Changeset 5623 in ntrip for trunk/BNC


Ignore:
Timestamp:
Jan 22, 2014, 4:15:39 PM (8 years ago)
Author:
mervart
Message:
 
Location:
trunk/BNC/txt
Files:
10 added
1 edited

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  • trunk/BNC/txt/frankfurt.tex

    r5622 r5623  
    509509
    510510\begin{frame}
     511  \frametitle{Combination using Kalman filtering}
     512  The combination is performed in two steps
     513  \begin{itemize}
     514  \item[1.] The satellite clock corrections that refer to different broadcast
     515    messages (different IODs) are modified in such a way that they all refer
     516    to common broadcast clock value (common IOD is that of the selected
     517    ``master'' analysis center).
     518  \item[2.] The corrections are used as pseudo-observations for Kalman filter
     519    using the following model (observation equation):
     520    \begin{displaymath}
     521    c_a^s = c^s + o_a + o_a^s
     522    \end{displaymath}
     523    where
     524    \begin{tabbing}
     525    $c_a^s$ ~~ \= is the clock correction for satellite s estimated by \\
     526               \> the analysis center a, \\
     527    $c^s$      \> is the resulting (combined) clock correction for
     528                  satellite s, \\
     529    $o_a$      \> is the AC-specific offset
     530                  (common for all satellites), and \\
     531    $o_a^s$      \> is the satellite and AC-specific offset.
     532    \end{tabbing}
     533  \end{itemize}
     534  The three types of unknown parameters $c^s$, $o_a$, $o_a^s$ differ in their
     535  stochastic properties: the parameters $c^s$ and $o_a$ are considered to be
     536  epoch-specific while the satellite and AC-specific offset $o_a^s$ is assumed
     537  to be a static parameter.
     538\end{frame}
     539
     540%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     541
     542\begin{frame}
    511543\frametitle{PPP -- Server-Side}
    512544  \begin{center}
     
    562594%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    563595
    564 \section{PPP AR}
    565 \subsection{Principles}
    566 
    567 \begin{frame}
    568 \frametitle{Principles of PPP with Ambiguity Resolution}
    569 \framesubtitle{Observation Equations}
    570 
    571 The PPPAR is in principle based on the processing the following two types of single-difference
    572 observations: \\
    573 The ionosphere-free linear combination
    574 \be\label{obs_IF}
    575 L^{ij}_3 = \varrho^{ij} - c\delta^{ij} + T^{ij} + \bar{N}^{ij}_3 ~,
    576 \ee
    577 where the ambiguity term is given by
    578 \be\label{amb_N3}
    579 \bar{N}^{ij}_3 =  N^{ij}_3 - l^{ij}_3
    580               = \frac{c\;f_2}{f^2_1-f^2_2}\;(n^{ij}_1-n^{ij}_2) + \lambda_3\;n^{ij}_1 - l^{ij}_3
    581 \ee
    582 and the Melbourne-W\"{u}bbena linear combination
    583 \be\label{obs_MW}
    584 L^{ij}_w = \lambda_5\;n^{ij}_5 - l^{ij}_w
    585 \ee
    586 the uncalibrated bias $l^{ij}_3$ is the corresponding linear combination of biases
    587 $l^{ij}_1,l^{ij}_2$, the uncalibrated bias $l^{ij}_w$ is the corresponding linear combination of
    588 biases $p^{ij}_1,p^{ij}_2,l^{ij}_1,l^{ij}_2$.
    589 \end{frame}
    590 
    591 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    592 
    593 \subsection{Parameters provided by Server}
    594 
    595 \begin{frame}
    596 \frametitle{Principles of PPP with Ambiguity Resolution}
    597 \framesubtitle{Parameters provided by Server}
    598 In addition to orbit corrections, the server(s) has(have) to provide the values
    599 \bdm
    600 c\delta^{ij} ~,~  l^{ij}_w ~,~ l^{ij}_3 ~~~ \mb{or} ~~~~ (c\delta^{ij} + l^{ij}_3) ~,~ l^{ij}_w
    601 \edm
    602 Corrections $l^{ij}_w,l^{ij}_3$ depend on the set of fixed single-difference ambiguities on the
    603 server-side. This set of fixed ambiguities is not unique - it depends on the constraints applied on
    604 the ambiguities.
    605 
    606 There is a difference between correction $l^{ij}_w$ and the narrow-lane correction $l^{ij}_3$. The
    607 wide-lane correction $l^{ij}_w$ depends {\em only} on the ambiguities estimated at the
    608 server-side. The narrow-lane correction $l^{ij}_3$ depends on the ambiguities and {\em also} on the
    609 satellite clock corrections $\delta^{ij}$ estimated at the server-side.
    610 
    611 \end{frame}
    612 
    613 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    614 
    615 \begin{frame}
    616 \frametitle{Principles of PPP with Ambiguity Resolution}
    617 \framesubtitle{How many servers?}
    618 All three corrections
    619 \bdm
    620 c\delta^{ij} ~~~  l^{ij}_w ~~~ l^{ij}_3
    621 \edm
    622 may be estimated together by a single server run (in which case the $c\delta^{ij}$ and $l^{ij}_3$
    623 are indistinguishable and are combined into $c\delta^{ij}+l^{ij}_3$) Or, each of them may be
    624 estimated by a separate server run.
    625 
    626 \vspace*{2mm}
    627 Current approach:
    628 \begin{itemize}
    629 \item PPPNB server: estimates $c\delta^{ij}$
    630 \item PPPAR server: uses $c\delta^{ij}$ from PPPNB server and estimates $l^{ij}_w,l^{ij}_3$
    631 \end{itemize}
    632 
    633 \vspace*{2mm}
    634 Advantages: PPPAR corrections are compatible with PPPNB corrections (the client may decide between
    635 PPP and PPPAR).
    636 
    637 \vspace*{2mm}
    638 Disadvantages: additional delay
    639 
    640 \vspace*{2mm}
    641 An alternative approach to consider: separate server run for $l^{ij}_w$.
    642 
    643 \end{frame}
    644 
    645 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    646 
    647 \begin{frame}
    648 \frametitle{Principles of PPP with Ambiguity Resolution}
    649 \framesubtitle{How to disseminate the corrections?}
    650 
    651 \begin{enumerate}
    652 \item The corrections are valid (accurate) on the single- (between satellites) difference
    653   level but it is more practical to send the zero-difference (satellite-specific) corrections.
    654 \item The corrections are specific for the observation types used for their estimation - e.g. if
    655   the C/A code on the first carrier and the P-code on the second carrier have been used at the
    656   server side, the client can use the $l^{ij}_w$ correction only if it uses the same two types of
    657   code observations.
    658 \end{enumerate}
    659 
    660 The corrections $l^{ij}_w,l^{ij}_3$ are actually the combinations of the phase (and in case of
    661 $l^{ij}_w$ also code) biases:
    662 \begin{eqnarray*}
    663 l^{ij}_w & = & \frac{1}{f_1-f_2} \bigl( f_1~l^{ij}_1 - f_2~l^{ij}_2 \bigr) -
    664   \frac{1}{f_1+f_2} \bigl( f_1~p^{ij}_1 + f_2~p^{ij}_2 \bigr) ~
    665 \\
    666 l^{ij}_3 & = & \frac{1}{f^2_1-f^2_2} \bigl( f^2_1~l^{ij}_1 - f^2_2~l^{ij}_2 \bigr)
    667 \end{eqnarray*}
    668 RTCM suggests to send $p^{ij}_1,p^{ij}_2,l^{ij}_1,l^{ij}_2$ directly ...
    669 
    670 \end{frame}
    671 
    672 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    673 
    674 \begin{frame}
    675 \frametitle{Principles of PPP with Ambiguity Resolution}
    676 \framesubtitle{How to disseminate the corrections (continuation)?}
    677 
    678 In principle there are altogether 5 values which can be sent by server(s):
    679 \bdm
    680 c\delta^{ij},p^{ij}_1,p^{ij}_2,l^{ij}_1,l^{ij}_2
    681 \edm
    682 PPPNB server estimates the $c\delta^{ij}$ and the ionosphere-free
    683 linear combination of the code biases
    684 \bdm
    685 p^{ij}_3 =  \frac{1}{f^2_1-f^2_2} \bigl( f^2_1~p^{ij}_1 - f^2_2~p^{ij}_2 \bigr)
    686 \edm
    687 PPPAR server estimates the $l^{ij}_w$ and $l^{ij}_3$. Assuming that we know the differential code
    688 bias
    689 \bdm
    690 d^{ij}_{p1p2} = p^{ij}_1 - p^{ij}_2
    691 \edm
    692 The four values
    693 \bdm
    694 p^{ij}_3 ~~~ l^{ij}_w ~~~~ l^{ij}_3 ~~~~ d^{ij}_{p1p2}
    695 \edm
    696 can be converted into four biases
    697 $p^{ij}_1,p^{ij}_2,l^{ij}_1,l^{ij}_2$.
    698 
    699 \end{frame}
    700 
    701 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    702 
    703 \begin{frame}
    704   \frametitle{Precise Point Positioning with PPP (cont.)}
    705   BNC provides a good framework for the PPP client (observations, orbits, and
    706   corrections stand for disposal).
    707 
    708   Main reasons for the PPP module in BNC have been:
    709   \begin{itemize}
    710   \item monitoring the quality of incoming data streams (primarily the PPP
    711     corrections)
    712   \item providing a simple easy-to-use tool for the basic PPP positioning
    713   \end{itemize}
    714 
    715   The PPP facility in BNC is provided in the hope that it will be useful.
    716   \begin{itemize}
    717   \item The mathematical model of observations and the adjustment algorithm are
    718     implemented in such a way that they are (according to our best knowledge)
    719     correct without any shortcomings, however,
    720   \item we have preferred simplicity to transcendence, and
    721   \item the list of options the BNC users can select is limited.
    722   \item[$\Rightarrow$] Commercial PPP clients may outperform BNC in some
    723     aspects.
    724   \end{itemize}
    725   We believe in a possible good coexistence of the commercial software and
    726   open source software.
    727 \end {frame}
     596\begin{frame}
     597  \frametitle{Principle of our PPP-RTK Algorithm}
     598  For a dual-band GPS receiver, the observation equations may read as
     599  \begin{eqnarray*}
     600  P^i & = & \varrho^i + c\;\delta - c\;\delta^i + T^i + b_P              \\
     601  L^i & = & \varrho^i + c\;\delta - c\;\delta^i + T^i + b^i
     602  \end{eqnarray*}
     603  where
     604  \begin{tabbing}
     605  $P^i$, $L^i$ ~~~~~~~ \= are the ionosphere-free code and phase measurements, \\
     606  $\varrho^i$          \> is the travel distance between the satellite
     607                          and the receiver,                               \\
     608  $\delta$, $\delta^i$ \> are the receiver and satellite clock errors,    \\
     609  $T^i$                \> is the tropospheric delay,                      \\
     610  $b_P$                \> is the code bias, and                           \\
     611  $b^i$                \> is the phase bias (including initial
     612                          phase ambiguity).
     613  \end{tabbing}
     614  The single-difference bias $b^{ij} = b^i - b^j$ is given by
     615  \begin{displaymath}
     616  b^{ij} = \displaystyle\frac{\lambda_5-\lambda_3}{2}\;(n_5^{ij} + b_5^{ij})
     617              + \lambda_3\;(n_1^{ij} + b_1^{ij})
     618  \end{displaymath}
     619   where
     620  \begin{tabbing}
     621  $n_1^{ij}$, $n_5^{ij}$ ~~~~ \= are the narrow-lane and wide-lane integer ambiguities \\
     622  $b_1^{ij}$                 \> is the narrow-lane (receiver-independent) SD bias \\
     623  $b_5^{ij}$                 \> is the wide-lane (receiver-independent) SD bias
     624  \end{tabbing}
     625\end{frame}
     626
     627%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     628
     629\begin{frame}
     630  \frametitle{Principle of our PPP-RTK Algorithm (cont.)}
     631  Receiver-independent single-difference biases $b_1^{ij}$ and $b_5^{ij}$ have
     632  to be estimated on the server-side.
     633  \begin{itemize}
     634  \item Narrow-lane bias $b_1^{ij}$ may be combined with satellite clock
     635    corrections $\Longrightarrow$ \textbf{modified satellite clock corrections.}
     636  \item Wide-lane bias have to be transmitted from the server to the client
     637    (this bias is stable in time and can thus be transmitted in lower rate).
     638  \end{itemize}
     639
     640  On the client-side the biases $b_1^{ij}$ and $b_5^{ij}$ are used as known
     641  quantities. It allows fixing the integer ambiguities $n_5^{ij}$ and
     642  $n_1^{ij}$. The technique is called Precise Point Positioning with Ambiguity
     643  Resolution (PPP~AR) or PPP~RTK, or zero-difference ambiguity
     644  fixing (the latter term not fully correct because the ambiguities are
     645  actually being fixed on single-difference level).
     646\end{frame}
     647
     648%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     649
     650\begin{frame}
     651  \frametitle{Performance}
     652  \begin{center}
     653    \includegraphics[width=0.75\textwidth]{kir0.png}
     654  \end{center}
     655  \vspace*{-5mm}
     656  \begin{block}{Standard deviations (N,E,U)}
     657  \vspace*{3mm}
     658  \begin{small}
     659  \hspace*{2cm}
     660  \begin{tabular}{l|ccc|ccc}
     661  \mbox{} & \multicolumn{3}{c|}{10-60 min} & \multicolumn{3}{c}{30-60 min} \\
     662  float & 0.034 & 0.026 & 0.026         & 0.010 & 0.009 & 0.011  \\
     663  fix   & 0.007 & 0.003 & 0.016         & 0.007 & 0.003 & 0.012
     664  \end{tabular}
     665  \end{small}
     666  \end{block}
     667\end{frame}
     668
     669%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     670
     671\begin{frame}
     672  \frametitle{Challenges}
     673  PPP~RTK works and provides mm-accuracy results, what is this symposium
     674  about?
     675
     676  \pause
     677  There are still both principal and technical problems and challenges:
     678  \begin{itemize}
     679  \item Principal problems:
     680    \begin{itemize}
     681    \item Convergence time: PPP~RTK in the form outlined above provides
     682      accuracy similar (or even slightly better) to RTK but the convergence
     683      time is longer.
     684    \item There is a degradation in accuracy with the age of corrections.
     685    \item Glonass ambiguity resolution: is it possible to resolve Glonass
     686      ambiguities? (yes, it is possible but it implicates introducing new
     687      parameters - does it really improve the results?)
     688    \item ...
     689    \end{itemize}
     690  \item Technical problems:
     691    \begin{itemize}
     692    \item Availability of data in real time (reference network, high-precision
     693          satellite orbits).
     694    \item Very high CPU requirements on the server-side.
     695    \item Solution robustness on the server-side
     696          (problems with reliable DD ambiguity resolution).
     697    \item ...
     698    \end{itemize}
     699  \end{itemize}
     700\end{frame}
     701
     702%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     703
     704\begin{frame}
     705  \frametitle{Challenges (cont.)}
     706  \begin{block}{Longer convergence time}
     707  In case of a standard RTK the very short convergence time is being achieved
     708  thanks to the combined DD ambiguity resolution on both $L_1$ and $L_2$ when
     709  the differential ionospheric bias can either be neglected (short baselines)
     710  or its influence is mitigated (stochastic ionosphere estimation with
     711  constraints).
     712
     713  On the contrary, the outlined PPP~RTK algorithm is in principle based on
     714  processing single (ionosphere-free) linear combination and resolving only
     715  one set of (narrow-lane) initial phase ambiguities.
     716  \end{block}
     717  \begin{block}{Possible solutions}
     718  \begin{itemize}
     719  \item third carrier
     720  \item multiple GNSS (Glonass ambiguity resolution?)
     721  \item processing original carriers (instead of ionosphere-free linear
     722    combination) and modeling the ionosphere?
     723  \item ?
     724  \end{itemize}
     725  \end{block}
     726\end{frame}
     727
     728%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     729
     730\begin{frame}
     731  \frametitle{Challenges (cont.)}
     732  \begin{block}{Age of corrections 0 s}
     733  \begin{center}
     734    \includegraphics[width=0.6\textwidth]{age1.png}
     735  \end{center}
     736  \end{block}
     737\end{frame}
     738
     739%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     740
     741\begin{frame}
     742  \frametitle{Challenges (cont.)}
     743  \begin{block}{Age of corrections up to 35 s}
     744  \begin{center}
     745    \includegraphics[width=0.6\textwidth]{age2.png}
     746  \end{center}
     747  \end{block}
     748\end{frame}
     749
     750%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     751
     752\begin{frame}
     753  \frametitle{Real-Time Data Availability}
     754  \framesubtitle{IGS network: very good global coverage:}
     755  \vspace*{-5.5cm}
     756  \begin{center}
     757    \includegraphics[width=0.9\textwidth]{map.pdf}
     758  \end{center}
     759\end{frame}
     760
     761%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     762
     763\begin{frame}
     764  \frametitle{Real-Time Data Availability (cont.)}
     765  \begin{tabular}{cc}
     766  \includegraphics[width=0.4\textwidth]{100A_lat.png} &
     767  \includegraphics[width=0.4\textwidth]{101A_lat.png} \\
     768  \includegraphics[width=0.4\textwidth]{102A_lat.png} &
     769  \includegraphics[width=0.4\textwidth]{104A_lat.png}
     770  \end{tabular}
     771
     772  Gaps in reference network data may degrade the PPP~RTK server performance
     773  considerably!
     774\end{frame}
     775
     776%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     777
     778\begin{frame}
     779  \frametitle{Technical issues}
     780  \begin{block}{CPU-requirements on the server-side}
     781  Processing a global reference network is a very CPU-intensive
     782  task. Numerically stable forms of the Kalman filter (square-root, UDU
     783  factorization etc.) require very fast hardware.
     784
     785  Possible solutions:
     786  \begin{itemize}
     787  \item Processing optimization (estimating various kinds of parameters in
     788        different rates)
     789  \item Parallel processing
     790  \item Advanced hardware (GPS Solutions uses GPU-accelerated library)
     791  \end{itemize}
     792  \end{block}
     793  \begin{block}{Reliable DD ambiguity resolution on the server-side}
     794  Reliable double-difference ambiguity resolution on the server-side remains
     795  the crucial issue of the PPP~RTK technique.
     796  \end{block}
     797  \begin{block}{Dissemination of PPP~RTK corrections}
     798  \begin{itemize}
     799  \item data links
     800  \item formats (standardization?)
     801  \item optimization of correction rates (bandwidth)
     802  \end{itemize}
     803  \end{block}
     804\end{frame}
     805
     806%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     807
     808\begin{frame}
     809  \frametitle{Satellite orbits}
     810
     811  Predicted part of the IGS ultra-rapid orbits (available in real-time) is
     812  sometimes not sufficient for the processing of a global reference network
     813  (with narrow-lane ambiguity resolution). We have been forced to implement
     814  the real-time orbit determination capability in our main processing tool
     815  RTNet (Real-Time Network software).
     816  \begin{center}
     817    \includegraphics[width=0.75\textwidth]{rtnet_pod.png}
     818  \end{center}
     819\end{frame}
     820
     821%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     822
     823\begin{frame}
     824  \frametitle{Regional versus global PPP~RTK services}
     825  Currently we are routinely running both regional and global PPP~RTK service
     826  demonstrators in real-time (some of the results will be shown below).
     827  \begin{itemize}
     828  \item in principal there is no difference between a global and regional
     829    service as far as the data processing, algorithms etc. is concerned
     830  \item global PPP~RTK service has at least the following two advantages
     831     \begin{itemize}
     832     \item[1.] a single correction stream can serve all users
     833     \item[2.] all satellites are tracked permanently (helps ambiguity
     834               resolution)
     835     \end{itemize}
     836  \item global PPP~RTK service is much more challenging (data availability,
     837    CPU-requirements on the server-side, DD ambiguity resolution on long
     838    baselines, the highest requirements for the accuracy of the satellite
     839    orbits)
     840  \end{itemize}
     841
     842\end{frame}
     843
     844%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     845
     846\begin{frame}
     847  \frametitle{Services monitoring}
     848  Reliable, production-quality PPP~RTK service requires sophisticated
     849  monitoring tools.
     850  \begin{tabular}{cc}
     851  \includegraphics[width=0.6\textwidth]{monitor1.png} & \\[-1.5cm]
     852  & \hspace*{-3cm} \includegraphics[width=0.6\textwidth]{monitor2.png}
     853  \end{tabular}
     854
     855\end{frame}
     856
     857%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     858
     859\begin{frame}
     860  \frametitle{Results}
     861  \begin{center}
     862    \includegraphics[width=0.9\textwidth]{tsunami.pdf}
     863  \end{center}
     864\end{frame}
     865
     866%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     867
     868\begin{frame}
     869  \frametitle{Results (cont.)}
     870  \begin{center}
     871    \includegraphics[width=0.9\textwidth]{nrcan.png}
     872  \end{center}
     873\end{frame}
     874
    728875
    729876%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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