%% Upslope Area Toolbox User Guide
%% What is "Upslope Area"?
% Imagine you are standing on the side of hill somewhere in the rain. Some of
% the water that falls uphill from your position will flow directly toward and
% then past your shoes. Some of the water, though, will flow downhill in a
% different direction, away from you. The area of land above you that drains
% directly through where you are standing is called the _upslope area_ of your
% If you were standing at the very top of the hill, the upslope area there would
% be 0; no water flows to you from anywhere else. On the other hand, if you
% stood at the deepest point in a crater with high rims all the way around, the
% upslope area would be the entire area of the crater.
% Upslope area is an important hydrology measurement used to study water
% drainage networks, the motion of sediments and contaminants, erosion,
%% About Digital Elevation Models
% A _digital elevation model_ (or DEM) is a computer representation of surface
% topography. A _raster DEM_ is a rectilinear grid of values, each of which
% represents the height of a surface at the corresponding grid location.
% High-quality, high-resolution DEMs are now widely available and being used for
% a wide variety of terrain analysis. DEM data for the United States can be
% obtained through the U.S. Geological Survey (USGS) and its data providers.
% The file milford_ma_dem.mat contains an example DEM covering a portion of
% Massachusetts in the United States. You can load this MAT-file and display
% the elevation matrix |Z| as follows:
imshow(Z, ) % imshow is in the Image Processing Toolbox
% Use |surf| and other MATLAB graphics functions to display a small portion of
% the DEM as a surface.
Zsub = Z(220:250, 170:215);
set(gca, 'YDir', 'reverse')
% You can see a bright hill on the left side and what looks like it might be a
% pond (the flat, dark region) in the middle. But the height effect is greatly
% exaggerated here. If you look at the |description| variable in the MAT-file
% you can see that the x- and y-resolution of the pixels in the DEM data is 30
% meters. You can get a better idea of the true appearance of the surface by
% setting the DataAspectRatio accordingly.
set(gca, 'DataAspectRatio', [1 1 30])
%% Modeling Surface Flows
% When analyzing water flow using a DEM, an essential step in the analysis is
% to determine the direction of flow at each point in the DEM grid. Consider,
% for example, the 3-by-3 matrix of height values below:
E = [10 10.5 11; 10 9 8.9; 10.3 8.5 8.4]
% The center point has a height of 9. It's eastern neighbor has the same height.
% It has two downhill neighbors to the south and the southeast. How should
% determine a direction of water flow for this point?
% The Upslope Area Toolbox provides functions that compute water flow direction
% using the _D-infinity_ method described in Tarboton, "A new method
% for the determination of flow directions and upslope areas in grid digital
% elevation models," _Water Resources Research_, vol. 33, no. 2, pages 309-319,
% February 1997.
% The function |pixelFlow| returns the flow direction for a given point in a
% DEM. The direction is returned as the angle (in radians) measured
% counter-clockwise from the east-pointing horizontal axis.
center_point_flow_in_degrees = pixelFlow(E, 2, 2) * (180/pi)
% So the flow from the center point is about 281 degrees, or south-southeast.
% You can use |demFlow| to compute the flow direction for all the points in a
R = demFlow(E)
% The |NaN| value in the lower right corner indicates that location has no
% downhill neighbors, so there is no downhill water flow from there.
%% Computing and Understanding the Flow Matrix
% Another important step in hydrological analysis is to answer this question for
% each point in the DEM: how much water flows into that point from each of its
% neighbor points? The _flow matrix_, computed by the function |flowMatrix|,
% answers this question for all the points in a DEM.
T = flowMatrix(E, R)
% Although the flow matrix is sparse, in this small example it is easier to look
% at the flow matrix values if you convert it to full.
T = full(T)
% Each of the nine rows and columns of |T| corresponds to one of the nine points
% in the 3-by-3 DEM, with the points in the DEM numbered columnwise. For
% example, the fourth row and the fourth column correspond to the DEM point
% |E(1,2)|. Similarly, the ninth row and the ninth column correspond to the DEM
% point |E(3,3)|.
% The 1's along the diagonal of |T| represent the idea that an equal unit volume
% of water is being added to the surface, presumably from rain, at each point in
% the DEM.
% Look at the columns of |T| to see where the rain water drains to. Here is the
% second column of |T|.
% The values in this column indicate that about 41% of the water flowing into
% DEM point #2 flows down into DEM point #5, because |T(5,2)| equals -0.4907.
% About 59% of the water flowing into DEM point #2 flows down into DEM point #6,
% because |T(6,2)| equals -0.5903.
% Look at the rows of |T| to see where a given DEM point receives its water
% from. For example, here is the ninth row of |T|:
% These values indicate that DEM point #9 receives all of the water flowing into
% DEM point #8 (|T(9,8)| equals -1.0), all of the water flowing into DEM point
% #6 (|T(9,6)| equals -1.0), and about 25% of the water flowing into DEM point
% #5 (|T(9,5)| equals -0.2513).
%% Computing Upslope Area
% The function |upslopeArea| computes the upslope area for every point in the
% DEM by solving a sparse linear system of equations based on the flow matrix.
% For example:
U = upslopeArea(E, T)
% Note that the upslope area of a point includes itself in this computation. An
% upslope area of 1.0 indicates that the only water flowing into that location
% is the unit amount assumed from rainfall. In our small 3-by-3 DEM example,
% the water falling at all points eventually flows downhill into the (3,3)
% location so that the upslope area at that point is 9.
% Now let's solve a real problem. Specifically, let's compute the upslope area
% for the data in milford_ma_dem.
R = demFlow(Z);
T = flowMatrix(Z, R);
U = upslopeArea(Z, T);
% It's hard to see much detail. Another visualization technique you can try is
% to display the logarithm of the upslope area. This technique shows much more
% You can also use the |visMap| function to superimpose the upslope area (shaded
% in green) over the original DEM data.
% Here's a zoomed-in view
axis([165 230 160 290])
%% Influence Maps
% The flow matrix can be used to construct other linear systems whose solutions
% give useful information. For example, you can ask this question: For a given
% DEM point P, what is the complete set of downhill DEM points that receive
% water from P? The _influence map_, a matrix computed by |influenceMap|,
% answers this question.
% In this example you will compute the influence map for the milford_ma DEM
% point (235, 185) and then display it using |visMap|.
I = influenceMap(Z, T, 235, 185);
visMap(I, Z, 235, 185)
% Zoom in
axis([165 230 220 290])
% You can see that water starting at the top of the hill (blue dot) flows to the
% east into the pond and then out through the southern end of the pond into a
% local minima (sink). (See the "Sinks" section in "Special Data
% Considerations" below.)
%% Dependence Maps
% The _dependence map_, a matrix computed by |dependenceMap|, is another
% quantity computed from the flow matrix. It shows the complete set of uphill
% DEM points that drain through a given DEM location. Just like the influence
% map, the dependence map can be visualized using |visMap|. The following
% example shows how to compute and visualize the dependence map for the DEM
% location (270, 189).
D = dependenceMap(Z, T, 270, 189);
visMap(D, Z, 270, 189)
% Zoom in
axis([65 325 70 350])
% There's a blue dot indicating the point that the green area is draining
% through, but at this scale it's too small to see, so zoom in further:
axis([155 225 235 305])
%% Special Data Considerations
% For the influence map example above, recall that the water flowed down from
% the top of the hill, into the pond, and out the southern end of the pond,
% where it appeared to simply stop. That's because the DEM data had a *local
% minimum* there. Here are the DEM data values immediately surrounding that
% You can see that the height at the middle point is 98, which is lower than all
% the DEM points surrounding it. This kind of local minimum is called a _sink_.
% For many kinds of topological analyses it is desirable to eliminate all sinks
% that are not located at the edge of the DEM. You can use the function
% |fillSinks| to eliminate these interior sinks.
% Let's repeat the influence map example on the original milford_ma DEM data,
% including this time a preprocessing step to eliminate interior sinks.
Zp = fillSinks(Z);
Rp = demFlow(Zp);
Tp = flowMatrix(Zp, Rp);
Ip = influenceMap(Zp, Tp, 235, 185);
visMap(Ip, Zp, 235, 185)
% Zoom in
axis([165 230 220 290])
% Now you can see that the water continues to flow. If you zoom out further you
% can see that the water continues to flow until it reaches a low, equal-height
% group of DEM points that is connected to the edge of the data set.
axis([100 320 200 473])
% A _plateau_ is a connected group of DEM points with equal height. For
% example, the DEM points in the pond shown here all have value 106.
axis([170 215 220 250])
% It isn't always clear how to compute water flow across a plateau. The function
% |demFlow| uses the _arrowing_ technique described in F. Meyer, "Skeletons and
% Perceptual Graphs," _Signal Processing_ 16 (1989) 335-363. (See Appendix A.2,
% Arrowing, on pages 360-361.) The technique works reasonably well in many
% cases, but you can see some plateau artifacts in the upslope area
% visualization for the milford_ma DEM.
% Zoom in
axis([110 210 20 150])
% The upslope area toolbox provides the function |postprocessPlateaus|, which
% replaces the upslope area values for a set of plateau points with the mean
% upslope area for the entire plateau. Here's how it would work for the
% milford_ma data.
Um = postprocessPlateaus(U, Z);
% Zoom in
axis([110 210 20 150])
% *Missing Data*
% Many DEM data sets have legitimate values only within a certain watershed,
% which is usually an irregular region. DEM arrays typically have a "fill
% value" outside the watershed region to indicate invalid data. Usually the
% fill value is something recognizable like -999, and usually the fill value is
% given in the description of the data set.
% To process such data sets using the Upslope Area Toolbox, replace the fill
% values with NaN using code like this:
% Z(Z == fill_value) = NaN;
% The toolbox function |borderNans| identifies NaN-valued DEM points that are
% connected to the "outside" edges of the DEM data set. The milford_ma DEM data
% contains border NaNs, which you can display as follows:
% Turn the axis box on so you can see the extent of the white pixels, which are
% the border NaNs.
% *Obtaining DEM Data*
% DEM data for the United States can be obtained from the U.S. Geological Survey
% (USGS). For example, DEM data and other datasets can be obtained from the
% <http://seamless.usgs.gov/index.php National Map Seamless Server>. You can
% download this data in BIL format, which can be read using the MATLAB function
% |multibandread|. There are several web sites that offer information and
% tutorials on getting data from the Seamless Server, including
% <http://www.yale.edu/ceo/Documentation/dem_import.pdf this one at Yale
% For locating DEM data covering other regions, you might try the listing of
% <http://www.terrainmap.com/rm39.html "Free Digital Elevation Model (DEM) and
% Free Satellite Imagery Download Links"> at <http://www.terrainmap.com/
% If you have a recent version (R2009b or later) of Mapping Toolbox, you can use
% the Web Map Service (WMS) features to obtain DEM data. The
% |wmsread| reference page> has an example showing how to obtain DEM data
% directly from the JPL WMS server.
% For sources of other geospatial data that can be read using functions in
% Mapping Toolbox, see <http://www.mathworks.com/support/tech-notes/2100/2101.html
% MathWorks Technical Note 2101 - Accessing Geospatial Data on the Internet for
% the Mapping Toolbox>.
% Copyright 2009 The MathWorks, Inc.