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%                    N E S P I / B I L I N . T E X 
%                    doc: Tue Aug 31 09:44:02 1999
%                    dlm: Wed Jun  7 23:34:49 2000
%                    (c) 1999 A.M. Thurnherr
%                    uE-Info: 141 19 NIL 0 0 72 3 2 4 ofnI
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% LINEAR CORRELATION => BILINEAR MODEL & METHOD

The results from the plume-bracketing method applied to the 1997 CTD
data indicate that the isopycnal hydrographic anomalies associated with
the \R particle plume are linearly correlated with the corresponding
nephelometry anomalies within individual profiles. Keeping the
assumption of a linear background $T/\rho$ relationship, the
temperature profile within a particle peak can therefore be written as
 %
\begin{equation}
T(z) = A + B\rho(z) + C\Delta n(z),      \label{eqn:bilin}
\end{equation}
 %
\ls{bilin_C}where the nephelometry anomalies are denoted by $\Delta
n$. \rExpr{bilin} defines a bilinear model which can be fitted to
simultaneous profiles of temperature, density, and nephelometry
anomalies. The regression coefficients $A$\ls{bilin_A} and
$B$\ls{bilin_B} define the background $T/\rho$ relationship within the
particle layer (similar to the linear interpolation between \sfT and
\sfB of the plume-bracketing method), and the isopycnal temperature
anomalies are given by
 %
\begin{equation}
	\Delta_\rho T(z) = C \Delta n(z) = T(z) - (A + B\rho(z)).
		\label{eqn:spiceprof}
\end{equation}
 %
To fit \rexpr{bilin} to a particle peak, the $\chi^2$\ls{chisq} merit
function
 %
\begin{equation}
\chi^2 = \sum_{k=1}^N 
	\left( \frac{T_k - (A + B\rho_k + C\Delta n_k)}{\sigma} \right)^2
		\label{eqn:chi2}
\end{equation}
 %
(where $T_k$,\ls{T_k} $\rho_k$\ls{rho_k} and $\Delta n_k$\ls{Delta_n_k}
are the individual temperature, density, and nephelometry anomaly
measurements) is minimized using a generalized least-squares algorithm
\citeEG{books/press+93}\REM{chapter 15.4}. This method will be called the
\emph{bilinear method}.

% ERROR PDF

\begin{figure}[t]

\placeFig{NESPI/RFCbilinerrs}

\caption[Residuals of bilinear method]{Distribution of the residuals
resulting from the bilinear method applied to the \F CTD data set; 
histogram bin size is \protect\degC{10^{-3}} (twice the temperature
sensor resolution); the curve shows the corresponding normal
distribution with bullets plotted at multiple standard deviations
(\protect\RMS{\sigma=3.1}) from the mean.} 

\labelFig{PDF}

\end{figure}

Minimizing \rexpr{chi2} yields the same regression coefficients
regardless of the (assumed constant) value of $\sigma$ which quantifies
the combined measurement uncertainties (expected RMS error). If the
residuals are normally distributed $\sigma$ is the corresponding
standard deviation which can be used to calculate confidence intervals
for the regression coefficients. To determine the distribution of the
residuals, a frequency histogram was derived from the \F CTD data set
\callout{\rfig{PDF}}. The distribution is approximately normal with
standard deviation \RMS{\sigma=3.1}.\REM{avg -sf res *.bilinfit} (The
residuals of the plume-bracketing method are similarly distributed with
standard deviation \RMS{\sigma=3.5}.\REM{avg -sf res *.pbmfit})

% GOODNESS-OF-FIT

\REM{A further use of the measurement uncertainty $\sigma$ is that it
allows the goodness-of-fit to be estimated independently (e.g.\
\citeNP{books/press+93} chapter 15.1). The problem here is that the RMS
error varies by more than a factor two between the individual profiles.
This may be due to the fact that the measurement uncertainty is a
combination of nephelometer, temperature, pressure, and conductivity
uncertainties. It appears likely, for example, that the temperature
measurement error in density space depends on the stratification.
Therefore, the mean $\sigma$ value is \emph{not} appropriate to assess
the individual goodness-of-fit. Strictly speaking, this is also true
for the calculation of confidence intervals (e.g.\ \rfigNP{variability}
below). One possible way around this is to take \rfigNP{PDF} to
indicate that the errors of individual profiles are normally
distributed as well (they are much less normal, however, at least in
part due to the smaller size of the data sets) but then take the RMS
error of each profile to assess the goodness-of-fit. Doing that results
in, if I remember correctly, one or two rejected profiles and one or
two additional rejected peaks.}

% METHOD COMPARISON

In profiles where the spice and the nephelometry anomalies are linearly
correlated and where the background $T/\rho$ relationships within the
particle layers are linear the bilinear and the plume-bracketing
methods are equivalent. This is confirmed in the case of the 11 \F CTD
profiles to which the plume-bracketing method had been applied
\rsec{NESPI-bracket}. The weighted mean ($\pm1\sigma$) spice vs.\
nephel correlation coefficient of the bilinear method is
$r=-0.87(\pm0.03)$ and the respective 68.3\% confidence intervals of
the individual regression-slope estimates derived with the two methods
overlap.

% DEPTH RANGE SELECTION

Strictly speaking the two new methods are only equivalent if they are
applied to the same subset of a given profile. Because the background
hydrographic properties of the bilinear method are calculated from the
entire plume peak the brackets do not have a ``thickness'' and can be
selected in depth or in pressure space. The results from the 1997 CTD
data set (see \rsecNP{NESPI-R} below) indicate that the background
$T/S$ slope was slightly different above and below the hydrothermal
plume, indicating that the particle peaks should be tightly bracketed.
To avoid inconsistent application of the method the depth of the
particle peak limits were rounded to the next \m{25} outside the peaks.

% RELAXATION OF REQUIREMENT

In contrast to the plume-bracketing method, the bilinear method can be
applied to truncated profiles \rsec{NESPI-bracket}. Out of the 16
truncated \F CTD profiles only 2 have nephelometry peak values
\V{>\!0.3} (both are \V{<\!0.7}). The resulting weighted mean spice
vs.\ nephel correlation coefficient is $r=-0.75$ and the respective
slopes and RMS errors are consistent with the remainder of the data set
(see \rfigNP{bilinvar}).

% CROSS-CORRELATION COEFFICIENTS

\rExpr{bilin} cannot be fitted unambiguously to profiles characterized
by (nearly) linear relationships between $\rho(z)$ and $\Delta n(z)$
because the data do not constrain both regression coefficients $B$ and
$C$ independently in this case. In practice this can happen when the
bilinear method is applied to the upper half of a particle peak, i.e.\
in truncated profiles which do not extend significantly below the
particle maxima. To assess the independence of the estimates for $B$
and $C$, the cross-correlation coefficient $r_{BC}$\ls{r_BC} can be
inspected. In the case of the 1997 CTD profiles $-0.4<r_{BC}<0.1$,
while $r_{BC}<-0.8$ when the bilinear method is applied to the same
profiles artificially truncated at the nephelometry maxima. 

% TEMPERATURE DISCONTINUITY

Minimizing \rexpr{chi2} does not constrain the temperature anomalies to
vanish at the top and at the bottom of the particle plume. Non-zero
values can result if one of the linearity assumptions is violated or if
the nephelometry anomaly profile is offset by a constant amount, i.e.\
non-zero values indicate failure of the bilinear method. This is used
in \rsecNP{NESPI-Pacific} to determine the appropriate nephelometry
origin of a profile where the choice is ambiguous.

