Added text to description by Dave Humm. Closes #3860 (#4107)

* Added text to description by Dave Humm. Closes #3860

* Forgot to add to the change log file

* Update after review

* Updated after review

CHANGELOG.md

@@ -29,6 +29,7 @@ update the Unreleased link so that it compares against the latest release tag.
 -->
 
 ## [Unreleased]
+ - Added documentation to lronaccal and lrowaccal to describe why there are negative DNs in I/F calibrated images.
 
 
 - Fixed so required files are reported instead of continuing without them. [#4038](https://github.com/USGS-Astrogeology/ISIS3/issues/4038)

isis/src/lro/apps/lronaccal/lronaccal.xml

@@ -8,44 +8,89 @@
 
   <description>
     <p>
-    lronaccal performs radiometric corrections to images acquired by the Narrow Angle
-    Camera aboard the Lunar Reconnaissance Orbiter spacecraft.  The LRO NAC camera
-    will make observations simulteously with the HiRise camera.
+      lronaccal performs radiometric corrections to images acquired by the Narrow Angle
+      Camera aboard the Lunar Reconnaissance Orbiter spacecraft.
     </p>
 
     <p>
-    The LRO NAC detector has a total of 5064 pixels, divided among an A
-    channel and a B channel.  The pixels alternate between the two channels:
-    ABABABAB, etc.  Images from LROC NAC may or may not include all pixels in the
-    acquired image.  There are special summing modes that are utilized
-    on-board the spacecraft to average detector pixels to combine them into a
-    single output pixel value.  The value of the ISIS label keyord,
-    SpatialSumming, indicates the number of samples that were summed and
-    averaged to result in the pixel values stored in the file.  Note that this
-    will reduce the number of samples in the output image by a factor of at
-    most the SpatialSumming mode value.
+      The LRO NAC detector has a total of 5064 pixels, divided among an A
+      channel and a B channel.  The pixels alternate between the two channels:
+      ABABABAB, etc.  Images from LROC NAC may or may not include all pixels in the
+      acquired image.  There are special summing modes that are utilized
+      on-board the spacecraft to average detector pixels to combine them into a
+      single output pixel value.  The value of the ISIS label keyord,
+      SpatialSumming, indicates the number of samples that were summed and
+      averaged to result in the pixel values stored in the file.  Note that this
+      will reduce the number of samples in the output image by a factor of at
+      most the SpatialSumming mode value.
     </p>
     <p>
-    The LROC NAC camera has the ability to acquire images of differing sizes in
-    both line and sample.  The starting hardware detector pixel for the
-    acquired image is specified by the ISIS label keyword, SampleFirstPixel.
-    The first pixel in the detector is indicated by a value of 0.
+      The LROC NAC camera has the ability to acquire images of differing sizes in
+      both line and sample.  The starting hardware detector pixel for the
+      acquired image is specified by the ISIS label keyword, SampleFirstPixel.
+      The first pixel in the detector is indicated by a value of 0.
     </p>
 
     <p>
-    Dark current pixels are taken for each line from the masked pixels
-    that lie along each edge of the image.
+      Dark current pixels are taken for each line from the masked pixels
+      that lie along each edge of the image.
     </p>
 
     <p>
-    If SpatialSumming is 1 the dark current pixels are averaged together then
-    this average is subtracted from all image pixels.  If SpatialSumming is 2,
-    the dark current pixels for the A and B channel are averaged separately,
-    then the A channel dark average is subtracted from the A channel image
-    pixels and the B channel dark average is subtracted from the B channel
-    image pixels.
+      If SpatialSumming is 1 the dark current pixels are averaged together then
+      this average is subtracted from all image pixels.  If SpatialSumming is 2,
+      the dark current pixels for the A and B channel are averaged separately,
+      then the A channel dark average is subtracted from the A channel image
+      pixels and the B channel dark average is subtracted from the B channel
+      image pixels.
     </p>
 
+    <p>
+      The DN level in an uncalibrated image is the sum of the true signal from the scene, 
+      the bias, the dark current, and random noise in all 3 components. The random noise in
+      the true signal and dark current is called shot noise and the random noise in the bias
+      is called read noise. The true signal, bias, and dark current are defined as mean
+      values so that if the random noise were averaged down to insignificance by taking a 
+      very large number of images and averaging them, the resulting image would be the true 
+      scene, bias, and dark current with no systematic error. That implies the statistical 
+      distribution of the random noise has an average of zero, and therefore the random noise
+      has both positive and negative values, except for the trivial case of zero random noise.
+    </p>
+
+    <p>
+      The calibration equation is:
+      <pre>  reportedDN = ObservedDN - MeanBias - DarkCurrent </pre>
+
+        Where:
+        <pre>
+   ObservedDN = TrueDN + E
+   E is a randomly sampled value from (mu, sigma^2) and mu=0
+   TrueDN is the signal that would be reported in an idealized case of an instrument with zero noise.</pre>
+     </p>
+
+    <p>
+      Let's look at the case of a calibrated image for which the true signal
+      is zero, a dark image. In calibration the mean bias and dark current are
+      subtracted. The random noise term is then randomly sampled from a known
+      distribution with a mean of zero. Since the distribution has a mean of
+      zero, values for the random noise can be positive or negative.
+      Therefore, the addition of random noise to a pixel with true signal near
+      zero can result in negative DN values.
+    </p>
+
+    <p>
+      Negative reported DNs are possible when E < -1 * TrueDN. These are
+      pixels in a very dark image that happen to have a strongly negative
+      random noise value.
+    </p>
+
+    <p>
+      Note: ObservedDN and TrueDN both must be greater than or equal to zero.
+      For ObservedDN, it's because the hardware is not able to report negative
+      DN values . For TrueDN, it's because radiance and reflectivity cannot be
+      negative. The dimmest target is one that is completely dark, and for
+      that target TrueDN = 0.
+    </p>
   </description>
 
   <history>

isis/src/lro/apps/lrowaccal/lrowaccal.xml

@@ -15,7 +15,7 @@
     </p>
 
     <p>
-      Corrections are  applied in the following order: Dark, Flat-field, Radiometric, Special pixel mask, and Temperature. 
+      Corrections are applied in the following order: Dark, Flat-field, Radiometric, Special pixel mask, and Temperature. 
     </p>
 
     <h3>Dark Correction:</h3>
@@ -149,6 +149,54 @@
       </pre>
       If TEMPRATUREFILE is not set, the constants are loaded from $lro/calibration/WAC_TempratureConstants.????.pvl
     </p>
+
+    <p>
+      The DN level in an uncalibrated image is the sum of the true signal from the scene, 
+      the bias, the dark current, and random noise in all 3 components. The random noise in
+      the true signal and dark current is called shot noise and the random noise in the bias
+      is called read noise. The true signal, bias, and dark current are defined as mean
+      values so that if the random noise were averaged down to insignificance by taking a 
+      very large number of images and averaging them, the resulting image would be the true 
+      scene, bias, and dark current with no systematic error. That implies the statistical 
+      distribution of the random noise has an average of zero, and therefore the random noise
+      has both positive and negative values, except for the trivial case of zero random noise.
+    </p>
+
+    <p>
+      The calibration equation is:
+      <pre>  reportedDN = ObservedDN - MeanBias - DarkCurrent </pre>
+
+        Where:
+        <pre>
+   ObservedDN = TrueDN + E
+   E is a randomly sampled value from (mu, sigma^2) and mu=0
+   TrueDN is the signal that would be reported in an idealized case of an instrument with zero noise.</pre>
+     </p>
+
+    <p>
+      Let's look at the case of a calibrated image for which the true signal
+      is zero, a dark image. In calibration the mean bias and dark current are
+      subtracted. The random noise term is then randomly sampled from a known
+      distribution with a mean of zero. Since the distribution has a mean of
+      zero, values for the random noise can be positive or negative.
+      Therefore, the addition of random noise to a pixel with true signal near
+      zero can result in negative DN values.
+    </p>
+
+    <p>
+      Negative reported DNs are possible when E < -1 * TrueDN. These are
+      pixels in a very dark image that happen to have a strongly negative
+      random noise value.
+    </p>
+
+    <p>
+      Note: ObservedDN and TrueDN both must be greater than or equal to zero.
+      For ObservedDN, it's because the hardware is not able to report negative
+      DN values . For TrueDN, it's because radiance and reflectivity cannot be
+      negative. The dimmest target is one that is completely dark, and for
+      that target TrueDN = 0.
+    </p>
+
   </description>
 
   <category>