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\begin{document}

%xx Prefatory material

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\def\reportdate{30 June 2011}
\def\reportseq{3rd}

\begin{center} \bf

{\rm DRAFT Addendum to \ph{II} Final Report for the SBIR project on}

{\huge An Ensemble Kalman Filtering Approach \\
for Regional Ocean Data Assimilation}

~

Contract N00039-08-C-0017 \\
{\rm Amount: \$747,336;
awarded by Space and Naval Warfare Systems Command}

~

{\rm Prepared by}

Rui M. Ponte, Principal Investigator \\
{\rm Email: ponte@aer.com;
Phone: 781.761.2231}

~

{\rm \small Distribution Statement B: Distribution authorized to US
Government Agencies only: contains proprietary information, \reportdate.
Other requests for this document shall be referred to Space and Naval
Warfare Systems Command, 4301 Pacific Highway, OT1 1480D, San Diego,
CA 92110-3127.}

~

{\rm Submitted to}

Space and Naval Warfare Systems Command \\
{\rm 4301 Pacific Highway, OT1 1480D \\
San Diego, CA  92110-3127}

Attn: Edward Mozley, PMW 160 \\
{\rm Email: edward.mozley@navy.mil;
Phone: 858.537.0230}

~

{\rm Submitted by}

 Atmospheric and Environmental Research, Inc.\ \\
{\rm 131 Hartwell Avenue, Lexington, MA 02421-3126 \\
 Phone: 781.761.2288;
 Fax: 781.761.2299}

~

\reportdate

\end{center}

\rm

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%xx summary report

\pagenumbering{arabic}

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\markboth{Distribution authorized to US Government agencies only:
Contains proprietary information.}{Distribution authorized to US Government agencies only:
Contains proprietary information.}

\noindent {\bf DRAFT Addendum to the PHASE II Final Report (period ending \reportdate)}

\noindent {\bf CONTRACT NO.~N00039-08-C-0017}

\vskip 0.5truecm

\noindent {\bf Administrative Narrative}

 The Phase II Option was used to extend the Phase II work by transitioning
LETKF-NCOM to the group 7320 computers at Stennis.
 The goals of the Phase II Option are stated in the Phase II Final report.

A Phase II Option kick-off teleconference was held on July 21, 2010.
 Ed Mosely conveined the meeting that included Clark Rowley, Pete Spence,
Ross Hoffman and Mark Leidner.


\noindent {\bf Technical Progress}

\noindent {\bf T--I.~~Introduction}

Link to prior Phase II work, and provide the roadmap for the current report.

\noindent {\bf T--II.~~LETKF Deployment at Stennis Space Center}

Discuss the process, development required (i.e., parallelize NCOM i/o) and current status.

\noindent {\bf T--III.~~Single Observation Testing}

 Assimilating a single observation yields the finger print of an
analysis system.
 In the case of LETKF, such a finger print is dependent on the
background flow.
 The first four figures of this report present results from single
observation tests.

 Single observation tests were conducted to quantify the impact that
an observation can have on the analysis under controlled conditions.
 These tests were also conducted to validate the assimilation technique
for simple-to-understand cases.
 The single observation tests were also used to tune the various
localization parameters to values appropriate for ocean data assimilation and 
in particular, the Okinawa Trough NCOM ocean modeling domain in
July 2007.

 The results presented here use a single observation near the center
of the NCOM model domain, approximately 27 N latitude and 126 E longitude.
 \figr{1ob-whole-domain-T} shows sea surface temperature for the whole 
modeling domain, including ensemble mean background (top panel), mean analysis
from assimilating the observation (middle panel), and the anaysis 
increment (analysis minus background; bottom panel).
 The horizontal red line in the bottom panel is a cross section locator
at 27 N latitiude, centered west-to-east and passing through the observation 
location (vertical cross section is presented in \figr{xsec-incr}).

 The observation is a single SST measurement, set to be 0.5 C warmer 
than the background, with an assigned observation error standard
deviation of 0.5 C.
 0.5 C was chosen since it is a larger but realistic difference between
model and observation.
 As a consequence, the color scale on the the analysis increment 
panel is -0.5 to +0.5 C to allow for the possibility that the analysis
matches the observed value exactly at the observation location
(i.e., an increment of +0.5 C).

 The ensemble used for the background began from the same 36-member
climate ensemble described above in Section T--II.
 We found that without a few days of cycling data assimiation, the
36-member ensemble retained unwanted characteristics, peculiar to our
choice of the starting climate ensemble.
 After a few days of cycling data assimilation, we found that 
artifacts in the starting climate ensemble had been washed away
through the evolution of the model states during cycling data
assimilation.
 At time of these single observation tests, a set of model states, randomly scattered about the mean model
state that has been reconciled to a time series of observations 
through cycling data assimilation, populates the ensemble.

 The observation location was also chosen because the background ensemble standard
deviation is a bit larger than the average location, 0.5 C, whereas
typical values for other observation locations are 0.1-0.3 C.  
 A larger than normal ensemble standard deviation for a particular 
location indicates larger uncertainty in the background field.

 Given that the background uncertainty {\emph and} and observation
uncertainty are 0.5 C, we would expect the analysis increment at the 
observation location to be an equal compromise between the background
value and the observed value -- that is, an increment of 0.25 C.
 Magnified plots, showing just on the analysis patch around
the observation (\figr{1ob-zoom-T}), demonstrate that the increment is nearly
exactly 0.25 C at the observation point (see bottom panel of 
\figr{1ob-zoom-T}).
 This increment produces a modest change to the background (compare
top and middle panels of \figr{1ob-zoom-T}).
 But the pattern of the increment reveals the flow dependence of the
background error uncertainty.
 Because this observation falls on a relatively sharp gradient
in sea surface temperature whose position is uncertain among the ensemble
members, the influence of this observation follows the temperature
gradient both to the west and even further to the east of the
observation location.
 In fact, the flow dependent uncertainty follows the gradient far from
the observation location, since the position of the north and east edges of 
the adjacent pool of warm water is uncertain.
 This can be seen in the bottom panel of \figr{T-incr} where contours of
the analysis increment (black) are overlaid on top of the mean background
sea surface temperature.
 In this visualization, it is easy to see how the increments follow the 
temperature gradient on the north and to the east sides of the warm pool.

 \figr{xsec-incr} shows a vertical cross section of potential temperature
at along 27 N latitude that crosses through the observation point.
 Note that vertical localization restricts the influence to the top
18 meters of the ocean, an approximate mixed layer depth for the 
North Pacific diagnosed from a sampling of nine NCOM model states during
early July 2007 (not shown).

\noindent {\bf T--IV.~~Cycling Assimilation Results}

Present the analysis of ensemble variance and o-a/o-b statistics for  a 1-week experiment.
(Hopefully, this section can be updated for the final submission, 6/30, with results from longer experiments, or experiments that have more refined/appropriate localization and inflation settings.)

\noindent {\bf T--V.~~Conclusions}

Stress the unique value of flow-dependence and the excellenbt scalability on many computers.

\noindent {\bf T--VI.~~Acknowlegments}

Computer support people @ Stennis (Daniel Goolsby, Tim Campbell); Section 7320 people: Clark Rowley, Pete Spence.

\noindent {\bf T--VII.~~References}

\noindent {\bf Current Status and Future Activities}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\fitem{1ob-whole-domain-T}
{LETKF single observation assimilation test of one sea surface temperature (SST) observation.}
{{\sffamily\upshape\bfseries
 {\setlength{\tabcolsep}{1pt}
  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Single Observation Test, Whole Domain} \\ \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0736992_sstbackg_000_1obT.pdf}}}
         \\ \hline \hline
  \ylab{SST Analysis (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0736992_sstanaly_000_1obT.pdf}}}
         \\ \hline \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0736992_ptem0001_000_1obT.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{1ob-zoom-T}
{As in \figr{1ob-whole-domain-T}, but for the analysis patch local to the observation.
The observation location is marked with either a white or black X.}
{{\sffamily\upshape\bfseries
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  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Single Observation Test, Analysis Patch} \\ \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0736992_sstbackg_zoom_000_1obT.pdf}}}
         \\ \hline \hline
  \ylab{SST Analysis (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0736992_sstanaly_zoom_000_1obT.pdf}}}
         \\ \hline \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0736992_ptem0001_zoom_000_1obT-nolocator.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{1ob-T-incr}
{Flow-dependent background uncertainty to a single sea surface temperature (SST) observation.
 The top panel is the same as \figr{1ob-zoom-T} but with the addition of a 
 cross section locator (red; see \figr{1ob-xsec-T-incr} for cross section.) 
 The bottom panel shows the analysis increments (black contours) on top of the background
SST (colored by temperature) to show the relation between the flow structure of the 
background and the uncertainty of the ensemble.
To highlight the pattern of increments (bottom panel), the following geometric
list of temperature contours are used above and below zero:
 $0.02, 0.07, 0.16, 0.29, 0.46, 0.67, 0.92$.}
{{\sffamily\upshape\bfseries
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  \tbox{Single Observation Flow Sensitivity} \\ \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.80]
         {inc_0736992_ptem0001_zoom_000_1obT-locator.pdf}}}
         \\ \hline \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.80]
         {plt_0736992_sstbginc_zoom_000_1obT.pdf}}}
         \\ \hline
  \end{tabular}}}}

\clearpage

\fitem{1ob-xsec-T-incr}
{Vertical cross section (0-150 m) of LETKF analysis increments of potential temperature.
See \figr{1ob-zoom-T} for cross section location.}
{{\sffamily\upshape\bfseries
 {\setlength{\tabcolsep}{1pt}
  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{East-West Vertical Cross Section of a Single Observation} \\ \hline
  \ylab{Potential Tempeature (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 195 575 585, clip=true, scale=0.85]
         {inc_0736992_ptemXsec_000_1obT.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\fitem{horiz-localization}
{Progression of horizontal localization in LETKF demonstrated by
assimilation tests of a sea surface temperature (SST)
observation.}
{{\sffamily\upshape\bfseries
 {\setlength{\tabcolsep}{1pt}
  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Single Observation Localization Tests} \\ \hline
  \ylab{1500 km patch/no taper} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0736992_ptem0001_000_1obB.pdf}}}
         \\ \hline \hline
  \ylab{250 km patch/no taper} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0736992_ptem0001_zoom_000_1obN.pdf}}}
         \\ \hline \hline
  \ylab{250 km patch/111 km taper} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0736992_ptem0001_zoom_000_1obT-nolocator.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{whole-domain-T}
{LETKF sea surface temperature (SST) analysis
using all available observations, valid 00Z July 3, 2007.
Observation locations are marked with black dots (N=1164).}
{{\sffamily\upshape\bfseries
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  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Sea Surface Temperature} \\ \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0735984_sstbackg_000_1wkA.pdf}}}
         \\ \hline \hline
  \ylab{SST Analysis (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0735984_sstanaly_000_1wkA.pdf}}}
         \\ \hline \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0735984_ptem0001_000_1wkA.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{zoom-T}
{As in \figr{whole-domain-T}, but for the subregion identified by
 the red boxes in \figr{whole-domain-T}.} 
{{\sffamily\upshape\bfseries
 {\setlength{\tabcolsep}{1pt}
  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Test} \\ \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0735984_sstbackg_zoom_000_1wkA.pdf}}}
         \\ \hline \hline
  \ylab{SST Analysis (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {plt_0735984_sstanaly_zoom_000_1wkA.pdf}}}
         \\ \hline \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.52]
         {inc_0735984_ptem0001_zoom_000_1wkA-nolocator.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{T-incr}
{As in \figr{1ob-T-incr} but for an analysis using all available SST observations.}
{{\sffamily\upshape\bfseries
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  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Flow Sensitivity to Many Observations} \\ \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.80]
         {inc_0735984_ptem0001_zoom_000_1wkA-locator.pdf}}}
         \\ \hline \hline
  \ylab{SST Background (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.80]
         {plt_0735984_sstbginc_zoom_000_1wkA.pdf}}}
         \\ \hline
  \end{tabular}}}}

\fitem{xsec-T-incr}
{Vertical cross section of temperature increments at 22.75 N latitude.
See the bottom panel of \figr{T-incr} for the cross section location.}
{{\sffamily\upshape\bfseries
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  \begin{tabular}{|c|c|} \hline
  \ylab{\makebox[10mm]{\rule[-4mm]{0mm}{10mm}}} &
  \tbox{Vertical west-east Cross Section} \\ \hline
  \ylab{SST Increments (C)} &
  \tbox{\raisebox{0pt}{\includegraphics[bb=50 215 575 585, clip=true, scale=0.80]
         {inc_0735984_ptemXsec_000_1wkA.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\fitem{cycling-stats}
{Ensemble verification/characterization for SST from a 1-week cycling data assimilation test (28, 6-hour analysis cycles).}
{{\sffamily\upshape\bfseries
 {\setlength{\tabcolsep}{1pt}
  \begin{tabular}{|c|} \hline
  \tbox{Time Series of SST Ensemble Verification statistics} \\ \hline
  \tbox{\raisebox{0pt}{\includegraphics[bb=100 365 655 540, clip=true, scale=0.75]
         {1wkA-stats1-1-1.pdf}}}
         \\ \hline \hline
  \tbox{\raisebox{0pt}{\includegraphics[bb=100 365 655 540, clip=true, scale=0.75]
         {1wkA-stats2-1-1.pdf}}}
         \\ \hline \hline
  \tbox{\raisebox{0pt}{\includegraphics[bb=100 365 655 540, clip=true, scale=0.75]
         {1wkA-stats3-1-1.pdf}}}
         \\ \hline 
  \end{tabular}}}}

\clearpage

\end{list}

\end{document}
