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\title{SRTM/ASTER boundary analysis}

\author{Jim Regetz}

\date{Last update: 12 Jul 2011}

 \item add constant offset to ASTER?

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neighbors, also known as the Multiple Flow Direction (MFD) model.

However, to simplify post-processing and summarization, the results here

are based on an alternative Single Flow Direction (SFD, \emph{a.k.a.}

D8) model, which indicates the neighbor associated with the steepest

downslope gradient. Note that this is equivalent to what ArcGIS GRID

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\section{Latitudinal mean terrain profiles}

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% first compute some values to use in the text

contour-based data \textbf{\color{red}[todo: check!]}. Moreover, this

figure makes it clear that the Canada DEM has some major artifacts at

regular intervals. The spike especially at 60\N (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.

\paragraph{Flow direction} With the exception of edge effects at the

margins of the input rasters, mean ASTER-derived flow is northward at

all latitudes, and SRTM-derived flow is northward at nearly all

latitudes (Figure \ref{mean-flowdir}). This seems reasonable considering

that most pixels in this Canada test region fall in the Arctic drainage.

For unknown reasons, the Canada DEM produces southward mean flow

direction at numerous latitudes, and generally seems to have a slightly

more eastward tendency. As was the case with aspect, note that a general

north-south bias is evident (Figure \ref{rose-diag-flowdir}), again

likely due to use of an unprojected raster at high latitudes.

The various fused layer profiles look as one would expect (Figure

\ref{mean-flowdir}), although the overall lack of latitudinal

variability in mean flow direction in SRTM, ASTER, and all three derived

layers makes it hard to say much more than that. 

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\section{Informal correlation analysis}

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\paragraph{Elevation}

Correlations between ASTER and SRTM are quite high, typically >0.999,

and RMSEs are on the order of 10-15 meters (Figures \ref{corr-elevation}

and \ref{rmse-elevation}). Spikes (downward for correlation, upward for

RMSE) occur at some latitudes, which I speculate are associated with

the observed extreme spikes in the ASTER DEM itself. As expected, the

multiresolution spline and Gaussian weighted average both produce layers

that gradually become less similar to ASTER and more similar to SRTM

moving south from 60\N, but in slightly different ways. This gradual

transition is less evident when considering associations with the Canada

DEM (bottom panels of Figures \ref{corr-elevation} and

\ref{rmse-elevation}), which in general is less correlated with SRTM,

and even less with ASTER, than those two layers are with each others.

\paragraph{Slope}

The patterns of slope layer similarity are much like those described

above for elevation, although the correlations are somewhat lower

($\sim$0.94 between SRTM and ASTER (Figure \ref{corr-slope})). RMSEs

between SRTM and ASTER are approximately 2 at all latitudes where the

data co-occur (Figure \ref{rmse-slope}). Another difference, echoing a

pattern previously noted in the profiles of mean slope itself, is that

Gaussian weighted averaging produces a layer that exhibits a gradual

transition from ASTER to SRTM, whereas the multiresolution spline yields

an abrupt transition. Not surprisingly, the simple fused layer is even

worse, producing not only a sudden transiton but also abberant values

right at the fusion seam; note downward (upward) spikes in correlation

(RMSE) at 60\N in the first column of plots in Figures \ref{corr-slope}

and \ref{rmse-slope}.

\paragraph{Aspect}

Because aspect values are on a circular scale, I calculated modified

versions of the Pearson correlation coefficient using the

\textbf{circular} R package function \texttt{cor.circular}. I believe

this implements the formula described by Jammalamadaka \& Sarma (1988):

\begin{equation}

r = \frac

  {\sum_{i=1}^{n} \sin(x_i-\bar{x}) \sin(y_i-\bar{y})}

  {\sqrt{\sum_{i=1}^{n} \sin(x_i-\bar{x})^2}

   \sqrt{\sum_{i=1}^{n} \sin(y_i-\bar{y})^2}}

\label{eq-ccor}

\end{equation}

where $x_i$ and $y_i$ are aspect values (in radians) for pixel $i$, and

$\bar{x}$ and $\bar{y}$ are circular means calculated as in Equation

\ref{eq-cmean}.

To calculate RMSEs for aspect, I did not use trigonometric properties in

the same way as for computing correlation, but instead imposed a simple

correction whereby all pairwise differences were computed using the

shorter of the two paths around the compass wheel (Equation

\ref{eq-circ-rmse}). For example, the difference between 0$^\circ$ and

150$^\circ$ is 150$^\circ$, but the difference between 0$^\circ$ and

250$^\circ$ is 110$^\circ$.

\begin{eqnarray}

\label{eq-circ-rmse}

&RMSE = \sqrt{\frac{\sum_{i=1}^{n} (\Delta_i^2)}{n}}\\ \nonumber

&\mbox{where }

  \Delta_i = \mbox{argmin} \left (|x_i-y_i|, 360-|x_i - y_i| \right )

\end{eqnarray}

Circular correlations between SRTM and ASTER were surprisingly low,

typically hovering around 0.5, but dipping down towards zero at numerous

latitudes (Figure \ref{corr-flowdir}). The corresponding RMSE is close

to 70 at all latitudes, surprisingly high considering that the maximum

difference between any two pixels is 180$^\circ$ (Figure

\ref{rmse-flowdir}). Comparison of SRTM and ASTER with the Canada DEM

yields similar patterns.

For both of the blended layers (multiresolution spline and Gaussian

weighted average), aspect values calculated in the zone of ASTER/SRTM

overlap appear to be \emph{less} similar to the component DEMs than

those to data sources are to each other. As evident in the upper middle

and upper right panels of Figure \ref{corr-flowdir}, correlations

between fused aspect and aspect based on the original SRTM and ASTER

images are substantially \emph{negative} for many latitudes in the zone

of overlap. I don't have a good feel for what's going on here, although

it part that may be because I don't have a great intuition for how the

circular correlation statistic might be responding to patterns in the

data. Note that the latitudinal profile of aspect RMSEs is less

worrisome (upper middle and upper right panels of Figure

\ref{rmse-flowdir}) and more closely resembles the profile of slope

RMSEs discussed above, particularly as regards the more gradual

transition in case of Gaussian weighted averaging compared to the

multiresolution spline.

\paragraph{Flow direction}

As with aspect, circular correlation coefficients and adjusted RMSEs

were calculated for each latitudinal band. Flow direciton correlations

between SRTM and ASTER are slightly lower than for aspect, typically

around 0.40 but spiking negative at a handful of latitudes (Figure

\ref{corr-flowdir}). RMSEs hover consistenly around $\sim$75, slightly

lower than was the case with aspect (Figure \ref{rmse-flowdir}).

Aside from an expected artifact at the southern image edge, and an

unexpected (and as-yet unexplained) negative spike at $\sim$59.95\N, the

correlation profiles are fairly well-behaved for both blended layers,

and don't show the same odd behavior as was the case for aspect. Again

it is clear that the multiresultion spline results in a much more abrupt

transition than does the Gaussian weighted average. The RMSE flow

direction profiles echo this pattern (Figure \ref{rmse-flowdir}), and

indeed look almost the same as those computed using aspect (Figure

\ref{rmse-aspect}).

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% FIGURES

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\clearpage

%

% Figure: Boundary analysis region

%

\begin{figure}[h]

  \caption{Focal area}

  \centering

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

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

  library(raster)

  library(maps)

  demdir <- "/home/regetz/media/temp/terrain/dem/"

  dem <- raster(file.path(demdir, "fused_300straddle.tif"))

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

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

  grid(col="darkgray")

  axis(1)

  axis(2)

  box()

  plot(extent(dem), col="red", add=TRUE)

  arrows(-110, 57, -113, 59.7, col="red", length=0.1)

  text(-110, 57, labels="focal region", col="red", font=3, pos=1)

@

  \label{focal-area}

\end{figure}

%

% Figure: Fusion methods

%

\begin{figure}

  \caption{SRTM/ASTER DEM fusion configurations}

  \centering

  \subfloat[Simple fusion]{

    \label{blend-simple}

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

  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")

@

  }\\

  \subfloat[Multiresolution spline]{

    \label{blend-multires}

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

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

  rect(-75, 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(-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(-150, 0.9, labels="SRTM", pos=4, col="blue")

  text(-37.5, 0.8, labels="overlap", pos=3, font=3)

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

@

  }\\

  \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}

%

% Figure: Mean latitudinal profiles

%

\begin{figure}

  \centering

  \caption{Mean elevation (m)}

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

    \label{mean-elevation}

\end{figure}

\begin{figure}

  \centering

  \caption{Mean slope (degrees)}

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

    \label{mean-slope}

\end{figure}

\begin{figure}

  \centering

  \caption{Circular mean aspect (azimuth)}

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

    \label{mean-aspect}

\end{figure}

\begin{figure}

  \centering

  \caption{Circular mean flow direction}

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

    \label{mean-flowdir}

\end{figure}

%

% Figure: ASTER vs SRTM elevation patterns

%

\begin{figure}

  \caption{ASTER vs SRTM elevation comparisons}

  \centering

  \subfloat[Boxplots of the arithmetic difference in elevations (ASTER -

  SRTM), summarized across a series of ASTER elevation bins. Boxes

  include the median (horizontal band), 1st and 3rd quartiles (box

  extents), and $\pm$1.5$\times$IQR (whiskers). Gray numbers above each

  whisker indicates how many thousands of pixels are included in the

  corresponding summary. Dashed red line indicates the median difference

  across all pixels south of 60\N.]{

    \includegraphics[width=\linewidth]{../dem/aster-srtm-bins.png}

    \label{aster-srtm-bins}

  }\\

  \subfloat[Pixel-wise plot of ASTER vs SRTM for all values in the 150

  latitudinal rows of overlap south of 60\N. Dashed blue line indicates

  the 1:1 diagonal, and the parallel red line is offset lower by the

  observed median difference between ASTER and SRTM

  (\Sexpr{delta.median}).]{

    \includegraphics[width=\linewidth]{../dem/aster-srtm-scatter.png}

    \label{aster-srtm-scatter}

  }

  \label{aster-srtm}

\end{figure}

%

% Figures: Aspect rose diagrams

%

%

% Figures: Flow direction rose diagrams

%

% Figures: Correlations and RMSEs

    \label{corr-flowdir}

    \label{rmse-flowdir}

\clearpage

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