# Source Area Model (Point)

## Footprint Model

The Footprint Model plugin calculates various morphometric parameters based on digital surface models. These morphometric parameters are:

• Mean building height (zH). The average building heights from ground in meters.
• Standard deviation of building heights (z).
• Maximum building heights (zHmax).
• Plan area index (λP). Fraction of buildings related to total ground area.
• Frontal area index (λF). Fraction of building walls in wind direction related to total ground area.
• Roughness length (z0). A parameter of some vertical wind profile equations that model the horizontal mean wind speed near the ground; in the log wind profile, it is equivalent to the height at which the wind speed theoretically becomes zero.
• Zero displacement height (zd). The height in meters above the ground at which zero wind speed is achieved as a result of flow obstacles such as trees or buildings.

The morphometric parameters above are used to describe the roughness of a surface and are included in various local and mesoscale climate models. Footprint models can be used to determine the likely position and influence of the surface area which is contributing to a turbulent flux measurement at a specific point in time and space with imposed boundary conditions (e.g. meteorological conditions, sources/sinks of passive scalars or surface characteristics). The principle of footprint models is that the measured flux is the integral of all contributing surface elements, with a ‘footprint function’ describing the relative fractional contribution of a discretisized area.

The model employed here is the analytical footprint model of Kormann and Meixner (2001). The mathematical basis of the model includes a stationary gradient diffusion formulation, height independent cross-wind dispersion, power law profiles of mean wind velocity and eddy diffusivity and a power law solution of the two-dimensional advection-diffusion equation. The final solution of the footprint function is calculated by fitting the power laws (mean wind and eddy diffusivity) to Monin-Obukhov similarity profiles. As with all models the limitations should be appreciated which include (but are not limited to) assumptions of Monin-Obukhov similarity theory, the use of power law profiles, assumptions of horizontally homogeneous flow and assumptions of stationarity during the meteorological or scalar variable input period (i.e. their averaging period; typically 30 – 60 minutes).

Preferably, a ground and 3D object DSM and a DEM should be used as input data. The 3D objects are usually buildings but can also be 3D vegetation (i.e. trees and bushes). A 3D object DSM with no ground heights makes it also possible to derive the parameters.

### Location

The Image Morphometric Parameters (Point) is located at

• UMEP
• Pre-Processor
• Urban Morphology
• Image Morphometric Parameters (Point)

## Running the Plugin

When you run the plugin, you will see the dialog shown below. It consists of four sections. The top section let you select a point on the map canvas by either clicking at a location or by selecting a "n" existing point from a point vector layer. The next section set the parameters for the area of interest where the morphometric parameters are calculated. You also set the search interval in degrees. The next section helps you to specify the input data regarding buildings and ground. The bottom section is for specifying output and for running the calculations.

### Select Point on Canvas

Click on this button to create a point from where the calculations will take place. When you click the button, the plugin will be disabled until you have clicked the map canvas.

### Use Existing Single Point Vector Layer

Tick this in is you want to use a point from a vector layer that already exist and is loaded to the QGIS-project. The Vector point layer drop down list will be enabled and include all point vector layer available.

### Raster DSM (only Building) Exist

Tick this in if a 3D-object DSM without ground heights is available. 3D objects (e.g. buildings should be meters over ground.

### Raster DSM (3D Objects and Ground)

A raster DSM (e.g. geoTIFF) consisting of ground and e.g. building height (meters above sea level).

### Raster DEM (only Ground

A DEM (e.g. geoTIFF) consisting of pixels with ground heights (meters above sea level).

### Raster DSM (only 3D Objects)

A DSM (e.g. geoTIFF) consisting of pixels with object (e.g. buildings or vegetation) heights above ground. Pixels where no objects are present should be set to zero.

### Use Input File on Specify Input Parameters

An input text file (.txt or .csv) containing the required inputs to the model (see below) with associated time stamp. An example:

iy id it imin z_0_input z_d_input z_m_input wind sigv Obukhov ustar dir 2014.00000 1.00000 0.00000 30.00000 1.16710 8.16968 42.13032 5.86567 1.48047 -5457.96436 0.84604 193.86502 2014.00000 1.00000 3.50000 30.00000 1.40072 9.80501 40.49499 3.49045 0.96155 1081.72595 0.50463 185.58742 2014.00000 1.00000 4.00000 30.00000 1.37383 9.61679 40.68321 4.02067 0.98699 854.99005 0.48490 189.04443

[Header: year, day of year, hour, minutes of averaging period, roughness length for momentum, zero plane displacement height for momentum, effective measurement height of sensor, wind speed, standard deviation of lateral wind, Obukhov length, friction velocity, wind direction] – For further descriptions and units see below

### Roughness Length for Momentum

First order estimation of roughness length for momentum (z0<\sub>) from the concerned wind direction. [m]

### Zero Displacement Height for Momentum

First order estimation of the zero-plane displacement height for momentum length (zd<\sub>) from the concerned wind direction. [m]

### Effective Height of the Measurement Data

Effective height of measurement tower = location of sensor above ground level – first order estimation of zero plane displacement height in the concerned wind direction (e.g. above). [m]

### Wind Speed

Horizontal wind speed aligned to the prevailing wind direction. [m s-1]

### Standard Deviation (sigma) of Cross Wind

Standard deviation of the wind in the y direction (lateral wind). [m s-1]

### Obukhov Length

Indication of atmospheric stability for use in Monin-Obukhov similarity theory. [m]

### FrictioN Velocity

Shear stress represented in units of velocity for non-dimensional scaling. [m s-1]

### Wind Direction

Prevailing wind direction during averaging period [Degrees].

### Maximum Fetch Considered in Meters

The furthest distance upwind considered in the calculation of the footprint function. [m]

### Roughness Calculation Method

Here, options to choose methods for roughness calculations regarding zero displacement height (zd) and roughness length (z0) are available. For more information, see the online UMEP manual:

‘RT’ – Rule of thumb (c.f. Grimmond and Oke 1998); ‘Rau’ – Raupach 1994; Bot – Bottema 1998; Mac – MacDonald et al. 1998; Mho – Millward-Hopkins et al. 2011; Kan – Kanda et al. 2013

### File Prefix

A prefix that will be included in the beginning of the output files.

### Output Folder

A specified folder where result will be saved.

### Run

This starts the calculations.

### Close

This button closes the plugin.

## Output

Two different outputs are generated:

The first is a raster grid which represents the fractional contribution of each pixel in the array to turbulent fluxes measured at the sensor (i.e. the footprint function). Each pixel of this grid will be of the same order to the input grid. Because the user can determine the maximum fetch extent that is considered, each pixel in the footprint function is weighted as a percentage of the pixel of maximum contribution. If the footprint model is set to run for more than one time period (i.e. integrated over time), the weighted footprint functions are summed. No reweighing procedure is applied to this summation.

The second output is a text file, which specifies the time dimensions of measurements, the initial aerodynamic and meteorological parameters which were input to the model and finally the weighted geometry in the footprint and thus the newly calculated roughness length (z0<\sub>) and displacement height (zd<\sub>) according to the user specified method. This is of the form:

“iy id it imin z_0_input z_d_input z_m_input wind sigv Obukhov ustar dir fai pai zH zMax zSdev zd z0”

[year, day of year, hour, minutes of averaging period, roughness length for momentum, zero plane displacement height for momentum, effective measurement height of sensor, wind speed, standard deviation of lateral wind, Obukhov length, friction velocity, wind direction, building frontal area weighted according to footprint function, building plan area weighted according to footprint, average height of buildings weighted according to footprint, maximum building height, standard deviation of building heights, footprint specific displacement height for specified method, footprint specific roughness length for specified method]

## Remarks

• All DSMs need to have the same extent and pixel size.