Face Area Calculations

7. Demonstrate the Need for Area Correction During Zonal Averaging Calculations Etc.#

When performing zonal averaging, it is common practice to calculate the intersection of faces with lines of constant latitude and longitude.

However, this approach can introduce small errors if the correct area adjustments are not applied.

Below is a demonstration of the impact of such errors:

# Define the latitude and longitude points
lon1, lon2, lon3, lon4 = -97.5, -87.5, -82.5, -92.5
lat1, lat2, lat3, lat4 = 32.0, 31.0, 35.0, 36.0

# Define the rectangle center and midlines
rect_center_lon = -90
rect_center_lat = 34

# Define intersection points computed with rect_center defined above
int_14 = (-95.0, 34.0)
int_23 = (-83.75, 34.0)
int_12 = (-90.0, 31.25)
int_34 = (-90.0, 35.75)

# Define nodes for polygons
node_lon = np.array(
    [
        int_14[0],
        rect_center_lon,
        int_12[0],
        lon1,  # Polygon 1
        int_12[0],
        rect_center_lon,
        int_23[0],
        lon2,  # Polygon 2
        int_23[0],
        rect_center_lon,
        int_34[0],
        lon3,  # Polygon 3
        int_34[0],
        rect_center_lon,
        int_14[0],
        lon4,  # Polygon 4
    ]
)

node_lat = np.array(
    [
        int_14[1],
        rect_center_lat,
        int_12[1],
        lat1,  # Polygon 1
        int_12[1],
        rect_center_lat,
        int_23[1],
        lat2,  # Polygon 2
        int_23[1],
        rect_center_lat,
        int_34[1],
        lat3,  # Polygon 3
        int_34[1],
        rect_center_lat,
        int_14[1],
        lat4,  # Polygon 4
    ]
)

# Define face-node connectivity (each polygon is a separate face)
face_node_connectivity = np.array(
    [
        [0, 1, 2, 3],  # Polygon 1
        [4, 5, 6, 7],  # Polygon 2
        [8, 9, 10, 11],  # Polygon 3
        [12, 13, 14, 15],  # Polygon 4
    ]
)

# Create UXArray grid for four small polygons
faces = ux.Grid.from_topology(
    node_lon=node_lon,
    node_lat=node_lat,
    face_node_connectivity=face_node_connectivity,
    fill_value=-1,
)

# Compute the area of each polygon
small_areas = faces.compute_face_areas(quadrature_rule="gaussian", order=4)
corrected_small_areas = faces.compute_face_areas(
    quadrature_rule="gaussian", order=4, latitude_adjusted_area=True
)
faces.plot(
    backend="bokeh", title="Four Polygons With One Edge on a Line of Constant Latitude"
)

This table demonstrates that when intersections with lines of constant latitude and longitude are performed, small discrepancies in area calculation can occur. While these errors may seem minor for individual polygons, they can accumulate over a larger dataset, leading to noticeable inaccuracies in zonal averaging. Applying a correction ensures the computed areas remain consistent and accurate. It must be noted that the correction is positive for some faces and negative for others causing the total sum to remain the same.

Polygon	Original Area	Corrected Area	Percentage Difference
Polygon 1	0.00387988	0.00385858	-0.5488%
Polygon 2	0.00329592	0.00325432	-1.2621%
Polygon 3	0.00245684	0.00249844	1.6932%
Polygon 4	0.00178601	0.00180730	1.1922%

Total Area Comparison#

Total Area (4 Polygons)	Corrected Total Area	Percentage Difference
0.01141865	0.01141865	0.0000%

# Print area of all 4 faces:
print("Area of each polygon:")
for i, area in enumerate(small_areas[0]):
    print(
        f"Polygon {i + 1}: {area:.8f}, Corrected Area: {corrected_small_areas[0][i]:.8f}, Percentage Difference: {(corrected_small_areas[0][i] - area) / area * 100:.4f}%"
    )

# Print the sum of areas of 4 faces is same for corrected and uncorrected areas
print(
    f"Total area of 4 polygons: {small_areas[0].sum():.8f}, Corrected total area of 4 polygons: {corrected_small_areas[0].sum():.8f}, Percentage Diff: {(corrected_small_areas[0].sum() - small_areas[0].sum()) / small_areas[0].sum() * 100:.4f}%"
)

Area of each polygon:
Polygon 1: 0.00387988, Corrected Area: 0.00385858, Percentage Difference: -0.5488%
Polygon 2: 0.00329592, Corrected Area: 0.00325432, Percentage Difference: -1.2621%
Polygon 3: 0.00245684, Corrected Area: 0.00249844, Percentage Difference: 1.6932%
Polygon 4: 0.00178601, Corrected Area: 0.00180730, Percentage Difference: 1.1922%
Total area of 4 polygons: 0.01141865, Corrected total area of 4 polygons: 0.01141865, Percentage Diff: -0.0000%

In the code snippet below, we compute the area of larger faces that do not contain edges along lines of constant latitude and compare it with the sum of the areas calculated above. This comparison demonstrates the necessity of applying area corrections when a face includes an edge that follows a line of constant latitude. Without such corrections, errors in area estimation may arise, potentially impacting calculations that rely on precise spatial measurements.

large_node_lon = np.array([lon1, lon2, lon3, lon4])
large_node_lat = np.array([lat1, lat2, lat3, lat4])
large_face_node_connectivity = np.array([[0, 1, 2, 3]])

large_face = ux.Grid.from_topology(
    node_lon=large_node_lon,
    node_lat=large_node_lat,
    face_node_connectivity=large_face_node_connectivity,
    fill_value=-1,
)

large_area = large_face.compute_face_areas(quadrature_rule="gaussian", order=4)
large_face.plot()
print(f"Area of the face is: {large_area[0][0]:.8f}")
large_face.plot(
    backend="bokeh",
    title="Large Face With No Edges on a Line of Constant Latitude (No Area Correction)",
)

Area of the face is: 0.01143206

percentage_diff = (large_area[0][0] - small_areas[0].sum()) / large_area[0][0] * 100
print(
    f"Percentage difference between large face and sum of small faces: {percentage_diff:.4f}%"
)

Percentage difference between large face and sum of small faces: 0.1173%

Face Area Calculations

Contents

Face Area Calculations#

1. Calculate Total Face Area#

2. Quadrature and Order Options for `Grid.calculate_total_face_area` Function#

3. Getting Area of Individual Faces#

4. Calculate Area of a Single Triangle in Cartesian Coordinates#

5. Calculate Area from Multiple Faces in Spherical Coordinates#

6. Area Correction#

Where:#

Assumptions:#

The following code:#

7. Demonstrate the Need for Area Correction During Zonal Averaging Calculations Etc.#

Total Area Comparison#

Face Area Calculations

Contents

Face Area Calculations#

1. Calculate Total Face Area#

2. Quadrature and Order Options for Grid.calculate_total_face_area Function#

3. Getting Area of Individual Faces#

4. Calculate Area of a Single Triangle in Cartesian Coordinates#

5. Calculate Area from Multiple Faces in Spherical Coordinates#

6. Area Correction#

Where:#

Assumptions:#

The following code:#

7. Demonstrate the Need for Area Correction During Zonal Averaging Calculations Etc.#

Total Area Comparison#

2. Quadrature and Order Options for `Grid.calculate_total_face_area` Function#