Post on 25-Dec-2015
Spatial Analysis Using GridsGIS in Water Resources
Fall 2014
by Ayse Kilic with materials fromDavid G. Tarboton, Utah State University and from ESRI
software
Spatial Analysis Using Grids
• Continuous surfaces or spatial fields representation of geographical information
• Grid data structure for representing numerical and categorical data
• Map algebra raster calculations • Interpolation • Calculate 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
Learning Objectives
• http://resources.arcgis.com/en/help/main/10.2/#/Raster_dataset_zones_and_regions/009t00000008000000/ Raster and Images, starting from "Introduction/What is raster data" to end of " Fundamentals of raster data/Rasters with functions"
Readings – at http://resources.arcgis.com/en/help/
Readings – at http://resources.arcgis.com/ • What is the ArcGIS Spatial Analyst extension and Essential ArcGIS Spatial
Analyst extension vocabulary http://resources.arcgis.com/en/help/main/10.2/index.html#/What_is_the_ArcGIS_Spatial_Analyst_extension/005900000001000000/
x
dx)y,x(f)y(f
x
y
f(x,y)
Two fundamental ways of representing geography are discrete (separated) objects and fields.
The discrete object view represents the real world as objects with well defined boundaries in empty space.
The field view represents the real world as a finite number of variables, each one defined at each possible position.
(x1,y1)
Points Lines Polygons
Continuous surface
Numerical representation of a spatial surface (field)
Grid or
Raster
TIN Contour and flowline
Fields can represent any number of two or three dimensional surfaces (any kind of gridded data)
Discrete (vector) and continuous (raster) data
Images from http://resources.arcgis.com/en/help/main/10.1/index.html#/Discrete_and_continuous_data/009t00000007000000/
Discrete is also called thematic, categorical, or discontinuous data
Discrete
Continuous
Continuous
Raster and Vector Data
Point
Line
Polygon
Vector Raster
Raster data are described by a cell grid, one value per cell
Zone of cells
Line as a Sequence of Cells
How do we define a line (stream) using grid cells?
Polygon as Zone of Grid Cells
http://resources.arcgis.com/en/help/main/10.2/index.html#/How_features_are_represented_in_a_raster/009t00000006000000/
When we use grid cells, we have to break down polygon shapes like lakes into individual grid cellsWe start on the left with a accurate picture of well-defined boundaries and we end up with a pixelated view on the right
Raster and Vector are two methods of representing geographic data in GIS
• Both represent different ways to encode and generalize geographic phenomena
• Both can be used to code both fields and discrete objects
• In practice a strong association between raster and fields and vector and discrete objects
A grid defines geographic space as a mesh of identically-sized square cells. Each cell holds a numeric value that measures a
geographic attribute (like elevation) for that unit of space.
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Elevation (meter) for part of Upper Klamath Basin, OR
The grid values
There are two types of grids: integer and floating pointUse integer grids to represent discrete data (land use classes)Use floating-point grids to represent continuous data (elevation) Grids may have associated attribute table
Attribute tables describe overall characteristics for the entire grid
The grid• Grid is defined by
extent, spacing, and perhaps a no data value
• Extent is – the Number of rows,
Number of columns– Top-left and bottom-
right coordinates
• Spacing is the Cell size (X and Y)
• NODATA is outside the extent or is undefined
Numberof
rows
Number of Columns
(X,Y) Cell sizeNODATA cell
NODATA Cells
Networks Using Cells
The same number identifies cells that are in the same reach
Grids Using Floating Point (Decimal Numbers)
Continuous data surfaces using floating point or decimal numbersExamples are ET (evapotranspiration), elevation, precipitation
Integer valued grids to represent zones
Examples are Landuse classes, soil types, political units
Value Attribute Table (VAT) to summarize categorical (integer) grid data
Attributes of grid zones
How are values for Rasters determined?
from Michael F. Goodchild. (1997) Rasters, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u055/u055.html, posted October 23, 1997
There are three ways that a “Value” for a grid cell is determined
Cell size of raster data
The cell must be small enough to capture the required detail but large enough so computer storage and analysis can be performed efficiently.
Discrete data
The level of detail (of features/phenomena) represented by a raster is dependent on the cell size. (Cell size matters!!!) Estimates of area
Integer Raster Generalization How do we determine the value for the cell?
Central point ruleLargest share rule
Whatever is happening at the centerWinner takes it all!
Map Algebra/Raster Calculation
Precipitation-
Losses (Evaporation + Infiltration)
=Runoff5 2
2 3
2 43 3
7 65 6
-
=
Cell by cell evaluation of mathematical functions
Example
5-3= 2
Runoff generation processes
Infiltration excess overland flowaka Horton overland flow
Partial area infiltration excess overland flow
Saturation excess overland flow
PP
P
qrqs
qo
PP
P
qo
f
PP
P
qo
f
f
Runoff generation at a point depends on
• Rainfall intensity or amount• Antecedent moisture conditions• Soils and vegetation type• Depth to water table (topography)• Time scale of interest
These factors vary spatially. That suggests using a spatial approach (GIS) to runoff estimation
Cell based generation of runoff and routing discharge
Radar Precipitation grid
Soil and land use grid
Runoff grid from raster-based calculations implementing runoff generation formulas
We use a DEM to show accumulation and routing of runoff within watersheds
Raster calculation – some complications
+
Sometimes different layers have different:
• Cell size
• Extent
• Projection
We need to tell Arc how to combine these data layers. We may need to reproject, and resample to a common cell size.
Raster calculation – some complications
Analysis extent
+
=
Analysis cell size
Analysis mask (Domain)
An Analysis Mask is used to tell Arc how to resample or interpolate (and reproject) inputs to the target extent, cell size, and projection
Spatial Snowmelt Raster Calculation Example
The grids below depict initial snow depth and average temperature over a day for an area.
40 50 55
42 47 43
42 44 41
100 m
100
m
(a) Initial snow depth (cm)
4 6
2 4
150 m
150
m
(b) Temperature (oC)
One way to calculate decrease in snow depth due to melt is to use a temperature index model that uses the formula
TmDD oldnew
Here Dold and Dnew give the snow depth at the beginning and end of a time step, T gives the temperature and m is a melt factor. Assume melt factor m = 0.5 cm/OC/day. Calculate the snow depth at the end of the day.
40 50 55
4347
414442
42
100 m
100
m
4
2 4
6
150 m
150
m
Lets Experiment with this in ArcGIS
temp.asc
ncols 2nrows 2xllcorner 0yllcorner 0cellsize 150NODATA_value -99994 62 4
snow.asc
ncols 3nrows 3xllcorner 0yllcorner 0cellsize 100NODATA_value -999940 50 5542 47 4342 44 41
New depth calculation using Raster Calculator
“snow100” - 0.5 * “temp150”
Environment SettingsTell Arc how to resample or interpolate (and reproject) inputs to the target extent, cell size
Example and Pixel Inspector
38 52
41 39
The Result for Snow Depth on Day 2
• Outputs are on 150 m grid.
• How were values obtained ?
When input layers have different cell sizes
40 50 55
4347
414442
42
100
m
4
2 4
6150
m
38 52
41 39
When we have two different input cell sizes, we have to specify one of them for the output cell size. If you specify using the layer having the maximum cell size, then the input layers having the smaller cell size will be resampled. The default resampling in Arc uses Nearest Neighbor
The NN resampling grabs the input cell that is closest to the center of the output cell
150
m
See next slide
Output raster
Nearest Neighbor (NN) Resampling using Cell size based on “Maximum of Inputs”
40 50 55
4347
414442
42
100
m
40 55
42 41
The NN resampling grabs the input cell that is closest to the center of the output cell
There are Nine 100 m cells . Only 4 of them will be selected for the calculations (because we use NN). Note that some spatial information is lost during resampling.
Red symbols are centers of the 150 m cells
Resampling the 100 meter input to 150 meters using NN
40 55
42 41
Nearest Neighbor (NN) Resampling using Cell size based on “Maximum of Inputs”
4
2 4
6150
m
40-0.5*4 = 38
55-0.5*6 = 5238 52
41 39
42-0.5*2 = 41
41-0.5*4 = 39
However, only 4 of original snow depths were utilized because we use NN.
Now we can make calculations with all the inputs and outputs at 150 meter
Nearest Neighbor (NN) Resampling using Cell size based on “Maximum of Inputs”
40 50 55
4347
414442
42
100
m
4
2 4
6150
m
40-0.5*4 = 38
55-0.5*6 = 5238 52
41 39
42-0.5*2 = 41
41-0.5*4 = 39
We could have combined the resampling and calculations into one step as shown here (this is what Arc Does)
There are Nine 100 m cells Only 4 of them will be selected for the calculations (because we use NN)
Red symbols are centers of the 150 m cells
From: Blöschl, G., (1996), Scale and Scaling in Hydrology, Habilitationsschrift, Weiner Mitteilungen Wasser Abwasser Gewasser, Wien, 346 p.
Cell size and sampling frequency need to agree with the behavior of our data
We need to sample more frequently to capture high frequency changes
We need to sample a longer series to capture the full data extent
We need to use a smaller cell size to capture the full range of values
Interpolation and Resampling• Estimate values between known values• A set of spatial analyst functions that predict values for a surface
from a limited number of sample points creating a continuous raster.
Apparent improvement in resolution may not be justified
Interpolation methods
• Nearest neighbor• Inverse distance
weight• Bilinear
interpolation• Kriging (best linear
unbiased estimator)• Spline• Cubic convolution
ii
zr
1z
)dyc)(bxa(z
iizwz
ii eei yxcz
Known data points are large red circles
(For resampling only)
Resampling Methods in Arc
4 6
2 4
5
3
3
54
4 6
2 4
Spacing & Support
Lets redo the calculations at 100 m resolution
Resample 150 m to 100 m to get consistent cell size
Black values are originalThe white values are resampled
Calculation with consistent 100 m cell size grid“snow100” - 0.5 * “temp100”
38 52
41 39
47.5
40.5
42.5
40.545
Cell by cell snowmelt calculations using 100 m cell size
40 50 55
4347
414442
42
100
m15
0 m
40-0.5*4 = 38
42-0.5*3 = 40.538 52
41 39
43-0.5*5 = 40.5
41-0.5*4 = 39
47.5
40.5 45 40.5
42.5
50-0.5*5 = 47.5
55-0.5*6 = 52
47-0.5*4 = 45
42-0.5*2 = 41
44-0.5*3 = 42.54
6
2 4
5
3
3
54
4
6
2 4 Two inputs on the left have the same 100 m resolution
Illustration of different Interpolation techniques
When you have to create a grid from a limited set of data points
Nearest Neighbor “Thiessen” Polygon Interpolation Spline Interpolation
Point to Raster Interpolation
Grayson, R. and G. Blöschl, ed. (2000)
Interpolation Comparison
ThiessenThin plate spline
Inverse Distance Squared Kriging (zero nugget, large range)
Sixth order polynomial Kriging (different nugget and range paramters)
Further Reading
Grayson, R. and G. Blöschl, ed. (2000), Spatial Patterns in Catchment Hydrology: Observations and Modelling, Cambridge University Press, Cambridge, 432 p.
Chapter 2. Spatial Observations and Interpolation
http://www.catchment.crc.org.au/special_publications1.html
Full text online at:
Spatial Surfaces used in Hydrology
Elevation Surface — the ground surface elevation at each point
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
Topographic Slope
• Used to determine how 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
• Calculated for each cell• Represents the rate of change of elevation for
each digital elevation model (DEM) cell. It's 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
Definition of Aspect
Δ𝑥
Δ 𝑦 𝛼 = aspect, angle defined as degrees clockwise from North
𝑥
𝑦
If I pour water on the ground, which direction does it flow?Identifies the direction where slope is maximum (slope is the steepest)Aspect identifies the downslope direction of the maximum rate of change in value from each cell to its neighbors
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)
This is my grid cell location
𝑇𝑎𝑛(𝛼)=∆ 𝑋∆𝑌
𝛼=𝐴𝑟𝑐𝑇𝑎𝑛∆𝑋∆𝑌
= Aspect
Invert that top equation and solve for α by Inverting the Tangent Function (ArcTan)
The other way to write ArcTan is Tan-1
Definition of Aspect
Aspect = Convert from radians to degrees (180/π)
Δ𝑥
Δ 𝑦 𝛼 = aspect, angle defined as degrees clockwise from North
𝑥
𝑦
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
dzd y
=(g+2h+i )− (a+2b+c )
8 ∆
ArcGIS “Slope” tool
gd
ah
eb
if
c∆
𝑎−𝑐2∆ 𝑑− 𝑓
2∆𝑔−𝑖2∆
2∆ xy
𝑎−𝑐2 ∆
+𝑑− 𝑓2∆
2+
𝑑− 𝑓2∆
+𝑔−𝑖2∆
22
The two equations for dz/dx and dz/dy are simplified from this first equation.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.
=
ArcGIS Aspect – the steepest downslope direction
dx
dz
dy
dz
dy/dz
dx/dzatan
In Arc, with grid cells it is easiest to calculate Aspect using the ratio of dz/dx to dz/dy.We use Arc tangent function to determine the aspect angle.
𝛼
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(
cingy_mesh_spa * 8
c) 2b (a-i) 2h (g
dy
dz
o8.21)401.0(atan
145.2o
m/m401.0
329.0229.0)/()/( 2222
dydzdxdzSlope
Mesh spacing=30 mSlope/Aspect at cell e?
Slope in degrees
Example for Aspect30
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(
cingy_mesh_spa * 8
c) 2b (a-i) 2h (g
dy
dz
o
dydz
dxdzAspect 8.34
329.0
229.0atan
/
/atan
oo 2.145180
145.2o
Mesh spacing=30 mSlope/Aspect at cell e?
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.
Specifically, 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 x < 0) then aspect = angle + 180 # flip the direction because x is negative if angle is in positive x directionelse 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
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 ee
eeatan
210
221 eeee
S
D∞ Example30
eo
e7 e8
o
70
871
9.145267
4852atan
ee
eeatan
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
210
221 eeee
S
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
CyberGIS Apps
http://sandbox.cigi.illinois.edu/home/
Big Spatial Data
77
A YouTube Videohttp://www.youtube.com/watch?v=hrJ_cZkG-Xs
Viewshed
From http://www.innovativegis.com/basis/mapanalysis/Topic15/Topic15.htm
The locations that are visible from a viewer location. Line of sight analysis. Useful for cell coverage and visual exposure analyses
Summary Concepts
• Grid (raster) data structures represent surfaces as an array of grid cells
• Raster calculation involves algebraic like operations on grids
• Interpolation and Generalization is an inherent part of the raster data representation
Summary Concepts (2) • 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.