, a response model on a 50 x 50 microradian grid
was constructed for each focal plane whose in-track profile was a
gaussian of the same half-width as the measurements and gaussian ends
for the cross-track response. Note that the in-track gaussians are
generally wider than the cross-track ones and increase with wavelength
(as expected). Fig. 6.15.1.2 shows a rendition of the model for array
1B. The response model contains 15750 bins (75 x 210).

For each bin and pixel, a cosine wave was generated according to cos(2
x. cos{ }) where tan^2^{} = tan^2^{} + tan^2^{} and & are the in-track
and cross-track off-axis angles for that location, x is the optical path
difference and is the monochromatic frequency. The result was summed
across all bins after multiplication by the response model. After
transformation, the ILS results (of which the black curve in Fig.
6.15.1.1 is one example).

# ***Self-apodization***

An ideal on-axis point detector in an FTS has an ILS which is the sinc
function (sin{x}/x). A (real) finite-size detector, on or off-axis, has
an ILS that displays self-apodization. The ILS's in Fig. 6.15.1.1 are
examples. For instance, in the sinc function the first negative lobes go
to --0.22 and all the lobes are symmetric about the center. Those shown
here do not quite meet these criteria. A convenient way of
characterizing the departure of the ILS from a sinc function is *via* a
magnitude and phase plot generated by downshifting the peak of the
measured or calculated ILS to zero frequency (baseband), transforming
and extracting the magnitude (= sqrt{real part^2^ + imaginary part^2^})
and the phase

(= tan^-1^{imaginary part/real part}). Fig. 6.15.1.3 is an example (for
the middle of filter 1A1 -- almost the worst case).

The plots are interpreted thus:

For a sinc function, the magnitude would be identically 1 from ZPD to
MPD and zero elsewhere. The phase would be zero everywhere. Since the
plots of Fig. 6.15.1.3 are manifestly not of this character, they
constitute *prima facie* evidence that the ILS is *not* a sinc function.
Furthermore, the "droop" of the magnitude at large PD is a measure of
the degree to which the secondary maxima are "sub sinc" and the
non-linearity of the phase is a measure of the *asymmetry* of the
ILS[^1].

The magnitude can be fitted with a low-order (\< 6^th^ ) polynomial
which can be used to construct a function to be divided into the
observed interferograms. Using a single function over an entire filter
somewhat over-corrects low frequencies and under-corrects high ones but
tests show that at TES OPD's and filter bandpasses, the effect is
essentially linear and is easily corrected in spectrum space. Obviously,
this assertion requires validation but, if it works, offers a simple way
of correcting for self-apodization.

It is also worth noting that over ranges where the magnitude "droop" is
less than about 10-15%, the departure from sinc is negligible. This
appears to be true for *all* short scans and for long scans at long
wavelengths.

![](media/image25.wmf)

Figure 6.15.1.1: red: measured CO~2~ laser line; black: predicted ILS
based on a field-of view model.

> ![](media/image26.wmf)

Figure 6.15.1.2: Model of the 1B array response (average for all pixels)

![](media/image27.wmf)

Figure 6.15.1.3: Example of self-apodization plots.

# **6.15.2 External Gas Cell Data**

The external gas cell measurements combining with model simulations are
used to spectrally characterize the instrument, including the frequency
calibrations for the measured spectra and the instrument line shape. The
gas cell measurements were made in TV19 interferometer only
measurements, and in the integrated system tests ST1, ST2, ST4 and ST5.
Table 6.15.2.1 lists the gases and the cell pressures etc. used in ST5
part A. A background blackbody at 340 K provided the radiation source.
The radiation went through a 40 cm long gas cell before entering to the
instrument. The gas pressures were carefully chosen for targeted filters
(spectral ranges) so that the majority of the absorption lines are
detectable but not saturated.

## Table 6.15.2.1. ST5 -- A Gas Cell Measurements

  ------ ----- ------------------ ----------------- --------- --------- --------
  Test   Run   Measurement        Target Filter     Cell Temp Cell      BB Temp
  ID     Cnt                                        (K)       Press     (K)
                                                              (Torr)    

  372    461   Empty 320 K BB     1A2, 1B1,1B2, 2B1                     317.6

  373    462   Empty 340 K BB,    1A2, 1B1, 1B2,                        336.9
               OBRCS, and Cold BB 2B1                                   

  374    463   CO~2~ 150 Torr 340 1B1, 1B2          286.7     149.8     337.1
               K BB                                                     

  375    464   CO~2~ 10 Torr 340  1A2, 2B1          288.6     10.3      336.14
               K BB                                                     

  376    465   Empty 320 K BB     1A2, 1B1, 1B2,                        317.7
                                  2A3, 2A4, 2B1                         

  377    466   Empty 340 K BB,    1A2, 1B1, 1B2,                        336.96
               OBRCS, and Cold BB 2A3, 2A4, 2B1                         

  378    467   N~2~O 150 Torr 340 1B1,1B2,2A3,2A4   290.6     150.0     336.8
               K BB                                                     

  379    468   N~2~O 5 Torr 340 K 1A2, 2A1, 2A2     291.5     5.2       337.04
               BB                                                       

  380    469   Empty 320 K BB     1A2, 2A1, 2A2                         317.63

  381    470   Empty 320 K BB     1A1                                   317.8

  382    471   Empty 340 K BB,    1A2, 2A1, 2A2                         337.0
               OBRCS, and Cold BB                                       

  383    472   Empty 340 K BB,    1A1                                   336.72
               OBRCS, and Cold BB                                       

  384    473   CO 5 Torr 340 K    1A1               291.8     4.9       337.13
  ------ ----- ------------------ ----------------- --------- --------- --------

# The gas cell measurement and the empty cell measurements at two background blackbody temperatures are used to derive the measured radiances (A&S DFM 337). A gas cell model simulation (LBLRTM with HITRAN 2000) is then made to compare with the measurement. As a first step, the pixel-dependent apparent laser frequencies were retrieved from measurement-model comparison (A&S DFM 464, 337 &447). This step greatly reduced the measurement-model radiance residual. 

There are a number of reasons possibly contributing to the remaining
measurement-model residuals. A&S DFM 512 listed and analyzed several
reasons. This DFM summarized some preliminary results from the studies
on the ST5 gas cell data to assess the role of ILS on the
measurement-model residuals.

Two important conclusions can be drawn from this study. First, The
smaller signal to noise ratio of the instrument compared to the original
prediction, due to variety of reasons including not applying electrical
filter, seems to "hide" the effect of ILS on measurement-model
residuals. The noises in the measured spectra dominate over the
systematic errors introduced by not-well-known ILS. Second, when
applying the theoretical ILS based on the geometry of the detectors to
the model radiance, the measurement-model residuals are reduced. This
implies that the theoretical ILS reasonably describes the ILS for the
real instrument.

**6.16 Field of View (FOV) Response**

The TES Field-of-View (FOV) was measured to assess both the spatial
response of the pixels as well as the relative boresight between the
four detector arrays. Measurements of the TES FOV were made using an
external target projector equipped with a chopper. The target projector
source was a slit 0.75 milliradians wide and approximately 12
milliradians long. The slit was oriented with the long dimension
parallel to the long dimension of the pixels for assessing the FOV for
limb scans and the relative detector alignment perpendicular to the long
axis of the pixels (along track). For assessing the relative alignment
along the long axis of the pixels, the slit was oriented with the long
dimension perpendicular to the long dimension of the pixels(cross
track). In both cases, the slit image was scanned across the array by
moving the source in the target projector in a direction perpendicular
to the slit long axis. A schematic diagram of the scan directions is
shown in 6.16.1.

To measure the pixel response, the amplitude of the chopped signal was
computed. The chopped amplitude was defined difference in the mean
values of the chopper open and chopper closed phases after the
zero-path-difference signal and noise spikes were removed. The resulting
response curves where then used directly for boresight location.
Additional details regarding relative boresight can be found in section
5.3: Shear/Co-Boresight.

For limb FOV use, the curves were further processed to remove the
effects of the finite slit size and target projector diffraction. The
slit and diffraction effects were deconvolved from the FOV curves using
a Richardson-Lucy algorithm. More details of the deconvolution are
documented in TES A&S DFM #479. Examples of the deconvolved FOV curves
for each detector array are shown in figures 6.16.2 to 6.16.5.

The pixel response curve are approximately Gaussian with a broader base.
The along-track full-width at half-maximum were approximately 0.75
milliradians for 1A1, with the longer wavelength filters being slightly
larger due to diffraction within TES. Cross-track widths were the
expected 7.5 milliradians. Inter-pixel spacing was the expected 0.75
milliradians. Full-width at half-maximum or each pixel and inter-pixel
spacing are tabulated in TES DFM #1432. The arrays exhibited some
variation in peak response across the pixels, which is documented in A&S
DFM 498.

![](media/image28.jpeg){width="4.113194444444445in"
height="4.113194444444445in"}

Figure 6.16.1

![](media/image29.png){width="4.595138888888889in"
height="3.6659722222222224in"}

> Figure 6.16.2
>
> ![](media/image30.png){width="4.5375in" height="3.6194444444444445in"}

Figure 6.16.3

> ![](media/image31.png){width="4.5375in" height="3.6145833333333335in"}
>
> Figure 6.16.4
>
> ![](media/image32.png){width="4.498611111111111in"
> height="3.588888888888889in"}

Figure 6.16.5

**7.0 CONCLUSIONS -- IMPACTS ON ALGORITHM REQUIREMENTS**

The analysis of calibration data has led to a number of changes to the
algorithm and software requirements. In some cases, these changes
required additional algorithm requirements, e.g., the calculation of
shear from spectral phase measurements, and in other cases, has led to a
relaxation of requirements, e.g., the lack of non-linearity in the
detector responses. Table 7.0 summarizes which subsystems are impacted
by the system test findings and whether or not a new requirement was
necessary.

The existence of channel shifts requires two changes to the Level 1
software and algorithm requirements. The first is the implementation of
the channel detection and correction algorithm (Sec 6.1) into the L1A
software. The second is a partial check of the correction in Level 1B
software through analysis of the spectral phase for all detectors in a
focal plane.

This latter requirement is implemented in software with the TES
performance monitoring algorithms. These algorithms monitor the shear
using the spectral phase of the OBRCS radiance measurements (Sec 6.3) .
In addition, the time variations of the spectral signal magnitude, which
is calculated in regions sensitive to ice build-up, is also monitored
(Sec 6.4). Changes in shear and ice-build up both affect signal levels
and therefore the signal-to-noise. If the signal levels drop below a
predefined threshold, then TES personnel will be alerted and an
appropriate action will be pursued, e.g., warming the detectors to
remove ice.

There are several results that have no appreciable impact on Level 1B
processing or algorithms. The analysis of the stability of channeling
from the ST data set suggest that channeling will have no impact on
system performance or L1B processing (Sec 6.5). The decision to use a
single gain setting obviates the need for gain calibration (Sec 6.7).
The lack of significant non-linearities in the detector response removes
the need for a correction algorithm (Sec 6.12). Other results, such as
the FFT sizes and spectral frequency ranges (Sec. 6.2) were needed so
that default values used in the algorithm could be replaced with
measured values.

The impact of the velocity variations of the interferometer arm on the
spectra through the electrical filters has lead to an operational
decision to turn those filters off. If a correction can be developed for
the electrical filters, then a significant software development will be
required in Level 1B to implement that algorithm. In the absence of
correction algorithm, the NESR has been increased by over a factor of
two (Sec 6.6). However, no special processing is required in that case.

There are several results of the calibration analysis that affect both
Level 1B and Level 2 processing. Our standard method for calculating the
NESR has been validated with the ST data (Sec 6.14). However, the
increase in NESR from the predicted values requires a reevaluation of
the "microwindows" selected for the Level 2 retrievals. In addition, the
measured spectral spikes must also be included in the Level 2
microwindow selection process (Sec 6.8). The time-variability of the
spike frequency locations will be monitored in Level 1B. The calculation
of the field-of-view (FOV) will be used to reevaluate microwindows for
TES limb retrievals (Sec 6.16). In addition, the FOV is used to
calculate the theoretical ILS function, which is used by both L1B and
L2.

From the analysis of the ILS, no correction for the ILS will be applied
in Level 1B processing or modeling of the ILS in Level 2 for nadir
spectra. In the case of the limb mode and higher frequencies (\>1500
cm\^-1) (where the effect of the ILS is more important), additional
study will be required.

Ultimately the impact of the system test characterization on the
expected accuracy and precision of TES retrievals must be assessed. The
One Day Test (ODT), which is currently in progress, is using the
expected NESR from the system tests to assess retrieval precision and
vertical resolution. The ODT simulates spectra, with added noise, and
performs retrievals for nadir and limb target scenes over 16 orbits.
Further retrieval studies will need to be performed that include the
effects of calibration systematic errors, such as radiometric accuracy.

Table 7.0 Impacts of System Test Results on Data Processing Subsystems

+-----------------------+--------+-----+-----+-----+------------------+
| System Test Result    | S      | L1A | L1B | L2  | New              |
|                       | ection |     |     |     |                  |
|                       |        |     |     |     | Requirement      |
|                       | Number |     |     |     |                  |
+-----------------------+--------+-----+-----+-----+------------------+
| Channel Shifts        | 6.1    | X   | X   |     | Yes              |
+-----------------------+--------+-----+-----+-----+------------------+
| FFT Size/Filter       | 6.2    |     | X   |     | No               |
| Ranges                |        |     |     |     |                  |
+-----------------------+--------+-----+-----+-----+------------------+
| Shear/Co-Boresight    | 6.3    |     | X   | X   | Yes              |
+-----------------------+--------+-----+-----+-----+------------------+
| Time Trend of Signal  | 6.4    |     | X   |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Channeling Stability  | 6.5    |     |     |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Electrical Filter     | 6.6    |     |     |     | Yes, if          |
|                       |        |     |     |     | correction is    |
|                       |        |     |     |     | implemented      |
+-----------------------+--------+-----+-----+-----+------------------+
| Gain                  | 6.7    |     | X   |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Spectral Spikes       | 6.8    |     | X   |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Cal. Pointing Angles  | 6.9    | X   |     |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| PCS Calibration       | 6.10   | X   |     |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Non-Linearity         | 6.11   |     |     |     | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Radiometric           | 6.12   |     | X   | X   | No               |
| Calibration           |        |     |     |     |                  |
+-----------------------+--------+-----+-----+-----+------------------+
| NESR                  | 6.13   |     | X   | X   | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Frequency Calibration | 6.14   |     | X   | X   | No               |
+-----------------------+--------+-----+-----+-----+------------------+
| Instr. Line Shape     | 6.15   |     | X   | X   | No               |
| (ILS)                 |        |     |     |     |                  |
+-----------------------+--------+-----+-----+-----+------------------+
| FOV Response          | 6.16   |     | X   | X   | No               |
+-----------------------+--------+-----+-----+-----+------------------+

[^1]: A linear term in the phase is simply a measure of the shift of the
    data point closest to the peak from the true peak itself (and,
    indeed, is the basis for a powerful method of determining the true
    peak location)