Handling Gridded Data:Topography and Projections

GIS in Water ResourcesFall 2014

by Ayse Kilic with materials fromDavid G. Tarboton, Utah State University and from ESRI

software

• 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

• Map Projections– State Plane Coordinate System– UTM (Universal Transverse Mercator Coordinate System)

Learning Objectives

Spatial Surfaces used in Hydrology

Elevation Surface — the ground surface elevation at each point -- Expressed as a Digital Elevation Model for Gridded Data

Types of Elevation Data availableData 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)

•GCS_WGS_1984 •Decimal degrees •WGS 1984

90 m m16-bit signed Integer

NED 30 (National Elevation Data)

•GDC_North_America_1983 •Decimal degrees •NAD 1983

1 arcsec (30 m) m Float

NED 10•GDC_North_America_1983 •Decimal degrees •NAD 1983

1/3 arcsec (10 m) m Float

Lidar (DEM/DSM)

•NAD83_HARN_StatePlane_Oregon_North •Foot •NAD 1983 HARN

3 ft ft Float

Slope Handout

Determine the length, slope and azimuth of the line AB.

http://snr.unl.edu/kilic/giswr/2014/Slope.pdf

3-D detail of the Tongue river at the WY/Montana border from LIDAR.

Roberto GutierrezUniversity of Texas at Austin

LIDAR from aircraft or from the ground can provide amazing detail on elevation, including individual tree heights and hydraulic channels

Topographic Slope

• Used to determine how (quickly) water flows downhill and concentrates into streams

• Topographic slope can be determined from a DEM

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, )

• Calculates the maximum rate of change in value from that cell to its neighbors

• Calculates for each cell• Represents the rate of change of elevation for

each digital elevation model (DEM) cell (slope). Slope is the first derivative of a DEM

• The lower the slope value, the flatter the terrain; the higher the slope value, the steeper the terrain.

ArcGIS “Slope” tool

Definition of X, Y, and Z in 3D space

Z axis is the direction that elevation changes (up or down)

Y axis is the direction that Y has a changing value (North-South in ArcGIS)X axis is the direction that

X has a changing value (East-West in ArcGIS)

Origin is the location of the point of interest (pixel or grid cell)

X, and Y are horizontal distancesZ is the vertical distance

The X, Y, Z axes are at right angles to one another

Definition of Slope

Run is the horizontal distance calculated using X and YRise is the vertical distance calculated using Z (elevation)Slope ranges (-900, +900) or (-infinity %, +infinity %)

(0)

* 100 (%)

𝑇𝑎𝑛(𝛳 )=𝑅𝑖𝑠𝑒𝑅𝑢𝑛

How Slope works

Run is the horizontal distance calculated using X and YRise is the vertical distance calculated using Z (elevation)

(0)

* 100 (%)

When the angle () is 45 degrees, the rise is equal to the run, and the percent rise is 100 percent

When the slope angle ) approaches vertical (90 degrees), the percent rise (slope) begins to approach infinity.

Pythagorean theoremUsed to calculate Run where a =ΔY and b = ΔX

The calculated “c” is the “Run”

The rate of change in the y direction for cell e is calculated with the following algorithm

ArcGIS “Slope” tool

a b c

d e f

g h ix

y

gd

ah

eb

if

c∆

2∆ xy

The rate of change in the x direction for cell e is calculated with the following algorithm

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

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.

ArcGIS “Slope” tool

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.

dzdy

=−(g+2h+i )− (a+2b+c )

8∆g

da

he

bi

fc

∆

𝑎−𝑐2∆ 𝑑− 𝑓

2∆𝑔−𝑖2∆

2∆ xy

𝑎−𝑐2 ∆

+𝑑− 𝑓

2∆2

+

𝑑− 𝑓2∆

+𝑔−𝑖2∆

22

= -

m

mdydzdxdzSlope 22 )/()/(

The negative sign in front of the equations is because we are computing uphill slope

Definition of Azimuth

Δ𝑥

Δ 𝑦 𝛼 = Azimuth, angle defined as degrees clockwise from North

𝑥

𝑦Y axis is the direction that Y has a changing value (South to North in ArcGIS)

X axis is the direction that X has a changing value (West to East in ArcGIS)

This is my grid cell location

𝑇𝑎𝑛(𝛼)=∆ 𝑋∆𝑌

𝛼=𝐴𝑟𝑐𝑇𝑎𝑛∆𝑋∆𝑌

= Azimuth  

Solve for α by Inverting the Tangent Function (ArcTan)

The other way to write ArcTan is Tan-1

Definition of Azimuth

Azimuth= Convert from radians to degrees (180/π)

Δ𝑥

Δ 𝑦 𝛼 = Azimuth, angle defined as degrees clockwise from North

𝑥

𝑦

Azimuth is the angle between North and any desired direction you want to travel

ArcGIS Aspect – the steepest downslope directionIf I pour water on the ground, which direction does it flow?

Aspect is the azimuth associated with the steepest downhill slope. 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).

dx

dz

dy

dz

dy/dz

dx/dzatan𝛼 = Aspect

Example for topographic slope

30

80 74 63

69 67 56

60 52 48

a b c

d e f

g h i

m/m 229.030*8

)4856*263()6069*280(

acing x_mesh_sp* 8

i) 2f (c - g) 2d (a

dx

dz

m/m329.030*8

)6374*280()4852*260(

acing y_mesh_sp* 8

c) 2b (a -i) 2h (g

dy

dz

o8.21180

)401.0(atan Slope

145.2o

m/m401.0

329.0)229.0()/()/( 2222

dydzdxdzSlope

Mesh spacing=30 mSlope/Aspect at cell e?

Note that this is the slope in Uphill direction (it is a positive number)

Converts slope from m/m to degrees (180/π)

Example for Aspect

30

80 74 63

69 67 56

60 52 48

a b c

d e f

g h i

m/m 229.030*8

)4856*263()6069*280(

cingx_mesh_spa * 8

i) 2f (c - g) 2d (a

dx

dz

m/m329.030*8

)6374*280()4852*260(

acing y_mesh_sp* 8

c) 2b (a -i) 2h (g

dy

dz

o

dydz

dxdzAspect 8.34

180

329.0

229.0atan

/

/atan

oo 2.1451808.34

Mesh spacing=30 mAspect at cell e?

One more adjustment: The above Aspect is in the direction of increasing elevation (increasing dz/dx). We need to add 180o to this calculated aspect to get the direction of decreasing z (i.e., the steepest downhill slope)

145.2o

-34.8o

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 vectorx = x component of a vectoratan(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 isangle = 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 = angleendif

32

16

8

64

4

128

1

2

D8 steepest single flow direction(Eight Direction Pour Point Model)

ESRI Direction encoding (ArcGIS)

In a gridded system, water can only flow to one of eight adjacent cells

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

80 74 63

69 67 56

60 52 48

30

45.0230

4867

50.0

30

5267

Slope:

Hydrologic Slope (Flow Direction Tool) Find Direction of Steepest Descent (ArcGIS)

80 74 63

69 67 56

60 52 48

30

For diagonal direction, the denominator for slope includes square root of 2

Slope:

?

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.

Flowdirection.

Steepest directiondownslope

1

2

1

234

5

67

8

Proportion flowing toneighboring grid cell 3is 2/(1+2)

Proportionflowing toneighboringgrid cell 4 is

1/(1+2)

The D Algorithm

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): 309-319.) (http://www.engineering.usu.edu/cee/faculty/dtarb/dinf.pdf)

Steepest direction downslope

1

2

1

2 3

4

5

6 7

8

0

The D Algorithm

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

10

211 atan

zz

zz

2

10

2

21

zzzz

S

zo z1

z2z3

D∞ Example30

zo

z7 z8

o

zz

zz

9.145267

4852atan

atan70

871

14.9o284.9o

517.0

30

5267

30

4852S

22

80 74 63

69 67 56

60 52 48

The tool is available at http://hydrology.usu.edu/taudem/taudem5/documentation.html

2

10

2

21

zzzz

S

z1

z2z3z4

z5

z6

Automating Processes using Model Builder

Using a DEM tif file as input

Elevation (m) for Upper Klamath Lake Basin, OR

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

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

Slope (Degree) for Upper Klamath Lake Basin, OR

Aspect (Degree) for Upper Klamath Lake Basin, OR

Percentage Drop (Degree) for Upper Klamath

Flow Direction Integer raster whose values range from 1 to 255

32

16

8

64

4

128

1

2

Hillshade

AzimuthThe 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).

AltitudeThe altitude is the slope or angle of the illumination source above the horizon. The units are in degrees, from 0 (on the horizon) to 90 (overhead). The default is 45 degrees.

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 and calculating the illumination values of each cell in relation to neighboring cells.

Hillshade

Map Projection Parameters

39

State Plane Coordinate Systems

• Like UTM but customized to minimize error within States-1930

• NAD27 – feet • NAD83 – feet or meters

State Plane Coordinate Systems

40

• TM for North South oriented states• Lambert Conformal Conic- East West orientation• Depending on TM or LCC, zones baselines and meridians positioned differently• Baselines are placed under the zones Southern Border and measured North from

there

State Plane zones

State plane zones of Minnesota (1/6 of the zone width) and details of standardparallel placement (Lambert Conformal Conic)

State Plane Coordinate System-overview

• Used by state/local governments– Units are feet (now appearing in meters)

• Panhandle of Alaska: oblique Mercator (at angle)• N/S states (Missouri): Transverse Mercator projection• E/W states: Lambert Conformal Conic for an East-West

State (California, Nebraska)• Doesn’t make sense?

– States divided into smaller zones– Distortion is extremely small for local area

• Coordinates: SW corner is 0, 0 .• Mapping large areas (over >1 zone) is trouble!

What are the coordinates of the origin (fo, lo) and the corresponding (Xo, Yo) ?

(fo, lo) = (39° 50’ N, 100° W)(lo, fo) = (100° W, 39° 50’ N)(Xo, Yo) = (1640416.667, 0.000) 

Don’t get confused. “X” is always associated with Longitude, “Y” with Latitude

Zones in the Texas State Plane

46

• Line of intersection at a central meridian. • Distances measured with respect to the central

meridian and from the equator in meters• 60 zones around the Earth from East to West• False Northing and False Easting

Universal Transverse Mercator Coordinate System

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• False Easting to avoid negative values: Central meridian is denoted as 500,000 not “0”

• Measurements are shown as positive• So if a point is 11,254 meters west of the

central meridian, it is shown as 500,000-11,254 or 488,746.

• False northing. In the southern hemisphere, 10,000,000 is the value assigned to the equator. So something 120,000 meters south of the equator is depicted as 9,880,000

Universal Transverse Mercator Coordinate System

Word in UTM GridsUTM zones – lower 48 states

Universal Transverse Mercator (UTM Zone 14)

Elevation (feet)

UTM (Location of N at Memorial football stadium)

(Xo, Yo) = (693,500 4,521,383) Meters

Base map (streets) is from ArcGIS.com

ArcGIS.Com ready to use maps including elevation services

http://www.arcgis.com/features/maps/earth.html

Land Cover

Soils

Elevation

Elevation Services

http://elevation.arcgis.com/arcgis/services

Summary Concepts• The elevation surface represented by a grid digital

elevation model is used to derive slope important for surface flow

• The eight direction pour point model approximates the surface flow using eight discrete grid directions.

• The D vector surface flow model approximates the surface flow as a flow vector from each grid cell apportioned between down slope grid cells.