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\paragraph{Region of analysis} For testing purposes, I used a narrow

band along the 60N boundary in Canada (Figure \ref{focal-area}). The

latitudinal extent of this band is 59.875N - 60.125N (i.e., 300 3"

pixels straddling 60N), and the longitudinal extent is 136W to 96W

(i.e., 48000 pixels wide).

  \centering

<<echo=FALSE,results=hide,fig=TRUE,height=5,width=7>>=

  par(omi=c(0,0,0,0))

  map("world", xlim=c(-140,-70), ylim=c(40, 70), fill=TRUE, col="grey",

      mar=c(4,0,0,0))

  \label{focal-area}

\paragraph{Fusion approaches} This document includes latitudinal

characterizations of terrain values based on three different variants of

a fused ASTER/SRTM DEM. I also explored (and briefly describe below) two

additional approaches to fusing the layers, but do not include further

assessment of these here.

\subparagraph{Simple fusion} Naive concatentation of SRTM below 60N with

ASTER above 60N, without applying any modifications to deal with

boundary artifacts (Figure \ref{blend-simple}).

\subparagraph{Multiresolution spline} Application of Burt \& Adelson's

(1983) method for blending overlapping images using multiresolution

splines, as implmented in the \emph{enblend} software package. Data prep

and post-processing were handled in R (see Listing \ref{code-enblend}).

As presented here, the ASTER and SRTM inputs were prepared such that the

overlap zone is 75 latitudinal rows (6.75km) (Figure

\ref{blend-multires}).

\subparagraph{Gaussian weighted average} Blend of the two layers using

weighted averaging such that the relative contribution of the SRTM

elevation is zero at 60N, and increases according to a Gaussian function

of distance moving south away from 60N. As presented here, this function

was parameterized such that the ASTER and SRTM are equally weighted at a

distance of $\sim$2.4km (26 cells) from the boundary, and the influence of

ASTER is negligible by $\sim$5km from the boundary (Figure

\ref{blend-gaussian}).

\subparagraph{Others not shown} Fused with exponential ramp north of

60N. The first step is to take the pixel-wise difference between ASTER

and SRTM in the row immediately below 60N (i.e., the northmost extent of

SRTM). An exponentially declining fraction of this difference is then

then added back into the ASTER values north of 60N. This does a fine job

of eliminating the artificial shelf and thus the appearance of a seam

right along the 60N boundary, but it does not address the abrupt

transition in texture (i.e., the sudden appearance of high frequency

variability moving north across the boundary). Additionally, it

introduces vertical ``ridges'' running north from the boundary. These

arise because the calculated ramps are independent from one longitudinal

``column'' to the next, and thus any changes in the boundary difference

from one pixel to the next lead to adjacent ramps with different

inclines.

I also tried a simple LOESS predictive model. This involved first

calculating the difference between ASTER and SRTM everywhere south of

60N, and then fitting a LOESS line to these differences using the actual

ASTER elevation as a predictor. I then used the fitted model to predict

the ASTER-SRTM difference for ASTER cells north of 60N, and add a

declining fraction (based on a Gaussian curve) of this difference to the

ASTER values north of 60N. Conceptually, this amounts to applying an

ASTER-predicted correction to the ASTER elevation values, where the

correction term declines to zero according with increasing distance

north of the bonudary. However, this method alone didn't yield

particularly promising results in removing the 60N seam itself,

presumably because adding a predicted correction at the boundary does

not close the SRTM-ASTER gap nearly as efficiently as do corrections

that are based on the actual elevation values. I therefore haven't

pursued this any further, although it (or something like it) may prove

useful in combination with one of the other methods.

\begin{figure}

  \caption{SRTM/ASTER blend configurations}

  \subfloat[Simple fusion]{

    \label{blend-simple}

  par(omi=c(0,0,0,0), mar=c(4,4,0,2)+0.1)

  plot(0, xlim=c(-150, 150), ylim=c(0, 1), type="n", xaxt="n", yaxt="n",

      bty="n", xlab="Latitude", ylab=NA)

  rect(0, 0, 150, 0.75, col="red", density=5, angle=45)

  rect(-150, 0, 0, 0.75, col="blue", density=5, angle=135)

#  axis(1, at=seq(-120, 120, by=60), labels=seq(59.9, 60.1, by=0.05))

  axis(1, at=seq(-150, 150, by=75), labels=c(59.875, NA, 60, NA, 60.125))

  text(-150, 0.9, labels="SRTM", pos=4, col="blue")

  text(150, 0.9, labels="ASTER", pos=2, col="red")

    \label{blend-multires}

  rect(-75, 0, 0, 0.75, col="lightgray")

  #axis(1, at=seq(-120, 120, by=60), labels=seq(59.9, 60.1, by=0.05))

  axis(1, at=seq(-150, 150, by=75), labels=c(59.875, NA, 60, NA, 60.125))

  }\\

  \subfloat[Gaussian weighted average]{

    \label{blend-gaussian}

<<echo=FALSE,results=hide,fig=TRUE, height=3, width=6>>=

  par(omi=c(0,0,0,0), mar=c(4,4,1,2)+0.1)

  curve(exp(-0.001*x^2), from=-150, to=0, xlim=c(-150, 150), col="red",

      xaxt="n", bty="n", xlab="Latitude", ylab="Weighting")

  curve(1+0*x, from=0, to=150, add=TRUE, col="red")

  curve(1-exp(-0.001*x^2), from=-150, to=0, add=TRUE, col="blue")

  #axis(1, at=seq(-150, 150, by=60), labels=seq(59.875, 60.125, by=0.05))

  #axis(1, at=0, labels=60, col="red", cex=0.85)

  #axis(1, at=seq(-120, 120, by=60), labels=seq(59.9, 60.1, by=0.05))

  axis(1, at=seq(-150, 150, by=75), labels=c(59.875, NA, 60, NA, 60.125))

  text(-100, 0.6, labels="gaussian\nblend")

  text(-140, 0.9, labels="SRTM", col="blue", pos=4)

  text(-140, 0.1, labels="ASTER", col="red", pos=4)

@

  }

\end{figure}

\newpage

%\includepdfset{pagecommand=\thispagestyle{fancy}}

%\setkeys{Gin}{width=1.8\linewidth}

%\captionsetup[table]{position=top}

\paragraph{Latitudinal terrain profiles (means)}

\subparagraph{Elevation} As indicated in Figure \ref{mean-elevation},

the mean SRTM elevation is uniformly higher than the mean ASTER

elevation at all latitudes in the area of overlap, by $\sim$11 meters.

However, the shape of the profile itself is nearly identical between the

two. As point of reference, the Canada DEM shares a similarly shaped

profile, but with elevation values intermediate to the ASTER and SRTM.

Not surprisingly, simple fusion produces an artificial cliff in the mean

elevation profile. This artifact is completely removed by both the

multiresolution spline and gaussian weighted average methods. The

transition is, at least to the eye, slightly smoother in the former

case, although ultimately this would depend on the chosen zone of

overlap and on the exact parameterization of the weighting function.

\subparagraph{Slope} As indicated in Figure \ref{mean-slope},

the mean ASTER slope is uniformly steeper than the mean SRTM

elevation at all latitudes in the area of overlap, by nearly 1 degree.

However, the shape of the profile itself is nearly identical between the

two. Although this may partly reflect inherent SRTM vs ASTER

differences, my guess is that CGIAR postprocessing of the particular

SRTM product we're using has removed some of the high frequency

``noise'' that remains in ASTER.

Note that the he Canada DEM tends to be flatter than both ASTER and

SRTM (presumably because it is at least partially derived from

contour-based data (\textbf{check!})). Moreover, this figure makes it

clear that the Canada DEM has some major artifacts at regular intervals.

The spike especially at 60N (which coincides with provincial boundaries

across the entirety of western Canada) means we probably need to scuttle

our plans to use this DEM as a reference dataset for boundary analysis.

The simple fused layer exhibits a dramatic spike in slope at the

immediate 60N boundary, undoubtedly associated with the artificial

elevation cliff. This artifact is eliminated by both the multiresolution

spline and gaussian weighting. However, former exhibits a sudden step

change in slope in the SRTM-ASTER overlap region, whereas the transition

is smoothed out in the latter. This likely reflects the fact that the

multiresolution spline effectively uses a very narrow transition zone

for stitching together high frequency components of the input images,

and I surmise these are precisely the features responsible for the shift

in mean slope. 

\subparagraph{Aspect} As indicated in Figure \ref{mean-aspect},

the mean aspect of ASTER and SRTM are quite similar across all latitudes

in the area of overlap. However, the shape of the

profile itself is nearly identical between the

two. Although this may partly reflect inherent SRTM vs ASTER

differences, my guess is that CGIAR postprocessing of the particular

SRTM product we're using has removed some of the high frequency

``noise'' that remains in ASTER.

%TODO: recalc aspect using circular mean?

%TODO: run flow for enblend output? or just leave what we have, and

%discuss that? be sure to mention that we're doing flow for a smaller

%longitudinal range

%

% Latitudinal terrain profiles (means)

%

\begin{figure}[t!]

  \centering

  \caption{Latitudinal terrain profiles}

  \subfloat[Mean elevation (m)]{

    \includegraphics[page=1, trim=1in 1in 1in 1in, width=\linewidth]{../dem/elevation-assessment.pdf}

    \label{mean-elevation}

  }\\

  \newpage

  \subfloat[Mean slope (degrees)]{

    \includegraphics[page=1, trim=1in 1in 1in 1in, width=\linewidth]{../slope/slope-assessment.pdf}

    \label{mean-slope}

  }

\end{figure}

\begin{figure}[h!]

  \ContinuedFloat

  \centering

  \captionsetup{position=top}

  \subfloat[Aspect (direction, 0=East)]{

    \includegraphics[page=1, trim=1in 1in 1in 1in, width=\linewidth]{../aspect/aspect-assessment.pdf}

    \label{mean-aspect}

  }\\

  \subfloat[Flow (D8 direction, 0=East)]{

    \includegraphics[page=1, trim=1in 1in 1in 1in, width=\linewidth]{../flow/flowdir-assessment.pdf}

    \label{mean-flow}

\paragraph{Summary of findings}

\begin{itemize}

 \item CDEM itself has problems

  \begin{itemize}

   \item 60N coincides with provincial boundaries; there are clear 60N

     artifacts in this layer!

   \item other evident tiling artifacts too

  \end{itemize}

 \item Big SRTM vs ASTER differences:

  \begin{itemize}

   \item ASTER is generally lower elevation

   \item ASTER has more high-frequency variability ("texture"), which

     affects slope/aspect?

  \end{itemize}

 \item Exponential ramping

  \begin{itemize}

   \item fixes seam itself

   \item leaves dramatic transition in SRTM/ASTER textural differences

   \item introduces ridging

  \end{itemize}

 \item Blending

  \begin{itemize}

   \item fixes seam itself

   \item smooths transition in textural differences

  \end{itemize}

 \item Other comments

  \begin{itemize}

   \item Routing is a big ball of wax

  \end{itemize}

\end{itemize}

\paragraph{To do (possibly)}

\begin{itemize}

 \item add constant offset to ASTER? or maybe an offset that is a

 function of elevation? (revisit LOESS fit)

 \item smooth ASTER?

\end{itemize}

%TODO: include and discuss the RMSE/correlation plots

%\includepdf[landscape=true,pages=-]{../dem/elevation-assessment.pdf}

%\includepdf[landscape=true,pages=-]{../slope/slope-assessment.pdf}

%\includepdf[landscape=true,pages=-]{../aspect/aspect-assessment.pdf}

%\includepdf[landscape=true,pages=-]{../flow/flowdir-assessment.pdf}

\appendix

\lstinputlisting[language=R, caption={R wrapper code for multiresolution

    spline}, label=code-enblend]{../dem/enblend.R}