r25504 - /trunk/docs/latex/dispersion.tex -- September 01, 2014 - 16:33

Author: bugman
Date: Mon Sep  1 16:33:30 2014
New Revision: 25504

URL: http://svn.gna.org/viewcvs/relax?rev=25504&view=rev
Log:
Added a derivation of the R2eff/R1rho error estimate for the two-point 
measurement to the manual.

This is from 
http://thread.gmane.org/gmane.science.nmr.relax.devel/6929/focus=6993 and is 
for the
rate uncertainty of a 2-parameter exponential curve when only two data points 
have been collected.
The derivation has been added to the dispersion chapter of the manual.


Modified:
    trunk/docs/latex/dispersion.tex

Modified: trunk/docs/latex/dispersion.tex
URL: 
http://svn.gna.org/viewcvs/relax/trunk/docs/latex/dispersion.tex?rev=25504&r1=25503&r2=25504&view=diff
==============================================================================
--- trunk/docs/latex/dispersion.tex     (original)
+++ trunk/docs/latex/dispersion.tex     Mon Sep  1 16:33:30 2014
@@ -239,7 +239,7 @@
 \subsubsection{Fixed relaxation period experiments}
 
 For the fixed relaxation time period CPMG-type experiments, the 
$\Rtwoeff$/$\Ronerho$ values are determined by direct calculation using the 
formula
-\begin{equation}
+\begin{equation} \label{eq: R2eff two-point}
     \Rtwoeff(\nucpmg) = - \frac{1}{\Trelax} \cdot \ln \left( 
\frac{I_1(\nucpmg)}{I_0} \right) .
 \end{equation}
 
@@ -253,6 +253,44 @@
 \begin{equation} \label{eq: dispersion error}
     \sigma_{\Rtwo} = \frac{1}{\Trelax} \sqrt{ \left( 
\frac{\sigma_{I_1}}{I_1(\omega_1)} \right)^2  +  \left( 
\frac{\sigma_{I_0}}{I_0} \right)^2 } .
 \end{equation}
+
+The derivation of this is simple enough.
+Rearranging~\ref{eq: R2eff two-point},
+\begin{equation}
+    \Rtwo \cdot \Trelax = -\ln \left( \frac{I_1}{I_0} \right) .
+\end{equation}
+
+Using the rule
+\begin{equation}
+    \ln\left(\frac{X}{Y}\right) = \ln(X) - \ln(Y),
+\end{equation}
+
+where X and Y are normally distributed variables, then
+\begin{equation}
+    \Rtwo \cdot \Trelax = \ln(I_0) - \ln(I_1) ,
+\end{equation}
+
+and
+\begin{equation}
+    \Rtwo = - \frac{1}{\Trelax} \cdot \left( \ln(I_0) - \ln(I_1) \right) ,
+\end{equation}
+
+Using the estimate from 
\url{https://en.wikipedia.org/wiki/Propagation_of_uncertainty} that for
+\begin{equation}
+    f = a \ln(A),
+\end{equation}
+
+the variance of $f$ is
+\begin{equation}
+    \sigma_f^2 = \left( a * \frac{\sigma_A}{A} \right)^2 ,
+\end{equation}
+
+then the $\Rtwo$ variance is
+\begin{equation}
+    \sigma_{\Rtwo}^2 = \left( \frac{1}{\Trelax} \cdot \frac{\sigma_I0}{I0} 
\right)^2 + \left( \frac{1}{\Trelax} \cdot \frac{\sigma_I1}{I1} \right)^2.
+\end{equation}
+
+Rearranging gives~\ref{eq: dispersion error}.
 
 In a number of publications, the error formula from \citet{IshimaTorchia05} 
has been used.
 This is the collapse of Equation~\ref{eq: dispersion error} by setting 
$\sigma_{I_0}$ to zero: