```
import numpy as np
import shapely
import geopandas as gpd
import topojson as tp
import rasterio
import rasterio.warp
import rasterio.plot
import rasterio.mask
import sys
```

# 4 Geometry operations

## Prerequisites

This chapter requires importing the following packages:

It also relies on the following data files:

```
= gpd.read_file('data/seine.gpkg')
seine = gpd.read_file('data/us_states.gpkg')
us_states = gpd.read_file('data/nz.gpkg')
nz = rasterio.open('data/dem.tif')
src = rasterio.open('output/elev.tif') src_elev
```

## 4.1 Introduction

So far the book has explained the structure of geographic datasets (Chapter 1), and how to manipulate them based on their non-geographic attributes (Chapter 2) and spatial relations (Chapter 3). This chapter focuses on manipulating the geographic elements of geographic objects, for example by simplifying and converting vector geometries, and by cropping raster datasets. After reading it you should understand and have control over the geometry column in vector layers and the extent and geographic location of pixels represented in rasters in relation to other geographic objects.

Section 4.2 covers transforming vector geometries with âunaryâ and âbinaryâ operations. Unary operations work on a single geometry in isolation, including simplification (of lines and polygons), the creation of buffers and centroids, and shifting/scaling/rotating single geometries using âaffine transformationsâ (Section 4.2.1 to Section 4.2.4). Binary transformations modify one geometry based on the shape of another, including clipping and geometry unions, covered in Section 4.2.5 and Section 4.2.7, respectively. Type transformations (from a polygon to a line, for example) are demonstrated in Section Section 4.2.8.

Section 4.3 covers geometric transformations on raster objects. This involves changing the size and number of the underlying pixels, and assigning them new values. It teaches how to change the extent and the origin of a raster âmanuallyâ (Section 4.3.2), how to change the resolution in fixed âstepsâ through aggregation and disaggregation (Section 4.3.3), and finally how to resample a raster into any existing template, which is the most general and often most practical approach (Section 4.3.4). These operations are especially useful if one would like to align raster datasets from diverse sources. Aligned raster objects share a one-to-one correspondence between pixels, allowing them to be processed using map algebra operations (Section 3.3.3).

In the next chapter (Chapter 5), we deal with the special case of geometry operations that involve both a raster and a vector layer together. It shows how raster values can be âmaskedâ and âextractedâ by vector geometries. Importantly it shows how to âpolygonizeâ rasters and ârasterizeâ vector datasets, making the two data models more interchangeable.

## 4.2 Geometric operations on vector data

This section is about operations that in some way change the geometry of vector layers. It is more advanced than the spatial data operations presented in the previous chapter (in Section 3.2), because here we drill down into the geometry: the functions discussed in this section work on the geometric part (the geometry column, which is a `GeoSeries`

object), either as standalone object or as part of a `GeoDataFrame`

.

### 4.2.1 Simplification

Simplification is a process for generalization of vector objects (lines and polygons) usually for use in smaller scale maps. Another reason for simplifying objects is to reduce the amount of memory, disk space and network bandwidth they consume: it may be wise to simplify complex geometries before publishing them as interactive maps. The **geopandas** package provides the `.simplify`

method, which uses the GEOS implementation of the Douglas-Peucker algorithm to reduce the vertex count. `.simplify`

uses the `tolerance`

to control the level of generalization in map units (Douglas and Peucker 1973).

For example, a simplified geometry of a `'LineString'`

geometry, representing the river Seine and tributaries, using tolerance of `2000`

meters, can be created using the `seine.simplify(2000)`

command (Figure 4.1).

```
= seine.simplify(2000)
seine_simp ;
seine.plot(); seine_simp.plot()
```

The resulting `seine_simp`

object is a copy of the original `seine`

but with fewer vertices. This is apparent, with the result being visually simpler (Figure 4.1, right) and consuming about twice less memory than the original object, as shown in the comparison below.

```
print(f'Original: {sys.getsizeof(seine)} bytes')
print(f'Simplified: {sys.getsizeof(seine_simp)} bytes')
```

```
Original: 374 bytes
Simplified: 188 bytes
```

Simplification is also applicable for polygons. This is illustrated using `us_states`

, representing the contiguous United States. As we show in Chapter 6, for many calculations **geopandas** (through **shapely**, and, ultimately, GEOS) assumes that the data is in a projected CRS and this could lead to unexpected results when applying distance-related operators. Therefore, the first step is to project the data into some adequate projected CRS, such as US National Atlas Equal Area (EPSG:`9311`

) (on the left in Figure Figure 4.2), using `.to_crs`

(Section 6.7).

`= us_states.to_crs(9311) us_states9311 `

The `.simplify`

method from **geopandas** works the same way with a `'Polygon'`

/`'MultiPolygon'`

layer such as `us_states9311`

:

`= us_states9311.simplify(100000) us_states_simp1 `

A limitation with `.simplify`

, however, is that it simplifies objects on a per-geometry basis. This means the âtopologyâ is lost, resulting in overlapping and âholeyâ areal units as illustrated in Figure 4.2 (b). The `.toposimplify`

method from package **topojson** provides an alternative that overcomes this issue. By default it uses the Douglas-Peucker algorithm like the `.simplify`

method. However, another algorithm, known as Visvalingam-Whyatt, which overcomes some limitations of the Douglas-Peucker algorithm (Visvalingam and Whyatt 1993), is also available in `.toposimplify`

. The main advanatage of `.toposimplify`

is that it is topologically âawareâ: it simplifies the combined borders of the polygons (rather than each polygon on its own), thus ensuring that the overlap is maintained. The following code chunk uses `.toposimplify`

to simplify `us_states9311`

. Note that, when using the **topojson** package, we first need to calculate a âtopologyâ object, using function `tp.Topology`

, and then apply the sumplification function, such as `.toposimplify`

, to obtain a simplified layer. We are also using the `.to_gdf`

method to return a `GeoDataFrame`

.

```
= tp.Topology(us_states9311, prequantize=False)
topo = topo.toposimplify(100000).to_gdf() us_states_simp2
```

Figure 4.2 compares the original input polygons and two simplification methods applied to `us_states9311`

.

```
='lightgrey', edgecolor='black');
us_states9311.plot(color='lightgrey', edgecolor='black');
us_states_simp1.plot(color='lightgrey', edgecolor='black'); us_states_simp2.plot(color
```

**geopandas**

**topojson**

**geopandas**(middle), and

**topojson**(right), packages.

### 4.2.2 Centroids

Centroid operations identify the center of geographic objects. Like statistical measures of central tendency (including mean and median definitions of âaverageâ), there are many ways to define the geographic center of an object. All of them create single point representations of more complex vector objects.

The most commonly used centroid operation is the geographic centroid. This type of centroid operation (often referred to as âthe centroidâ) represents the center of mass in a spatial object (think of balancing a plate on your finger). Geographic centroids have many uses, for example to create a simple point representation of complex geometries, to estimate distances between polygons, or to specify the location where polygon text labels are placed. Centroids of the geometries in a `GeoSeries`

or a `GeoDataFrame`

are accessible through the `.centroid`

property, as demonstrated in the code below, which generates the geographic centroids of regions in New Zealand and tributaries to the River Seine (black points in Figure 4.3).

```
= nz.centroid
nz_centroid = seine.centroid seine_centroid
```

Sometimes the geographic centroid falls outside the boundaries of their parent objects (think of vector data in shape of a doughnut). In such cases âpoint on surfaceâ operations, created with the `.representative_point`

method, can be used to guarantee the point will be in the parent object (e.g., for labeling irregular multipolygon objects such as island states), as illustrated by the red points in Figure 4.3. Notice that these red points always lie on their parent objects.

```
= nz.representative_point()
nz_pos = seine.representative_point() seine_pos
```

The centroids and points in surface are illustrated in Figure 4.3.

```
# New Zealand
= nz.plot(color='white', edgecolor='lightgrey')
base =base, color='None', edgecolor='black')
nz_centroid.plot(ax=base, color='None', edgecolor='red');
nz_pos.plot(ax# Seine
= seine.plot(color='grey')
base =base, color='None', edgecolor='red')
seine_pos.plot(ax=base, color='None', edgecolor='black'); seine_centroid.plot(ax
```

### 4.2.3 Buffers

Buffers are polygons representing the area within a given distance of a geometric feature: regardless of whether the input is a point, line or polygon, the output is a polygon (when using positive buffer distance). Unlike simplification, which is often used for visualization and reducing file size, buffering tends to be used for geographic data analysis. How many points are within a given distance of this line? Which demographic groups are within travel distance of this new shop? These kinds of questions can be answered and visualized by creating buffers around the geographic entities of interest.

Figure 4.4 illustrates buffers of two different sizes (5 and 50 \(km\)) surrounding the river Seine and tributaries. These buffers were created with commands below, using the `.buffer`

method, applied to a `GeoSeries`

or `GeoDataFrame`

. The `.buffer`

method requires one important argument: the buffer distance, provided in the units of the CRS, in this case, meters (Figure 4.4).

```
= seine.buffer(5000)
seine_buff_5km = seine.buffer(50000)
seine_buff_50km ='none', edgecolor=['c', 'm', 'y']);
seine_buff_5km.plot(color='none', edgecolor=['c', 'm', 'y']); seine_buff_50km.plot(color
```

Note that both `.centroid`

and `.buffer`

return a `GeoSeries`

object, even when the input is a `GeoDataFrame`

.

` seine_buff_5km`

```
0 POLYGON ((657550.332 6852587.97...
1 POLYGON ((517151.801 6930724.10...
2 POLYGON ((701519.740 6813075.49...
dtype: geometry
```

In the common scenario when the original attributes of the input features need to be retained, you can replace the existing geometry with the new `GeoSeries`

by creating a copy of the original `GeoDataFrame`

and assigning the new buffer `GeoSeries`

to the `geometry`

column.

```
= seine.copy()
seine_buff_5km = seine.buffer(5000)
seine_buff_5km.geometry seine_buff_5km
```

name | geometry | |
---|---|---|

0 | Marne | POLYGON ((657550.332 6852587.97... |

1 | Seine | POLYGON ((517151.801 6930724.10... |

2 | Yonne | POLYGON ((701519.740 6813075.49... |

Alternative option is to add a secondary geometry column directly to the original `GeoDataFrame`

.

```
'geometry_5km'] = seine.buffer(5000)
seine[ seine
```

name | geometry | geometry_5km | |
---|---|---|---|

0 | Marne | MULTILINESTRING ((879955.277 67... | POLYGON ((657550.332 6852587.97... |

1 | Seine | MULTILINESTRING ((828893.615 67... | POLYGON ((517151.801 6930724.10... |

2 | Yonne | MULTILINESTRING ((773482.137 66... | POLYGON ((701519.740 6813075.49... |

You can then switch to either geometry column (i.e., make it the âactiveâ geometry column) using `.set_geometry`

, as in:

`= seine.set_geometry('geometry_5km') seine `

Letâs revert to the original state of `seine`

before moving on to the next section.

```
= seine.set_geometry('geometry')
seine = seine.drop('geometry_5km', axis=1) seine
```

### 4.2.4 Affine transformations

Affine transformations include, among others, shifting (translation), scaling and rotation, or any combination of these. They preserves lines and parallelism, by angles and lengths are not necessarily preserved. These transformations are an essential part of geocomputation. For example, shifting is needed for labels placement, scaling is used in non-contiguous area cartograms, and many affine transformations are applied when reprojecting or improving the geometry that was created based on a distorted or wrongly projected map.

The **geopandas** package implements affine transformation, for objects of classes `GeoSeries`

and `GeoDataFrame`

. In both cases, the method is applied on the `GeoSeries`

part, returning a just the `GeoSeries`

of transformed geometries.

Affine transformations of `GeoSeries`

can be done using the `.affine_transform`

method, which is a wrapper around the `shapely.affinity.affine_transform`

function. As documented, a 2D affine transformation requires a six-parameter list `[a,b,d,e,xoff,yoff]`

which represents the following equations for transforming the coordinates (Equation 4.1 and Equation 4.2)/

\[ x' = a x + b y + x_\mathrm{off} \tag{4.1}\]

\[ y' = d x + e y + y_\mathrm{off} \tag{4.2}\]

There are also simplified `GeoSeries`

methods for specific scenarios, such as:

`.translate(xoff=0.0, yoff=0.0)`

`.scale(xfact=1.0, yfact=1.0, origin='center')`

`.rotate(angle, origin='center', use_radians=False)`

`.skew(angle, origin='center', use_radians=False)`

For example, *shifting* only requires the \(x_{off}\) and \(y_{off}\), using `.translate`

. The code below shifts the y-coordinates of `nz`

by 100 \(km\) to the north, but leaves the x-coordinates untouched.

```
= nz.translate(0, 100000)
nz_shift nz_shift
```

```
0 MULTIPOLYGON (((1745493.196 610...
1 MULTIPOLYGON (((1803822.103 600...
2 MULTIPOLYGON (((1860345.005 595...
...
13 MULTIPOLYGON (((1616642.877 552...
14 MULTIPOLYGON (((1624866.279 551...
15 MULTIPOLYGON (((1686901.914 545...
Length: 16, dtype: geometry
```

**shapely**, and consequently **geopandas**, operations, typically ignore the z-dimension of geometries in operations. For example, `shapely.LineString([(0,0,0),(0,0,1)]).length`

returns `0`

(and not `1`

), since `.length`

ignores the z-dimension. In this book (like in most real-world spatial analysis applications), we deal only with two-dimensional geometries.

Scaling enlarges or shrinks objects by a factor, and can be applied either globally or locally. Global scaling increases or decreases all coordinates values in relation to the origin coordinates, while keeping all geometries topological relations intact. **geopandas** implements local scaling using the `.scale`

method. Local scaling treats geometries independently and requires points around which geometries are going to be scaled, e.g., centroids. In the example below, each geometry is shrunk by a factor of two around the centroids (Figure 4.5 (b)). To achieve that, we pass the `0.5`

and `0.5`

scaling factors (for x and y, respectively), and the `'centroid'`

option for the point of origin.

```
= nz.scale(0.5, 0.5, origin='centroid')
nz_scale nz_scale
```

```
0 MULTIPOLYGON (((1710099.077 603...
1 MULTIPOLYGON (((1778686.524 591...
2 MULTIPOLYGON (((1839927.904 582...
...
13 MULTIPOLYGON (((1593619.590 541...
14 MULTIPOLYGON (((1628907.395 542...
15 MULTIPOLYGON (((1665262.436 536...
Length: 16, dtype: geometry
```

When setting the `origin`

in `.scale`

, other than `'centroid'`

it is possible to use `'center'`

, for the bounding box center, or specific point coordinates, such as `(0,0)`

.

Rotating the geometries can be done using the `.rotate`

method. When rotating, we need to specify the rotation angle (positive values imply clockwise rotation) and the `origin`

points (using the same options as in `scale`

). For example, the following expression rotates `nz`

by 30 degrees counter-clockwise, around the geometry centroids.

```
= nz.rotate(-30, origin='centroid')
nz_rotate nz_rotate
```

```
0 MULTIPOLYGON (((1701904.887 597...
1 MULTIPOLYGON (((1779714.772 587...
2 MULTIPOLYGON (((1890843.462 582...
...
13 MULTIPOLYGON (((1616991.636 539...
14 MULTIPOLYGON (((1617733.547 542...
15 MULTIPOLYGON (((1665898.669 533...
Length: 16, dtype: geometry
```

Figure 4.5 shows the original layer `nz`

, and the shifting, scaling and rotation results.

```
# Shift
= nz.plot(color='lightgrey', edgecolor='darkgrey')
base =base, color='red', edgecolor='darkgrey');
nz_shift.plot(ax# Scale
= nz.plot(color='lightgrey', edgecolor='darkgrey')
base =base, color='red', edgecolor='darkgrey');
nz_scale.plot(ax# Rotate
= nz.plot(color='lightgrey', edgecolor='darkgrey')
base =base, color='red', edgecolor='darkgrey'); nz_rotate.plot(ax
```

### 4.2.5 Pairwise geometry-generating operations

Spatial clipping is a form of spatial subsetting that involves changes to the geometry columns of at least some of the affected features. Clipping can only apply to features more complex than points: lines, polygons and their âmultiâ equivalents. To illustrate the concept we will start with a simple example: two overlapping circles with a center point one unit away from each other and a radius of one (Figure 4.6).

```
= shapely.Point((0, 0)).buffer(1)
x = shapely.Point((1, 0)).buffer(1)
y shapely.GeometryCollection([x, y])
```

`x`

and `y`

Imagine you want to select not one circle or the other, but the space covered by both x and y. This can be done using the `.intersection`

method from **shapely**, illustrated using objects named `x`

and `y`

which represent the left- and right-hand circles (Figure 4.7).

` x.intersection(y)`

`x`

and `y`

More generally, clipping is an example of a âpairwise geometry-generating operationâ, where new geometries are generated from two inputs. Other than `.intersection`

(Figure 4.7), there are three other standard pairwise operators: `.difference`

(Figure 4.8), `.union`

(Figure 4.9), and `.symmetric_difference`

(Figure 4.10).

` x.difference(y)`

`x`

and `y`

(namely, `x`

âminusâ `y`

)` x.union(y)`

`x`

and `y`

` x.symmetric_difference(y)`

`x`

and `y`

Keep in mind that `x`

and `y`

are interchangeable in all predicates except for `.difference`

, where `x.difference(y)`

means `x`

minus `y`

, whereas `y.difference(x)`

means `y`

minus `x`

.

The latter examples demonstrate pairwise operations between individual `shapely`

geometries. The **geopandas** package, as is often the case, contains wrappers of these **shapely** functions to be applied to multiple, or pairwise, use cases. For example, applying either of the pairwise methods on a `GeoSeries`

or `GeoDataFrame`

, combined with a `shapely`

geometry, returns the pairwise (many-to-one) results (which is analogous to other operators, like `.intersects`

or `.distance`

, see Section 3.2.1 and Section 3.2.7, respectively).

Letâs demonstrate the âmany-to-oneâ scenario by calculating the difference between each geometry in a `GeoSeries`

and a âfixedâ `shapely`

geometry. To create the latter, letâs take `x`

and combine it with itself translated (Section 4.2.4) to a distance of `1`

and `2`

units âupwardsâ on the y-axis.

```
= gpd.GeoSeries([x])
geom1 = geom1.translate(0, 1)
geom2 = geom1.translate(0, 2)
geom3 = pd.concat([geom1, geom2, geom3])
geom geom
```

```
0 POLYGON ((1.00000 0.00000, 0.99...
0 POLYGON ((1.00000 1.00000, 0.99...
0 POLYGON ((1.00000 2.00000, 0.99...
dtype: geometry
```

Figure 4.11 shows the `GeoSeries`

`geom`

with the `shapely`

geometry (in red) that we will intersect with it.

```
= plt.subplots()
fig, ax ='lightgrey', edgecolor='black', ax=ax)
geom.plot(color='#FF000040', edgecolor='black', ax=ax); gpd.GeoSeries(y).plot(color
```

`GeoSeries`

with three circles, and a `shapely`

geometry that we will âsubtractâ from it (in red)Now, using `.intersection`

automatically applies the **shapely** method of the same name on each geometry in `geom`

, returning a new `GeoSeries`

, which we name `geom_inter_y`

, with the pairwise âintersectionsâ. Note the empty third geometry (can you explain the meaning of this result?).

```
= geom.intersection(y)
geom_inter_y geom_inter_y
```

```
0 POLYGON ((0.99518 -0.09802, 0.9...
0 POLYGON ((0.99518 0.90198, 0.98...
0 POLYGON EMPTY
dtype: geometry
```

Figure 4.12 is a plot of the result `geom_inter_y`

.

`='lightgrey', edgecolor='black'); geom_inter_y.plot(color`

`GeoSeries`

, after subtracting a `shapely`

geometry using `.intersection`

The `.overlay`

method (see Section 3.2.6) further extends this technique, making it possible to apply âmany-to-manyâ pairwise geometry generations between all pairs of two `GeoDataFrame`

s. The output is a new `GeoDataFrame`

with the pairwise outputs, plus the attributes of both inputs which were the inputs of the particular pairwise output geometry. See the âSet operations with overlayâ article in the **geopandas** documentation for examples of `.overlay`

.

### 4.2.6 Subsetting vs. clipping

In the last two chapters we have introduced two types of spatial operators: boolean, such as `.intersects`

(Section 3.2.1), and geometry-generating, such as `.intersection`

(Section 4.2.5). Here, we illustrate the difference between them. We do this using the specific scenario of subsetting points by polygons, where (unlike in other cases) both methods can be used for the same purpose and giving the same result.

To illustrate the point, we will subset points that cover the bounding box of the circles `x`

and `y`

from Figure 4.6. Some points will be inside just one circle, some will be inside both and some will be inside neither. The following code sections generates the sample data for this section, a simple random distribution of points within the extent of circles `x`

and `y`

, resulting in output illustrated in Figure 4.13. We create the sample points in two steps. First, we figure out the bounds where random points are to be generated.

```
= x.union(y).bounds
bounds bounds
```

`(-1.0, -1.0, 2.0, 1.0)`

Second, we use `np.random.uniform`

to calculate `n`

random x- and y-coordinates within the given bounds.

```
1)
np.random.seed(= 10
n = np.random.uniform(bounds[0], bounds[2], n)
coords_x = np.random.uniform(bounds[1], bounds[3], n)
coords_y = list(zip(coords_x, coords_y))
coords coords
```

```
[(0.2510660141077219, -0.1616109711934104),
(1.1609734803264744, 0.370439000793519),
(-0.9996568755479653, -0.5910955005369651),
(-0.0930022821044807, 0.7562348727818908),
(-0.5597323275486609, -0.9452248136041477),
(-0.7229842156936066, 0.34093502035680445),
(-0.4412193658669873, -0.16539039526574606),
(0.03668218112914312, 0.11737965689150331),
(0.1903024226920098, -0.7192261228095325),
(0.6164502020100708, -0.6037970218302424)]
```

Third, we transform the list of coordinates into a `list`

of `shapely`

points and then to a `GeoSeries`

.

```
= [shapely.Point(i) for i in coords]
pnt = gpd.GeoSeries(pnt) pnt
```

The result `pnt`

, which `x`

and `y`

circles in the background, is shown in Figure 4.13.

```
= pnt.plot(color='none', edgecolor='black')
base =base, color='none', edgecolor='darkgrey');
gpd.GeoSeries([x]).plot(ax=base, color='none', edgecolor='darkgrey'); gpd.GeoSeries([y]).plot(ax
```

Now, we can get back to our question: how to subset the points to only return the point that intersects with both `x`

and `y`

? The code chunks below demonstrate two ways to achieve the same result. In the first approach, we can calculate a boolean `Series`

, evaluating whether each point of `pnt`

intersects with the intersection of `x`

and `y`

(see Section 3.2.1) and then use it to subset `pnt`

to get the result `pnt1`

.

```
= pnt.intersects(x.intersection(y))
sel = pnt[sel]
pnt1 pnt1
```

```
0 POINT (0.25107 -0.16161)
7 POINT (0.03668 0.11738)
9 POINT (0.61645 -0.60380)
dtype: geometry
```

In the second approach, we can also find the intersection between the input points represented by `pnt`

, using the intersection of `x`

and `y`

as the subsetting/clipping object. Since the second argument is an individual `shapely`

geometry (`x.intersection(y)`

), we get âpairwiseâ intersections of each `pnt`

with it (see Section 4.2.5):

```
= pnt.intersection(x.intersection(y))
pnt2 pnt2
```

```
0 POINT (0.25107 -0.16161)
1 POINT EMPTY
2 POINT EMPTY
...
7 POINT (0.03668 0.11738)
8 POINT EMPTY
9 POINT (0.61645 -0.60380)
Length: 10, dtype: geometry
```

The subset `pnt2`

is shown in Figure 4.14.

```
= pnt.plot(color='none', edgecolor='black')
base =base, color='none', edgecolor='darkgrey');
gpd.GeoSeries([x]).plot(ax=base, color='none', edgecolor='darkgrey');
gpd.GeoSeries([y]).plot(ax=base, color='red'); pnt2.plot(ax
```

The only difference between the two approaches is that `.intersection`

returns all âintersectionsâ, even if they are empty. When these are filtered out, `pnt2`

becomes identical to `pnt1`

:

```
= pnt2[~pnt2.is_empty]
pnt2 pnt2
```

```
0 POINT (0.25107 -0.16161)
7 POINT (0.03668 0.11738)
9 POINT (0.61645 -0.60380)
dtype: geometry
```

The example above is rather contrived and provided for educational rather than applied purposes. However, we encourage the reader to reproduce the results to deepen your understanding for handling geographic vector objects in Python.

### 4.2.7 Geometry unions

Spatial aggregation can silently dissolve the geometries of touching polygons in the same group, as we saw in Section 2.2.2. This is demonstrated in the code chunk below, in which 49 `us_states`

are aggregated into 4 regions using the `.dissolve`

method.

```
= us_states[['REGION', 'geometry', 'total_pop_15']] \
regions ='REGION', aggfunc='sum').reset_index()
.dissolve(by regions
```

REGION | geometry | total_pop_15 | |
---|---|---|---|

0 | Midwest | MULTIPOLYGON (((-89.10077 36.94... | 67546398.0 |

1 | Norteast | MULTIPOLYGON (((-75.61724 39.83... | 55989520.0 |

2 | South | MULTIPOLYGON (((-81.38550 30.27... | 118575377.0 |

3 | West | MULTIPOLYGON (((-118.36998 32.8... | 72264052.0 |

Figure 4.15 compares the original `us_states`

layer with the aggregated `regions`

layer.

```
# States
= plt.subplots(figsize=(9, 2.5))
fig, ax =ax, edgecolor='black', column='total_pop_15', legend=True);
us_states.plot(ax# Regions
= plt.subplots(figsize=(9, 2.5))
fig, ax =ax, edgecolor='black', column='total_pop_15', legend=True); regions.plot(ax
```

What is happening with the geometries here? Behind the scenes, `.dissolve`

combines the geometries and dissolve the boundaries between them using the `.unary_union`

method per group. This is demonstrated in the code chunk below which creates a united western US using the standalone `unary_union`

operation. Note that the result is a `shapely`

geometry, as the individual attributes are âlostâ as part of dissolving (Figure 4.16).

```
= us_states[us_states['REGION'] == 'West']
us_west = us_west.geometry.unary_union
us_west_union us_west_union
```

To dissolve two (or more) groups of a `GeoDataFrame`

into one geometry, we can either (a) use a combined condition or (b) concatenate the two separate subsets and then dissove using `.unary_union`

.

```
# Approach 1
= (us_states['REGION'] == 'West') | (us_states['NAME'] == 'Texas')
sel = us_states[sel]
texas_union = texas_union.geometry.unary_union
texas_union # Approach 2
= us_states[us_states['REGION'] == 'West']
us_west = us_states[us_states['NAME'] == 'Texas']
texas = pd.concat([us_west, texas]).unary_union texas_union
```

The result is identical in both cases, shown in Figure 4.17.

` texas_union`

### 4.2.8 Type transformations

Transformation of geometries, from one type to another, also known as âgeometry castingâ, is often required to facilitate spatial analysis. Either the **geopandas** or the **shapely** packages can be used for geometry casting, depending on the type of transformation, and the way that the input is organized (whether and individual geometry, or a vector layer). Therefore, the exact expression(s) depend on the specific transformation we are interested in.

In general, you need to figure out the required input of the respective construstor function according to the âdestinationâ geometry (e.g., `shapely.LineString`

, etc.), then reshape the input of the âsourceâ geometry into the right form to be passed to that function. Or, when available, you can use a wrapper from **geopandas**.

In this section we demonstrate several common scenarios. We start with transformations of individual geometries from one type to another, using **shapely** methods:

`'MultiPoint'`

to`'LineString'`

(Figure 4.19)`'MultiPoint'`

to`'Polygon'`

(Figure 4.20)`'LineString'`

to`'MultiPoint'`

(Figure 4.22)`'LineString'`

to`'Polygon'`

(Figure 4.23)`'Polygon'`

s to`'MultiPolygon'`

(Figure 4.24)`'MultiPolygon'`

s to`'Polygon'`

s (Figure 4.25, Figure 4.26)

Then, we move on and demonstrate casting workflows on `GeoDataFrame`

s, where we have further considerations, such as keeping track of geometry attributes, and the possibility of dissolving, rather than just combining, geometries. As we will see, these are done either by âmanuallyâ applying **shapely** methods on all geometries in the given layer, or using **geopandas** wrapper methods which do it automatically:

`'MultiLineString'`

to`'LineString'`

s (using`.explode`

) (Figure 4.28)`'LineString'`

to`'MultiPoint'`

s (using`.apply`

) (Figure 4.29)`'LineString'`

s to`'MultiLineString'`

(using`.dissolve`

)`'Polygon'`

s to`'MultiPolygon'`

(using`.dissolve`

or`.agg`

) (Figure 4.30)`'Polygon'`

to`'(Multi)LineString'`

(using`.boundary`

or`.exterior`

) (demonstrated in a subsequent chapter, see Section 5.4.2)

Letâs start with the simple individual-geometry casting examples, to illustrate how geometry casting works on **shapely** geometry objects. First, letâs create a `'MultiPoint'`

(Figure 4.18).

```
= shapely.MultiPoint([(1,1), (3,3), (5,1)])
multipoint multipoint
```

`'MultiPoint'`

geometry used to demonstrate **shapely**type transformations

A `'LineString'`

can be created using `shapely.LineString`

from a `list`

of points. Thus, a `'MultiPoint'`

can be converted to a `'LineString'`

by extracting the individual points into a `list`

, then passing them to `shapely.LineString`

(Figure 4.19). The `.geoms`

property, mentioned in Section 1.2.5, give access to the indivudual parts that comprise a multi-part geometry; it is one of the **shapely** access methods to internal parts of a geometry.

```
= shapely.LineString(multipoint.geoms)
linestring linestring
```

`'LineString'`

created from the `'MultiPoint'`

in Figure 4.18Similarly, a `'Polygon'`

can be created using function `shapely.Polygon`

, which accepts a sequence of point coordinates. In principle, the last coordinate must be equal to the first, in order to form a closed shape. However, `shapely.Polygon`

is able to complete the last coordinate automatically, and therefore we can pass all of the coordinates of the `'MultiPoint'`

directly to `shapely.Polygon`

(Figure 4.20).

```
= shapely.Polygon([[p.x, p.y] for p in multipoint.geoms])
polygon polygon
```

`'Polygon'`

created from the `'MultiPoint'`

in Figure 4.18The source `'MultiPoint'`

geometry, and the derived `'LineString'`

and `'Polygon'`

geometries are shown in Figure 4.21. Note that we convert the `shapely`

geometries to `GeoSeries`

to be able to use the **geopandas** `.plot`

method.

```
;
gpd.GeoSeries(multipoint).plot();
gpd.GeoSeries(linestring).plot(); gpd.GeoSeries(polygon).plot()
```

`'LineString`

â and `'Polygon'`

casted from a `'MultiPoint'`

geometryConversion from `'MultiPoint'`

to `'LineString'`

(Figure 4.19) is a common operation that creates a line object from ordered point observations, such as GPS measurements or geotagged media. This allows spatial operations such as the length of the path traveled. Conversion from `'MultiPoint'`

or `'LineString'`

to `'Polygon'`

(Figure 4.20) is often used to calculate an area, for example from the set of GPS measurements taken around a lake or from the corners of a building lot.

Our `'LineString'`

geometry can be converted back to a `'MultiPoint'`

geometry by passing its coordinates directly to `shapely.MultiPoint`

(Figure 4.22).

` shapely.MultiPoint(linestring.coords)`

`'MultiPoint'`

created from the `'LineString'`

in Figure 4.19A `'Polygon'`

(exterior) coordinates can be passed to `shapely.MultiPoint`

, to go back to a `'MultiPoint'`

geometry, as well (Figure 4.23).

` shapely.MultiPoint(polygon.exterior.coords)`

`'MultiPoint'`

created from the `'Polygon'`

in Figure 4.20Using these methods, we can transform between `'Point'`

, `'LineString'`

, and `'Polygon'`

geometries, assuming there is a sufficient number of points (at least two to form a line, and at least three to form a polygon). When dealing with multi-part geometries using **shapely**, we can:

- Access single-part geometries (e.g., each
`'Polygion'`

in a`'MultiPolygon'`

geometry) using`.geoms[i]`

, where`i`

is the index of the geometry - Combine single-part geometries into a multi-part geometry, by passing a
`list`

of the latter to the constructor function

For example, here is how we combine two `'Polygon'`

geometries into a `'MultiPolygon'`

(while also using a **shapely** affine function `shapely.affinity.translate`

, which is underlying the **geopandas** `.translate`

method used earlier, see Section 4.2.4) (Figure 4.24):

```
= shapely.MultiPolygon([
multipolygon
polygon, buffer(1.5), 3, 2)
shapely.affinity.translate(polygon.centroid.
]) multipolygon
```

`'MultiPolygon'`

created from the `'Polygon'`

in Figure 4.20 and another polygonNow, here is how we can get back the `'Polygon'`

part 1 (Figure 4.25):

`0] multipolygon.geoms[`

^{st}âpartâ extracted from the

`'MultiPolygon'`

in Figure 4.24and part 2 (Figure 4.26):

`1] multipolygon.geoms[`

^{nd}âpartâ extracted from the

`'MultiPolygon'`

in Figure 4.24However, dealing with multi-part geometries can be easier with **geopandas**. Thanks to the fact that geometries in a `GeoDataFrame`

are associated with attributes, we can keep track of the origin of each geometry: duplicating the attributes when going from multi-part to single-part (using `.explode`

, see below), or âcollapsingâ the attributes through aggregation when going from single-part to multi-part (using `.dissolve`

, see Section 4.2.7).

Letâs demonstrate going from multi-part to single-part (Figure 4.28) and then back to multi-part (Section 4.2.7), using a small line layer. As input, we will create a `'MultiLineString'`

geometry composed of three lines (Figure 4.27).

```
= shapely.LineString([(1, 5), (4, 3)])
l1 = shapely.LineString([(4, 4), (4, 1)])
l2 = shapely.LineString([(2, 2), (4, 2)])
l3 = shapely.MultiLineString([l1, l2, l3])
ml ml
```

`'MultiLineString'`

geometry composed of three linesLetâs place it into a `GeoSeries`

.

```
= gpd.GeoSeries([ml])
geom geom
```

```
0 MULTILINESTRING ((1.00000 5.000...
dtype: geometry
```

Then, put it into a `GeoDataFrame`

with an attribute called `'id'`

:

```
= gpd.GeoDataFrame(geometry=geom, data=pd.DataFrame({'id': [1]}))
dat dat
```

id | geometry | |
---|---|---|

0 | 1 | MULTILINESTRING ((1.00000 5.000... |

You can imagine it as a road or river network. The above layer `dat`

has only one row that defines all the lines. This restricts the number of operations that can be done, for example it prevents adding names to each line segment or calculating lengths of single lines. Using **shapely** methods with which we are already familiar with (see above), the individual single-part geometries (i.e., the âpartsâ) can be accessed through the `.geoms`

property.

`list(ml.geoms)`

`[<LINESTRING (1 5, 4 3)>, <LINESTRING (4 4, 4 1)>, <LINESTRING (2 2, 4 2)>]`

However, specifically for the âmulti-part to single partâ type transformation scenarios, there is also a method called `.explode`

, which can convert an entire multi-part `GeoDataFrame`

to a single-part one. The advantage is that the original attributes (such as `id`

) are retained, so that we can keep track of the original multi-part geometry properties that each part came from. The `index_parts=True`

argument also lets us keep track of the original multipart geometry indices, and part indices, named `level_0`

and `level_1`

, respectively.

```
= dat.explode(index_parts=True).reset_index()
dat1 dat1
```

level_0 | level_1 | id | geometry | |
---|---|---|---|---|

0 | 0 | 0 | 1 | LINESTRING (1.00000 5.00000, 4.... |

1 | 0 | 1 | 1 | LINESTRING (4.00000 4.00000, 4.... |

2 | 0 | 2 | 1 | LINESTRING (2.00000 2.00000, 4.... |

For example, here we see that all `'LineString'`

geometries came from the same multi-part geometry (`level_0`

=`0`

), which had three parts (`level_1`

=`0`

,`1`

,`2`

). Figure 4.28 demonstrates the effect of `.explode`

in converting a layer with multi-part geometries into a layer with single part geometries.

```
='id');
dat.plot(column='level_1'); dat1.plot(column
```

`.explode`

`'MultiLineString'`

layer with one feature, into a `'LineString'`

layer with three features, using `.explode`

As a side-note, letâs demonstrate how the above **shapely** casting methods can be translated to **geopandas**. Suppose that we want to transform `dat1`

, which is a layer of type `'LineString'`

with three features, to a layer of type `'MultiPoint'`

(also with three features). Recall that for a single geometry, we use the expression `shapely.MultiPoint(x.coords)`

, where `x`

is a `'LineString'`

(Figure 4.22). When dealing with a `GeoDataFrame`

, we wrap the conversion into `.apply`

, to apply it on all geometries:

```
= dat1.copy()
dat2 = dat2.geometry.apply(lambda x: shapely.MultiPoint(x.coords))
dat2.geometry dat2
```

level_0 | level_1 | id | geometry | |
---|---|---|---|---|

0 | 0 | 0 | 1 | MULTIPOINT (1.00000 5.00000, 4.... |

1 | 0 | 1 | 1 | MULTIPOINT (4.00000 4.00000, 4.... |

2 | 0 | 2 | 1 | MULTIPOINT (2.00000 2.00000, 4.... |

The result is illustrated in Figure 4.29.

```
='level_1');
dat1.plot(column='level_1'); dat2.plot(column
```

`'LineString'`

layer with three features, into a `'MultiPoint'`

layer (also with three features), using `.apply`

and **shapely**methods

The opposite transformation, i.e., âsingle-part to multi-partâ, is achieved using the `.dissolve`

method (which we are already familiar with, see Section 4.2.7). For example, here is how we can get back to the `'MultiLineString'`

geometry:

`='id').reset_index() dat1.dissolve(by`

id | geometry | level_0 | level_1 | |
---|---|---|---|---|

0 | 1 | MULTILINESTRING ((1.00000 5.000... | 0 | 0 |

The next code chunk is another example, dissolving the `nz`

north and south parts into `'MultiPolygon'`

geometries.

```
= nz[['Island', 'Population', 'geometry']] \
nz_dis1 ='Island', aggfunc='sum') \
.dissolve(by
.reset_index() nz_dis1
```

Island | geometry | Population | |
---|---|---|---|

0 | North | MULTIPOLYGON (((1865558.829 546... | 3671600.0 |

1 | South | MULTIPOLYGON (((1229729.735 479... | 1115600.0 |

Note that `.dissolve`

not only combines single-part into multi-part geometries, but also dissolves any internal borders. So, in fact, the result may be single-part (in case when all parts touch each other, unlike in `nz`

). If, for some reason, we want to combine geometries into multi-part *without* dissolving, we can fall back to the **pandas** `.agg`

method (custom table aggregation), supplemented with a **shapely** function specifying how exactly we want to transform each group of geometries into a new single geometry. In the following example, for instance, we collect all `'Polygon'`

and `'MultiPolygon'`

parts of `nz`

into a single `'MultiPolygon'`

geometry with many separate parts (i.e., without dissolving), per group (`Island`

).

```
= nz \
nz_dis2 'Island') \
.groupby(
.agg({'Population': 'sum',
'geometry': lambda x: shapely.MultiPolygon(x.explode().to_list())
\
})
.reset_index()= gpd.GeoDataFrame(nz_dis2).set_geometry('geometry').set_crs(nz.crs)
nz_dis2 nz_dis2
```

Island | Population | geometry | |
---|---|---|---|

0 | North | 3671600.0 | MULTIPOLYGON (((1745493.196 600... |

1 | South | 1115600.0 | MULTIPOLYGON (((1557042.169 531... |

The difference between the last two results (with and without dissolving, respectively) is not evident in the printout: in both cases we got a layer with two features of type `'MultiPolygon'`

. However, in the first case internal borders were dissolved, while in the second case they were not. This is illustrated in Figure 4.30:

```
='lightgrey', edgecolor='black');
nz_dis1.plot(color='lightgrey', edgecolor='black'); nz_dis2.plot(color
```

**geopandas**

`.dissolve`

method)`.agg`

and a custom **shapely**-based function)

It is also worthwhile to note the `.boundary`

and `.exterior`

properties of `GeoSeries`

, which are used to cast polygons to lines, with or without interior rings, respectively (see Section 5.4.2).

## 4.3 Geometric operations on raster data

Geometric raster operations include the shift, flipping, mirroring, scaling, rotation or warping of images. These operations are necessary for a variety of applications including georeferencing, used to allow images to be overlaid on an accurate map with a known CRS (Liu and Mason 2009). A variety of georeferencing techniques exist, including:

- Georectification based on known ground control points
- Orthorectification, which also accounts for local topography
- Image registration is used to combine images of the same thing but shot from different sensors, by aligning one image with another (in terms of coordinate system and resolution)

Python is rather unsuitable for the first two points since these often require manual intervention which is why they are usually done with the help of dedicated GIS software. On the other hand, aligning several images is possible in Python and this section shows among others how to do so. This often includes changing the extent, the resolution and the origin of an image. A matching projection is of course also required but is already covered Section 6.8.

In any case, there are other reasons to perform a geometric operation on a single raster image. For instance, a common reason for aggregating a raster is to decrease run-time or save disk space. Of course, this approach is only recommended if the task at hand allows a coarser resolution of raster data.

### 4.3.1 Geometric intersections

In Section 3.3.1 we have shown how to extract values from a raster overlaid by coordinates or by a matching boolean mask. A different case is when the area of interest is defined by any general (possibly non-matching) raster B, to retrieve a spatial output of a (smaller) subset of raster A we can:

- Extract the bounding box polygon of B (hereby,
`clip`

) - Mask and crop A (hereby,
`elev.tif`

) using B (Section 5.2)

For example, suppose that we want to get a subset of the `elev.tif`

raster using another, smaller, raster. To demonstrate this, letâs create (see Section 1.3.2) that smaller raster, hereby named `clip`

. First, we need to create a \(3 \times 3\) array of raster values.

```
= np.array([1] * 9).reshape(3, 3)
clip clip
```

```
array([[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
```

Then, we define the transformation matrix, in such a way that `clip`

intersects with `elev.tif`

(Figure 4.32).

```
= rasterio.transform.from_origin(
new_transform =0.9,
west=0.45,
north=0.3,
xsize=0.3
ysize
) new_transform
```

```
Affine(0.3, 0.0, 0.9,
0.0, -0.3, 0.45)
```

Now, for subsetting, we will derive a `shapely`

geometry representing the `clip`

raster extent, using `rasterio.transform.array_bounds`

.

```
= rasterio.transform.array_bounds(
bbox 1], # columns
clip.shape[0], # rows
clip.shape[
new_transform
) bbox
```

`(0.9, -0.4499999999999999, 1.7999999999999998, 0.45)`

The four numeric values can be transformed into a rectangular `shapely`

geometry using `shapely.box`

(Figure 4.31).

```
= shapely.box(*bbox)
bbox bbox
```

`shapely`

geometry derived from a clipping raster bounding box coordinates, a preliminary step for geometric intersection between two rastersFigure 4.32 shows the alignment of `bbox`

and `elev.tif`

.

```
= plt.subplots()
fig, ax =ax)
rasterio.plot.show(src_elev, ax='none', ax=ax); gpd.GeoSeries([bbox]).plot(color
```

`elev.tif`

raster, and the extent of another (smaller) raster `clip`

which we use to subset itFrom here on, subsetting can be done using masking and cropping, just like with any vector layer other than `bbox`

, regardless whether it is rectangular or not. We elaborate on masking and cropping in Section 5.2 (check that section for details about `rasterio.mask.mask`

), but, for completeness, here is the code for the last step of masking and cropping:

```
= rasterio.mask.mask(
out_image, out_transform
src_elev,
[bbox], =True,
crop=True,
all_touched=0
nodata )
```

The resulting subset array `out_image`

contains all pixels intersecting with `clip`

*pixels* (not necessarily with the centroids!). However, due to the `all_touched=True`

argument, those pixels which intersect with `clip`

, but their centroid does not, retain their original values (e.g., `17`

, `23`

) rather than turned into âNo Dataâ (e.g., `0`

).

` out_image`

```
array([[[17, 18],
[23, 24]]], dtype=uint8)
```

Therefore, in our case, subset `out_image`

dimensions are \(2 \times 2\) (Figure 4.33; also see Figure 4.32).

```
= plt.subplots()
fig, ax =out_transform, ax=ax)
rasterio.plot.show(out_image, transform='none', ax=ax); gpd.GeoSeries([bbox]).plot(color
```

`elev.tif`

raster### 4.3.2 Extent and origin

When merging or performing map algebra on rasters, their resolution, projection, origin and/or extent have to match. Otherwise, how should we add the values of one raster with a resolution of `0.2`

decimal degrees to a second raster with a resolution of `1`

decimal degree? The same problem arises when we would like to merge satellite imagery from different sensors with different projections and resolutions. We can deal with such mismatches by aligning the rasters. Typically, raster alignment is done through resamplingâthat way, it is guaranteed that the rasters match exactly (Section 4.3.4). However, sometimes it can be useful to modify raster placement and extent âmanuallyâ, by adding or removing rows and columns, or by modifying the origin, that is, slightly shifting the raster. Sometimes, there are reasons other than alignment with a second raster for manually modifying raster extent and placement. For example, it may be useful to add extra rows and columns to a raster prior to focal operations, so that it is easier to operate on the edges.

Letâs demostrate the first operation, raster padding. First, we will read the array with the `elev.tif`

values:

```
= src_elev.read(1)
r r
```

```
array([[ 1, 2, 3, 4, 5, 6],
[ 7, 8, 9, 10, 11, 12],
[13, 14, 15, 16, 17, 18],
[19, 20, 21, 22, 23, 24],
[25, 26, 27, 28, 29, 30],
[31, 32, 33, 34, 35, 36]], dtype=uint8)
```

To pad an `ndarray`

, we can use the `np.pad`

function. The function accepts an array, and a tuple of the form `((rows_top,rows_bottom),(columns_left, columns_right))`

. Also, we can specify the value thatâs being used for padding with `constant_values`

(e.g., `18`

). For example, here we pad `r`

with one extra row and two extra columns, on both sides, resulting in the array `r_pad`

:

```
= 1
rows = 2
cols = np.pad(r, ((rows,rows),(cols,cols)), constant_values=18)
r_pad r_pad
```

```
array([[18, 18, 18, 18, 18, 18, 18, 18, 18, 18],
[18, 18, 1, 2, 3, 4, 5, 6, 18, 18],
[18, 18, 7, 8, 9, 10, 11, 12, 18, 18],
[18, 18, 13, 14, 15, 16, 17, 18, 18, 18],
[18, 18, 19, 20, 21, 22, 23, 24, 18, 18],
[18, 18, 25, 26, 27, 28, 29, 30, 18, 18],
[18, 18, 31, 32, 33, 34, 35, 36, 18, 18],
[18, 18, 18, 18, 18, 18, 18, 18, 18, 18]], dtype=uint8)
```

However, for `r_pad`

to be used in any spatial operation, we also have to update its transformation matrix. Whenever we add extra columns on the left, or extra rows on top, the raster *origin* changes. To reflect this fact, we have to take to âoriginalâ origin and add the required multiple of pixel widths or heights (i.e., raster resolution steps). The transformation matrix of a raster is accessible from the raster file metadata (Section 1.3.2) or, as a shortcut, through the `.transform`

property of the raster file connection. For example, the next code chunk shows the transformation matrix of `elev.tif`

.

` src_elev.transform `

```
Affine(0.5, 0.0, -1.5,
0.0, -0.5, 1.5)
```

From the transformation matrix, we are able to extract the origin.

```
= src_elev.transform[2], src_elev.transform[5]
xmin, ymax xmin, ymax
```

`(-1.5, 1.5)`

We can also get the resolution of the data, which is the distance between two adjacent pixels.

```
= src_elev.transform[0], src_elev.transform[4]
dx, dy dx, dy
```

`(0.5, -0.5)`

These two parts of information are enough to calculate the new origin (`xmin_new,ymax_new`

) of the padded raster.

```
= xmin - dx * cols
xmin_new = ymax - dy * rows
ymax_new xmin_new, ymax_new
```

`(-2.5, 2.0)`

Using the updated origin, we can update the transformation matrix (Section 1.3.2). Keep in mind that the meaning of the last two arguments is `xsize`

, `ysize`

, so we need to pass the absolute value of `dy`

(since it is negative).

```
= rasterio.transform.from_origin(
new_transform =xmin_new,
west=ymax_new,
north=dx,
xsize=abs(dy)
ysize
) new_transform
```

```
Affine(0.5, 0.0, -2.5,
0.0, -0.5, 2.0)
```

Figure 4.34 shows the padded raster, with the outline of the original `elev.tif`

(in red), demonstrating that the origin was shifted correctly and the `new_transform`

works fine.

```
= plt.subplots()
fig, ax =new_transform, cmap='Greys', ax=ax)
rasterio.plot.show(r_pad, transform= gpd.GeoSeries(shapely.box(*src_elev.bounds))
elev_bbox ='none', edgecolor='red', ax=ax); elev_bbox.plot(color
```

`elev.tif`

raster, and the extent of the original `elev.tif`

raster (in red)We can shift a raster origin not just when padding, but in any other use case, just by changing its transformation matrix. The effect is that the raster is going to be shifted (which is analogous to `.translate`

for shifting a vector layer, see Section 4.2.4). Manually shifting a raster to arbitrary distance is rarely needed in real-life scenarios, but it is useful to know how to do it at least for better understanding the concept of *raster origin*. As an example, letâs shift the origin of `elev.tif`

by `(-0.25,0.25)`

. First, we need to calculate the new origin.

```
= xmin - 0.25 # shift xmin to the left
xmin_new = ymax + 0.25 # shift ymax upwards
ymax_new xmin_new, ymax_new
```

`(-1.75, 1.75)`

To shift the origin in other directions we should change the two operators (`-`

, `+`

) accordingly.

Then, same as when padding (see above), we create an updated transformation matrix.

```
= rasterio.transform.from_origin(
new_transform =xmin_new,
west=ymax_new,
north=dx,
xsize=abs(dy)
ysize
) new_transform
```

```
Affine(0.5, 0.0, -1.75,
0.0, -0.5, 1.75)
```

Figure 4.35 shows the shifted raster and the outline of the original `elev.tif`

raster (in red).

```
= plt.subplots()
fig, ax =new_transform, cmap='Greys', ax=ax)
rasterio.plot.show(r, transform='none', edgecolor='red', ax=ax); elev_bbox.plot(color
```

`elev.tif`

raster (Figure 4.34) further shifted by `(0.25,0.25)`

, and the extent of the original `elev.tif`

raster (in red)### 4.3.3 Aggregation and disaggregation

Raster datasets vary based on their resolution, from high resolution datasets that enable individual trees to be seen, to low resolution datasets covering large swaths of the Earth. Raster datasets can be transformed to either decrease (aggregate) or increase (disaggregate) their resolution, for a number of reasons. For example, aggregation can be used to reduce computational resource requirements of raster storage and subsequent steps, while disaggregation can be used to match other datasets, or to add detail. As an example, we here change the spatial resolution of `dem.tif`

by a factor of `5`

(Figure 4.36).

Raster aggregation is, in fact, a special case of raster resampling (see Section 4.3.4), where the target raster grid is aligned with the original raster, only with coarser pixels. Conversely, raster resampling is the general case where the new grid is not necessarily an aggregation of the original one, but any other type of grid (such as a rotated and/or shifted one, etc.).

To aggregate a raster using **rasterio**, we go through two steps:

- Reading the raster values (using
`.read`

) into an`out_shape`

that is different from the original`.shape`

- Updating the
`transform`

according to`out_shape`

Letâs demonstrate it, using the `dem.tif`

file. Note the original shape of the raster; it has `117`

rows and `117`

columns.

`1).shape src.read(`

`(117, 117)`

Also note the transform, which tells us that the raster resolution is about 30.85 \(m\).

` src.transform`

```
Affine(30.849999999999604, 0.0, 794599.1076146346,
0.0, -30.84999999999363, 8935384.324602526)
```

To aggregate, instead of reading the raster values the usual way, as in `src.read(1)`

, we can specify `out_shape`

to read the values into a different shape. Here, we calculate a new shape which is downscaled by a factor of `5`

, i.e., the number of rows and columns is multiplied by `0.2`

. We must truncate any âpartialâ rows and columns, e.g., using `int`

. Each new pixel is now obtained, or âresampledâ, from \(\sim 5 \times 5 = \sim 25\) âoldâ raster values. It is crucial to choose an appropriate *resampling method* through the `resampling`

parameter. Here we use `rasterio.enums.Resampling.average`

, i.e., the new âlargeâ pixel value is the average of all coinciding small pixels, which makes sense for our elevation data in `dem.tif`

. See Section 4.3.4 for a list of other available methods.

```
= 0.2
factor = src.read(1,
r =(
out_shapeint(src.height * factor),
int(src.width * factor)
),=rasterio.enums.Resampling.average
resampling )
```

As expected, the resulting array `r`

has ~5 times smaller dimensions, as shown below.

` r.shape`

`(23, 23)`

Whatâs left to be done is the second step, to update the transform, taking into account the change in raster shape. This can be done as follows, using `.transform.scale`

.

```
= src.transform * src.transform.scale(
new_transform / r.shape[1]),
(src.width / r.shape[0])
(src.height
) new_transform
```

```
Affine(156.93260869565017, 0.0, 794599.1076146346,
0.0, -156.9326086956198, 8935384.324602526)
```

Figure 4.36 shows the original raster and the aggregated one.

```
;
rasterio.plot.show(src)=new_transform); rasterio.plot.show(r, transform
```

This is a good opportunity to demonstrate exporting a raster with modified dimensions and transformation matrix. We can update the raster metadata required for writing with the `update`

method.

```
= src.meta.copy()
dst_kwargs
dst_kwargs.update({'transform': new_transform,
'width': r.shape[1],
'height': r.shape[0],
}) dst_kwargs
```

```
{'driver': 'GTiff',
'dtype': 'float32',
'nodata': nan,
'width': 23,
'height': 23,
'count': 1,
'crs': CRS.from_epsg(32717),
'transform': Affine(156.93260869565017, 0.0, 794599.1076146346,
0.0, -156.9326086956198, 8935384.324602526)}
```

Then we can create a new file (`dem_agg5.tif`

) in writing mode, and write the values from the aggregated array `r`

into the 1^{st} band of the file (see Section 7.6.2 for a detailed explanation of writing raster files with **rasterio**).

```
= rasterio.open('output/dem_agg5.tif', 'w', **dst_kwargs)
dst 1)
dst.write(r, dst.close()
```

The `**`

syntax in Python is known as variable-length â*keyword* argumentsâ. It is used to pass a dictionary of numerous `parameter:argument`

pairs to named arguments of a function. In `rasterio.open`

writing mode, the âkeyword argumentsâ syntax often comes in handy, because, instead of specifying each and every property of a new file, we pass a (modified) `.meta`

dictionary based on another, template, raster.

Technically, keep in mind that the expression:

`rasterio.open('out.tif', 'w', **dst_kwargs)`

where `dst_kwargs`

is a `dict`

of the following form (typically coming from a template raster, possibly with few âupdatedâ properties using `.update`

, see above):

```
{'driver': 'GTiff',
'dtype': 'float32',
'nodata': nan,
...
}
```

is a shortcut of:

```
rasterio.open(
'out.tif', 'w',
driver=dst_kwargs['driver'],
dtype=dst_kwargs['dtype'],
nodata=dst_kwargs['nodata'],
...
)
```

â*Positional* argumentsâ is a related technique; see note in Section 6.8.

The opposite operation, namely disaggregation, is when we increase the resolution of raster objects. Either of the supported resampling methods (see Section 4.3.4) can be used. However, since we are not actually summarizing information but transferring the value of a large pixel into multiple small pixels, it makes sense to use either:

- Nearest neighbor resampling (
`rasterio.enums.Resampling.nearest`

), when want to keep the original values as-is, since modifying them would be incorrect (such as in categorical rasters) - Smoothing techniques, such as bilinear resampling (
`rasterio.enums.Resampling.bilinear`

), when we would like the smaller pixels to reflect gradual change between the original values, e.g., when the disaggregated raster is used for visualization purposes

To disaggregate a raster, we go through exactly the same workflow as for aggregation, only using a different scaling factor, such as `factor=5`

instead of `factor=0.2`

, i.e., *increasing* the number of raster pixels instead of decreasing. In the example below, we disaggregate using bilinear interpolation, to get a smoothed high-resolution raster.

```
= 5
factor = src.read(1,
r2 =(
out_shapeint(src.height * factor),
int(src.width * factor)
),=rasterio.enums.Resampling.bilinear
resampling )
```

As expected, the dimensions of the disaggregated raster are this time ~5 times *bigger* than the original ones.

` r2.shape`

`(585, 585)`

To calculate the new transform, we use the same expression as for aggregation, only with the new `r2`

shape.

```
= src.transform * src.transform.scale(
new_transform2 / r2.shape[1]),
(src.width / r2.shape[0])
(src.height
) new_transform2
```

```
Affine(6.169999999999921, 0.0, 794599.1076146346,
0.0, -6.169999999998726, 8935384.324602526)
```

The original raster `dem.tif`

was already quite detailed, so it would be difficult to see any difference when plotting it along with the disaggregation result. A zoom-in of a small section of the rasters works better. Figure 4.37 allows us to see the top-left corner of the original raster and the disaggregated one, demonstrating the increase in the number of pixels through disaggregation.

```
1)[:5, :5], transform=src.transform);
rasterio.plot.show(src.read(25, :25], transform=new_transform2); rasterio.plot.show(r2[:
```

Code to export the disaggregated raster would be identical to the one used above for the aggregated raster.

### 4.3.4 Resampling

Raster aggregation and disaggregation (Section 4.3.3) are only suitable when we want to change just the resolution of our raster by a fixed factor. However, what to do when we have two or more rasters with different resolutions and origins? This is the role of resamplingâa process of computing values for new pixel locations. In short, this process takes the values of our original raster and recalculates new values for a target raster with custom resolution and origin (Figure 4.38).

There are several methods for estimating values for a raster with different resolutions/origins (Figure 4.38). The main resampling methods include:

- Nearest neighbor: assigns the value of the nearest cell of the original raster to the cell of the target one. This is a fast simple technique that is usually suitable for resampling categorical rasters
- Bilinear interpolation: assigns a weighted average of the four nearest cells from the original raster to the cell of the target one. This is the fastest method that is appropriate for continuous rasters
- Cubic interpolation: uses values of the 16 nearest cells of the original raster to determine the output cell value, applying third-order polynomial functions. Used for continuous rasters and results in a smoother surface compared to bilinear interpolation, but is computationally more demanding
- Cubic spline interpolation: also uses values of the 16 nearest cells of the original raster to determine the output cell value, but applies cubic splines (piecewise third-order polynomial functions). Used for continuous rasters
- Lanczos windowed sinc resampling: uses values of the 36 nearest cells of the original raster to determine the output cell value. Used for continuous rasters
- Additionally, we can use straightforward summary methods, taking into account all pixels that coincide with the target pixel, such as average (Figure 4.36), minimum, maximum (Figure 4.38), median, mode, and sum

The above explanation highlights that only nearest neighbor resampling is suitable for categorical rasters, while all remaining methods can be used (with different outcomes) for continuous rasters.

With **rasterio**, resampling can be done using the `rasterio.warp.reproject`

function . To clarify this naming convention, note that raster *reprojection* is not fundamentally different from *resampling*âthe difference is just whether the target grid is in the same CRS as the origin (resampling) or in a different CRS (reprojection). In other words, reprojection is *resampling* into a grid that is in a different CRS. Accordingly, both resampling and reprojection are done using the same function `rasterio.warp.reproject`

. We will demonstrate reprojection using `rasterio.warp.reproject`

later in Section 6.8.

The information required for `rasterio.warp.reproject`

, whether we are resampling or reprojecting, is:

- The source and target
*CRS*. These may be identical, when resampling, or different, when reprojecting - The source and target
*transform*

Importantly, `rasterio.warp.reproject`

can work with file connections, such as a connection to an output file in write (`'w'`

) mode. This makes the function efficient for large rasters.

The target and destination CRS are straightforward to specify, depending on our choice. The source transform is also available, e.g., through the `.transform`

property of the source file connection. The only complicated part is to figure out the *destination transform*. When resampling, the transform is typically derived either from a *template* raster, such as an existing raster file that we would like our origin raster to match, or from a numeric specification of our target grid (see below). Otherwise, when the exact grid is not of importance, we can simply aggregate or disaggregate our raster as shown above (Section 4.3.3). (Note that when *reprojecting*, the target transform is not as straightforward to figure out, therefore we further use the `rasterio.warp.calculate_default_transform`

function to compute it, as will be shown in Section 6.8.)

Finally, the resampling method is specified through the `resampling`

parameter of `rasterio.warp.reproject`

. The default is nearest neighbor resampling. However, as mentioned above, you should be aware of the distiction between resampling methods, and choose the appropriate one according to the data type (continuous/categorical), the input and output resolution, and resampling purposes. Possible arguments for `resampling`

include:

`rasterio.enums.Resampling.nearest`

âNearest neighbor`rasterio.enums.Resampling.bilinear`

âBilinear`rasterio.enums.Resampling.cubic`

âCubic`rasterio.enums.Resampling.lanczos`

âLanczos windowed`rasterio.enums.Resampling.mode`

âMode. i.e., most common value`rasterio.enums.Resampling.min`

âMinimum`rasterio.enums.Resampling.max`

âMaximum`rasterio.enums.Resampling.med`

âMedian`rasterio.enums.Resampling.sum`

âMedian

Letâs demonstrate resampling into a destination grid which is specified through numeric constraints, such as the extent and resolution. These could have been specified manually (such as here), or obtained from a template raster metadata that we would like to match. Note that the resolution of the destination grid is ~10 times more coarse (300 \(m\)) than the original resolution of `dem.tif`

(~30 \(m\)) (Figure 4.38).

```
= 794650
xmin = 798250
xmax = 8931750
ymin = 8935350
ymax = 300 res
```

The corresponding transform based on these constraints can be created using the `rasterio.transform.from_origin`

function, as follows:

```
= rasterio.transform.from_origin(
dst_transform =xmin,
west=ymax,
north=res,
xsize=res
ysize
) dst_transform
```

```
Affine(300.0, 0.0, 794650.0,
0.0, -300.0, 8935350.0)
```

Again, note that in case we needed to resample into a grid specified by an existing âtemplateâ raster, we could skip this step and simply read the transform from the template file, as in `rasterio.open('template.tif').transform`

.

Now, we can move on to creating the destination file connection. For that, we also have to know the raster dimensions that can be derived from the extent and the resolution.

```
= int((xmax - xmin) / res)
width = int((ymax - ymin) / res)
height width, height
```

`(12, 12)`

Now we can create the destination file connection. We are using the same metadata as the source file, except for the dimensions and the transform, which are going to be different and reflecting the resampling process.

```
= src.meta.copy()
dst_kwargs
dst_kwargs.update({'transform': dst_transform,
'width': width,
'height': height
})= rasterio.open('output/dem_resample_nearest.tif', 'w', **dst_kwargs) dst
```

Finally, we reproject using function `rasterio.warp.reproject`

. Note that the source and destination are specified using `rasterio.band`

applied on either the file connection, reflecting the fact that we operate on a specific layer of the rasters. The resampling method being used here is nearest neighbor resampling (`rasterio.enums.Resampling.nearest`

).

```
rasterio.warp.reproject(=rasterio.band(src, 1),
source=rasterio.band(dst, 1),
destination=src.transform,
src_transform=src.crs,
src_crs=dst_transform,
dst_transform=src.crs,
dst_crs=rasterio.enums.Resampling.nearest
resampling )
```

```
(Band(ds=<open DatasetWriter name='output/dem_resample_nearest.tif' mode='w'>, bidx=1, dtype='float32', shape=(12, 12)),
Affine(300.0, 0.0, 794650.0,
0.0, -300.0, 8935350.0))
```

In the end, we close the file connection and create a new file `output/dem_resample_nearest.tif`

with the resampling result (Figure 4.38).

` dst.close()`

Here is another code section just to demonstrate a different resampling method, the maximum resampling, i.e., every new pixel gets the maximum value of all the original pixels it coincides with (Figure 4.38). Note that all arguments in the `rasterio.warp.reproject`

function call are identical to the previous example, except for the `resampling`

method.

```
= rasterio.open('output/dem_resample_maximum.tif', 'w', **dst_kwargs)
dst
rasterio.warp.reproject(=rasterio.band(src, 1),
source=rasterio.band(dst, 1),
destination=src.transform,
src_transform=src.crs,
src_crs=dst_transform,
dst_transform=src.crs,
dst_crs=rasterio.enums.Resampling.max
resampling
) dst.close()
```

The original raster `dem.tif`

, and the two resampling results `dem_resample_nearest.tif`

and `dem_resample_maximum.tif`

, are shown in Figure 4.38.

```
# Input
= plt.subplots(figsize=(4,4))
fig, ax =ax);
rasterio.plot.show(src, ax# Nearest neighbor
= plt.subplots(figsize=(4,4))
fig, ax open('output/dem_resample_nearest.tif'), ax=ax);
rasterio.plot.show(rasterio.# Maximum
= plt.subplots(figsize=(4,4))
fig, ax open('output/dem_resample_maximum.tif'), ax=ax); rasterio.plot.show(rasterio.
```

## 4.4 Exercises

## 4.5 References

*Cartographica: The International Journal for Geographic Information and Geovisualization*10 (2): 112â22. https://doi.org/bjwv52.

*Essential Image Processing and GIS for Remote Sensing*. Chichester, West Sussex, UK, Hoboken, NJ: Wiley-Blackwell.

*The Cartographic Journal*30 (1): 46â51. https://doi.org/fx74gh.