1. CIVE 853- GIS in Water Resources Fall 2017
Terrain Analysis
CIVE 853- GIS in Water Resources Fall 2017
by Ayse Kilic
with some materials from David G. Tarboton, Utah State University and from ESRI software
Revised Sept. 19, 2017

2. Learning Objectives Calculation of slope on a raster using
ArcGIS method based in finite differences
D8 steepest single flow direction
D steepest outward slope on grid centered triangular facets
Calculation of other Terrain parameters including Aspect, Flow Direction, Hillshade, Viewshed

3. Spatial Surfaces used in Hydrology
Elevation Surface — the ground surface elevation at each point -- Expressed as a Digital Elevation Model for Gridded Data

4. Types of Elevation Data available
Spatial reference
Pixel size
Z units
Bit Depth
GTOPO (Global Topography)
GCS_WGS_1984
Decimal degrees
WGS 1984
30 arcsec (1 km) m
16-bit signed/unsigned Integer
SRTM (Shuttle Radar Topography Mission)
30 m
16-bit signed Integer
NED 30 (National Elevation Data)
GDC_North_America_1983
NAD 1983
1 arcsec (30 m)
Float
NED 10
1/3 arcsec (10 m)
Lidar (DEM/DSM)
NAD83_HARN_StatePlane_Oregon_North
Foot
NAD 1983 HARN
3 ft ft
The NED is assembled from approximately 57,000 files of quadrangle-based source DEMs. As source data for NED production, nearly 54,000 DEMs are used for the continental United States and about 3,000 DEMs for Alaska, Hawaii, and the island territories. Production of 7.5-minute DEMs, especially at the 10-meter posting interval,

5. Shuttle Radar Topography Mission (SRTM)
The Shuttle Radar Topography Mission collected topographic data over nearly 80 percent of Earth's land surfaces, creating the first-ever near-global data set of land elevations.

6. SRTM http://srtm.csi.cgiar.org/

7. National Elevation Dataset (NED)
Scientists and resource managers use NED data for global change research, hydrologic modeling, resource monitoring, mapping and visualization applications.
Serves the elevation layer of The National Map,
Provides basic elevation information for earth science studies and mapping applications in the United States.

8. National Elevation Dataset (NED)
Digital Elevation Model with 1 arc-second (30m) cells
Seamless in 1° blocks for the United States (larger areas are available from ESRI on-line services)
10 billion data
Derived from USGS 1:24,000 quadrangle sheets
composed of the best available raster elevation data of the conterminous United States, Alaska, Hawaii, territorial islands, Mexico and Canada.

9. The NED are derived from diverse source data that are processed to a common coordinate system and unit of vertical measure.
These data are distributed in geographic coordinates in units of decimal degrees, and in conformance with the North American Datum of 1983 (NAD 83).
All elevation values are in meters and, over the continental United States, are referenced to the North American Vertical Datum of 1988 (NAVD 88).
The vertical reference will vary in other areas.
NED data are available nationally (except for Alaska) at resolutions of 1 arc-second (approx. 30 m) and 1/3 arc-second (approx. 10 m), and in limited areas at 1/9 arc-second (approx. 3 meters).

10. Measuring in Arc-Seconds
Some USGS DEM data is stored in a block that utilizes 1/3, one, three, five, or thirty arc-seconds of longitude and latitude to register cell values.
The geographic reference system treats the globe as if it were a sphere divided into 360 equal parts called “degrees”.
Each degree is subdivided into 60 minutes.
Each minute is composed of 60 seconds.
An arc-second represents the distance of latitude or longitude traversed on the earth's surface while traveling one second (1/3600th of a degree).
At the equator, an arc-second of longitude approximately equals an arc-second of latitude, which is 1/60th of a nautical mile (30.87 meters).

11. The planet Earth is about 13,000 kilometers (8000 miles) in diameter and 40,000 kilometers (25,000 miles) in circumference.
The figure on the left is at

12. Measuring in Arc-Seconds
At the equator, an arc-second of longitude approximately equals an arc-second of latitude, which is 1/60th of a nautical mile (30.87 meters).
Arc-seconds of latitude remain nearly constant, while arc-seconds of longitude decrease as one moves toward the earth's poles.
At 49 degrees north latitude, along the northern boundary of the Concrete sheet, an arc-second of longitude equals meters * (cos 49°) or meters.

13. 3-D detail of the Tongue river at the WY/Montana border from LIDAR.
LIDAR from aircraft or from the ground can provide amazing detail on elevation, including individual tree heights and hydraulic channels
Roberto Gutierrez
University of Texas at Austin

14. Topographic Slope
We would like to trace the flow of water. One of the beautiful thing that Arc does is to help convert a DEM into water courses (stream channels)
Used to determine how (quickly) water flows downhill and concentrates into streams
Topographic slope can be determined from a DEM

15. Topographic Slope
There are three alternative sets of inputs (choose one)
Surface derivative z (dz/dx, dz/dy)
Vector with x and y components (Sx, Sy). Slope in x and y direction.
Vector with magnitude (slope) and direction (aspect) (S, )

16. ArcGIS “Slope” tool
Calculates the maximum rate of change in the elevation value from that cell to its neighbors
Calculates slope for each cell
Slope is the first derivative of a DEM (dz/dx, dz/dy)
The smaller the slope value, the flatter the terrain; the larger the slope value, the steeper the terrain.

17. Definition of X, Y, and Z in 3D space
Z axis is the direction that elevation changes (up or down)
Origin is the location of the point of interest (pixel or grid cell)
Y axis is the direction that Y has a changing value (North-South in ArcGIS)
Fundamentally, to describe earth surface, you have to define, x, y and z
As a definition, x and y go in the horizontal direction
Z is the vertical direction.
In Arc, y goes north and south
In arc, x goes east and west
I am at Chase Hall and looking at my office at Hardin Hall.
I am looking towards South West.
X axis is the direction that X has a changing value (East-West in ArcGIS)
X, and Y are horizontal distances
Z is the vertical distance
The X, Y, Z axes are at right angles to one another

18. Recall: Xo, Yo are horizontal and vertical distances
A planar coordinate system is defined by a pair
of orthogonal (x,y) axes drawn through an origin Y X
Origin
(xo,yo)
(fo,lo)
The origin can be wherever the user wishes. However, there are standard locations.

19. Slope Handout Determine the length, slope and azimuth of the line AB.
Determine the length, slope and azimuth of the line AB.
How much distance a unit of water has to travel as it travels down a slope?

20. y z b a
Slope: A line in 3D space
The simplest way to think about slope is to consider two points in a three dimensional space and a line that connects them. Suppose there are two such points, “a” and “b”, and they have coordinates (xa, ya, za,) and (xb, yb, zb,).
The vector “ab” can also be thought of as having two components, a Rise, which represents the vertical difference, dz, between the two points “a” and “b”,
and a Run = ∆𝑥2+∆𝑦2 , which represents the horizontal distance between them.
With these definitions, the actual length of the line AB, considers the change in elevation, and is given by
UTM gives us x and y coordinates so that we can calculate run
Run is the horizontal distance
DEM gives us the z coordinate

21. Example of distance and slope from Hardin Hall to Nebraska Hall
Blue line is driving distance = 1.6 miles
Yellow line is actual distance B
Hardin Hall:
m E
m N
Z = 355 m amsl
𝑅𝑢𝑛= ∆𝑥2+∆𝑦2 = m =1.34 miles
Nebraska Hall:
m E
m N
Z = 352 m amsl
∆𝑥= − = 𝑚
∆𝑦= − = 𝑚
∆𝑧=355 −352=3 𝑚

22. Two additional definitions of Slope: degrees and %
𝑇𝑎𝑛 (𝛳)= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 50
100
𝑆𝑙𝑜𝑝𝑒=𝛳=𝐴𝑟𝑐𝑇𝑎𝑛( 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 ) (0)
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝑆𝑙𝑜𝑝𝑒= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 * 100 (%)
Profile: Looking side ways at a slope
The side of the house (the wall) have infinite slope because run = 0.
Run is the horizontal distance calculated using X and Y
Rise is the vertical distance calculated using Z (elevation)
Slope ranges (-900, +900) or (-infinity %, +infinity %)

23. For the previous campus example, the three expressions for slope are:1 2 3
𝑆𝑙𝑜𝑝𝑒=𝛳=𝐴𝑟𝑐𝑇𝑎𝑛( 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 ) = ArcTan( ) =
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝑆𝑙𝑜𝑝𝑒= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 * 100 = *100 = %
Run is the horizontal distance calculated using X and Y
Rise is the vertical distance calculated using Z (elevation)
Slope ranges (-900, +900) or (-infinity %, +infinity %)
The side of the house (the wall) have infinite slope because run = 0.

24. Special Cases 𝑆𝑙𝑜𝑝𝑒 𝛳 =𝐴𝑟𝑐𝑇𝑎𝑛( 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 ) (0)
𝑆𝑙𝑜𝑝𝑒(𝛳)= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 * 100 (%)
When the angle (𝛳) is 45 degrees, the rise is equal to the run, and the percent rise (slope) is 100 percent
When the slope angle (𝛳 ) approaches vertical (90 degrees), the percent rise (slope) begins to approach infinity.
The side of the house (the wall) have infinite slope because run = 0.
Run is the horizontal distance calculated using X and Y
Rise is the vertical distance calculated using Z (elevation)

25. ArcGIS “Slope” tool g d a h e b i f c y
Calculates slope for each cell. In this illustration, it is for Cell “e”
For each cell, the Slope tool calculates the maximum of the rate of change in value from that cell to each of its eight neighbors a b c d e f g h i
The rate of change in the x direction for cell e is calculated with the following algorithm
dz dx =− a+2d+g − c+2f+i 8∆ x
The rate of change in the y direction for cell e is calculated with the following algorithm
dz dy =− g+2h+i − a+2b+c 8∆ g d a h e b i f c ∆ 2∆ x y
The negative sign in front of the equations is because x increases to the right (east) and y increases to the north. Now dz/dx is + if z increases with increasing x (to the east).
see next slide for derivation of equation

26. Explanation of Previous Equations
The two equations for dz/dx and dz/dy are simplified from the first equation below.
The basis for that equation is illustrated in the Figure and represents an average of central finite differences over each of the three rows of cells, with the middle row counting twice as it appears in averages on each side.
dz dy =− g+2h+i − a+2b+c 8∆ g d a h e b i f c ∆
𝑎−𝑐 2∆
𝑑−𝑓 2∆
𝑔−𝑖 2∆ 2∆ x y
dz dx =− a+2d+g − c+2f+i 8∆
𝑎−𝑐 2∆ + 𝑑−𝑓 2∆ 𝑑−𝑓 2∆ + 𝑔−𝑖 2∆ 2 2
dz dx = -
The equation calculates the slope of each edge separately and then averages them (for the x direction)
The negative sign in front of the equations is because we are computing uphill slope

27. Definition of Azimuth
The orientation of the land surface is defined by its aspect, which is analogous to the azimuth used in land surveying
north Δ𝑥 Δ𝑦 𝛼
𝛼 = Azimuth, angle defined as degrees clockwise from North 𝑥 𝑦
Y axis is the direction that Y has a changing value (South to North in ArcGIS)
I am standing on a hill, which direction looks the steepest. Aspect of my driveway is North East.
If you go from top of the drive way to the bottom of drive way, you change location by delta X and delta y.
Can’t see elevation here anymore because you are looking down.
east
This is my grid cell location
X axis is the direction that X has a changing value (West to East in ArcGIS)

28. Calculation of Azimuth
Solve for α by Inverting the Tangent Function (ArcTan)
north
𝑇𝑎𝑛 (𝛼)= ∆𝑋 ∆𝑌 Δ𝑥 Δ𝑦 𝛼
𝛼 = Azimuth, angle defined as degrees clockwise from North 𝑥 𝑦
𝛼=𝐴𝑟𝑐𝑇𝑎𝑛 ∆𝑋 ∆𝑌 = Azimuth
east
The other way to write ArcTan is Atan or Tan-1
Azimuth=
Convert from radians to degrees (180/π)
Azimuth is the angle between North and any desired direction you want to travel

29. ArcGIS Aspect – the steepest downslope direction
If I pour water on the ground, which direction does it flow?
Aspect is the azimuth associated with the steepest downhill slope when we are facing downhill.
Therefore, we use slopes instead of distances in the tangent function.
In Arc, with grid cells it is easiest to calculate Aspect using the ratio of slopes (dz/dx) and (dz/dy). 𝛼
Here, we are ‘looking down from above’

30. Example for topographic slope30 80 74 63 69 67 56 60 52 48 a b c d e f g h i
145.2o
Mesh spacing=30 m
What are Slope/Aspect at cell e?
Note that this is the slope in Uphill direction (it is a positive number)
(elevation decreases in
east direction)
(elev. increases in
north direction)
Converts slope from m/m to degrees (180/π)

31. Example for Aspect 30 80 74 63 69 67 56 60 52 48 Mesh spacing=30 m
Aspect at cell e? 30 80 74 63 69 67 56 60 52 48 a b c d e f g h i
145.2o
-34.8o
-34.8 shows direction of slope towards north west. And this is uphill direction.
Because of the behavior of Pythagorean theorem, the slope is always positive meaning uphill.
Water flows downhill, therefore, I am interested in direction of slope downhill. Therefore, I turn 180 degree from looking uphill
so that I am looking downhill.
One more adjustment: The above Aspect is in the direction of increasing elevation (an increase in dz with distance). We need to add 180o to this calculated aspect to get the direction of decreasing z (i.e., the steepest downhill slope)

32. The Atan function is multivalued on the full circle and only unique in a range of 180 degrees.  To unambiguously determine the direction from two components you really need the atan2 function that keeps the sign on y and x components separately. For example, let y = y component of a vector x = x component of a vector atan(x/y) gives the direction of the vector as an angle (with the ratio x/y since angle here is measured from north). 
But x/y is the same value if y is positive and x negative, or x positive and y negative.  So once you take the ratio x/y, if you get a negative number you do not know which (y or x) was negative. A way to resolve this is angle = atan(x/y) if(0 < angle < 180 and dz/dx < 0)  then   aspect = angle + 180  (flip the direction because dz/dx is negative) else   aspect = angle endif

33. D8 steepest single flow direction (Eight Direction Pour Point Model)
In reality, in a gridded system, water can only flow to one of eight adjacent cells
D8 steepest single flow direction
(Eight Direction Pour Point Model) 32 16 8 64 4
128 1 2
The direction of flow is determined by the direction of steepest descent:
Maximum_drop = (change_in_z-value / distance) * 100
This is maximum percentage drop. Defined as “Hydrologic slope” in ArcGIS
Forget the aspect calculation. And just look at slope in eight directions and take the maximum slope.
ESRI Direction encoding (ArcGIS)

34. Slope: Slope: Hydrologic Slope (Flow Direction Tool)
Find Direction of Steepest Descent (ArcGIS) 80 74 63 69 67 56 60 52 48 30 80 74 63 69 67 56 60 52 48 30
Hydrologic Slope (Flow Direction Tool)
Slope:
Slope:
For diagonal direction, the denominator for slope includes square root of 2

35. Limitation due to 8 grid directions.?
The “true” flow direction follows the red arrow. However, we can only choose one of the blue arrows because we have to use one of eight adjacent cells.

36. The D Algorithm (optional)
Tarboton, D. G., (1997), "A New Method for the Determination of Flow Directions and Contributing Areas in Grid Digital Elevation Models," Water Resources Research, 33(2): ) (

37. The D Algorithm z2 z3 zo z1 
If 1 does not fit within the triangle, the angle is chosen along the steepest edge or diagonal resulting in a slope and direction equivalent to D8

38. D∞ Example 30
284.9o z4 z3 z2 80 74 63 69 67 56 60 52 48 z5 zo z1
14.9o z6 z7 z8
The tool is available at

39. Automating Processes using Model Builder
Using a DEM tif file as input
The orange boxes are Arc tools

40. Elevation (m) for Upper Klamath Lake Basin, OR

41. Elevation Contours for Wood River Valley Watershed of Upper Klamath Lake Basin

42. Slope (%) for Upper Klamath Lake Basin, OR
(-infinity, + infinity)

43. Slope (Degree) for Upper Klamath Lake Basin, OR

44. Aspect (Degree) for Upper Klamath Lake Basin, OR

45. Flow Direction Integer raster whose values range from 1 to 255 32 16 864 4
128 1 2
What is the direction of 208?
Raster value of 256 is the same as 1 (towards East)

46. Hydrologic slope- Percentage Drop (%) for Upper Klamath

47. Hillshade
Hypothetical illumination of a surface by determining illumination values for each cell in a raster. It does this by setting a position for a hypothetical light source (sun) and calculating the illumination values of each cell in relation to neighboring cells.
Azimuth
The azimuth is the angular direction of the sun, measured from north in clockwise degrees from 0 to 360. An azimuth of 90º is east. The default azimuth is 315º (NW).
Altitude
The altitude is the slope or angle of the illumination source (sun) above the horizon. The units are in degrees, from 0 (on the horizon) to 90 (overhead). The default is 45 degrees.

48. Hillshade

49. Viewshed
The locations that are visible from a viewer location. Line of sight analysis. Useful for cell coverage and visual exposure analyses
From

50. ArcGIS.Com ready to use maps including elevation services
Elevation
Land Cover
Soils

51. Elevation Services

52. Summary Concepts
The elevation surface is represented by a gridded digital elevation model and is used to derive slope. This is important for surface flow
The eight direction pour point model approximates the surface flow direction using eight discrete grid directions.