- This chapter requires the following packages (**lwgeom** is also used, but does not need to be attached):

```{r 06-reproj-1, message=FALSE, warning=FALSE}

Section \@ref(crs-intro) introduced coordinate reference systems (CRSs), with a focus on the two major types: *geographic* ('lon/lat', with units in degrees longitude and latitude) and *projected* (typically with units of meters from a datum) coordinate systems.

This chapter builds on that knowledge and goes further.

It demonstrates how to set and *transform* geographic data from one CRS to another and, furthermore, highlights specific issues that can arise due to ignoring CRSs that you should be aware of, especially if your data is stored with lon/lat coordinates.

\index{CRS!geographic}

\index{CRS!projected}

In many projects there is no need to worry about, let alone convert between, different CRSs.

It is important to know if your data is in a projected or geographic coordinate system, and the consequences of this for geometry operations.

However, if you know the CRS of your data and the consequences for geometry operations (covered in the next section), CRSs should *just work* behind the scenes: people often suddenly need to learn about CRSs when things go wrong.

Having a clearly defined project CRS that all project data is in, plus understanding how and why to use different CRSs, can ensure that things don't go wrong.

Furthermore, learning about coordinate systems will deepen your knowledge of geographic datasets and how to use them effectively.

This chapter teaches the fundamentals of CRSs, demonstrates the consequences of using different CRSs (including what can go wrong), and how to 'reproject' datasets from one coordinate system to another.

In the next section we introduce CRSs in R, followed by Section \@ref(crs-in-r) which shows how to get and set CRSs associated with spatial objects.

Section \@ref(geom-proj) demonstrates the importance of knowing what CRS your data is in with reference to a worked example of creating buffers.

We tackle questions of when to reproject and which CRS to use in Section \@ref(whenproject) and Section \@ref(which-crs), respectively.

We cover reprojecting vector and raster objects in sections \@ref(reproj-vec-geom) and \@ref(reproj-ras) and modifying map projections in Section \@ref(mapproj).

\index{CRS!EPSG}

\index{CRS!WKT}

\index{CRS!proj4string}

Most modern geographic tools that require CRS conversions, including core R-spatial packages and desktop GIS software such as QGIS, interface with [PROJ](https://proj.org), an open source C++ library that "transforms coordinates from one coordinate reference system (CRS) to another".

CRSs can be described in many ways, including the following.

1. Simple yet potentially ambiguous statements such as "it's in lon/lat coordinates".

2. Formalised yet now outdated 'proj4 strings' such as `+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs`.

3. With an identifying 'authority:code' text string such as `EPSG:4326`.

Each refers to the same thing: the 'WGS84' coordinate system that forms the basis of Global Positioning System (GPS) coordinates and many other datasets.

But which one is correct?

The short answer is that the third way to identify CRSs is correct: `EPSG:4326` is understood by **sf** (and by extension **stars**) and **terra** packages R covered in this book, plus many other software projects for working with geographic data including [QGIS](https://docs.qgis.org/3.16/en/docs/user_manual/working_with_projections/working_with_projections.html) and [PROJ](https://proj.org/development/quickstart.html).

`EPSG:4326` is future-proof.

Furthermore, although it is machine readable, unlike the `proj4string` representation "EPSG:4326" is short, easy to remember and highly 'findable' online (searching for EPSG:4326 yields a dedicated page on the website [epsg.io](https://epsg.io/4326), for example).

The more concise identifier `4326` is understood by **sf**, but **we recommend the more explicit `AUTHORITY:CODE` representation to prevent ambiguity and to provide context**.

The longer answer is that none of the three descriptions are sufficient and more detail is needed for unambiguous CRS handling and transformations: due to the complexity of CRSs, it is not possible to capture all relevant information about them in such short text strings.

For this reason, the Open Geospatial Consortium (OGC, which also developed the simple features specification that the **sf** package implements) developed an open standard format for describing CRSs that is called WKT (Well Known Text).

This is detailed in a [100+ page document](https://portal.opengeospatial.org/files/18-010r7) that "defines the structure and content of a text string implementation of the abstract model for coordinate reference systems described in ISO 19111:2019" [@opengeospatialconsortium_wellknown_2019].

The WKT representation of the WGS84 CRS, which has the **identifier** `EPSG:4326` is as follows:

The output of the command shows how the CRS identifier (also known as a Spatial Reference Identifier or [SRID](https://postgis.net/workshops/postgis-intro/projection.html)) works: it is simply a look-up, providing a unique identifier associated with a more complete WTK representation of the CRS.

This raises the question: what happens if there is a mismatch between the identifier and the longer WKT representation of a CRS?

On this point @opengeospatialconsortium_wellknown_2019 is clear, the verbose WKT representation takes precedence over the [identifier](https://docs.opengeospatial.org/is/18-010r7/18-010r7.html#37):

The convention of referring to CRSs identifiers in the form `AUTHORITY:CODE`, which is also used by geographic software written in other [languages](https://jorisvandenbossche.github.io/blog/2020/02/11/geopandas-pyproj-crs/), allows a wide range of formally defined coordinate systems to be referred to.^[

Several

other ways of referring to unique CRSs can be used, with five identifier types (EPSG code, PostGIS SRID, INTERNAL SRID, PROJ4 string, and WKT strings accepted by [QGIS](https://docs.qgis.org/3.16/en/docs/pyqgis_developer_cookbook/crs.html?highlight=srid) and other identifier types such as a more verbose variant of the `EPSG:4326` identifier, `urn:ogc:def:crs:EPSG::4326` @opengeospatialconsortium_wellknown_2019.

The most commonly used authority in CRS identifiers is *EPSG*, an acronym for the European Petroleum Survey Group which published a standardized list of CRSs (the EPSG was [taken over](http://wiki.gis.com/wiki/index.php/European_Petroleum_Survey_Group) by the oil and gas body the [Geomatics Committee of the International Association of Oil & Gas Producers](https://www.iogp.org/our-committees/geomatics/) in 2005).

Other authorities can be used in CRS identifiers.

`ESRI:54030`, for example, refers to ESRI's implementation of the Robinson projection, which has the following WKT string (only first 8 lines shown):

```{r, eval=FALSE}

```{r, eval=FALSE, echo=FALSE}

WKT strings are exhaustive, detailed, and precise, allowing for unambiguous CRSs storage and transformations.

They contain all relevant information about any given CRS, including its datum and ellipsoid, prime meridian, projection, and units.^[

Before the emergence of WKT CRS definitions, `proj4string` was the standard way to specify coordinate operations and store CRSs.

These string representations, built on a key=value form (e.g, `+proj=longlat +datum=WGS84 +no_defs`), are, however, currently discouraged in most cases.

PROJ version 6 and further still allows to use `proj4string`s to define coordinate operations, but some `proj4string` keys are no longer supported or are not advisable to use (e.g., `+nadgrids`, `+towgs84`, `+k`, `+init=epsg:`) and only three datums (i.e., WGS84, NAD83, and NAD27) can be directly set in `proj4string`.

Importantly, `proj4string`s are not used to store CRSs anymore.

Longer explanations of the evolution of CRS definitions and the PROJ library can be found in @bivand_progress_2021, Chapter 2 of @pebesma_spatial_2022, and [blog post by Floris Vanderhaeghe](https://inbo.github.io/tutorials/tutorials/spatial_crs_coding/).

As outlined in the [PROJ documentation](https://proj.org/development/reference/cpp/cpp_general.html) there are different versions of the WKT CRS format including WKT1 and two variants of WKT2, the latter of which (WKT2, 2018 specification) corresponds to the ISO 19111:2019 [@opengeospatialconsortium_wellknown_2019].

Let's look at how CRSs are stored in R spatial objects and how they can be queried and set.

First we will look at getting and setting CRSs in **vector** geographic data objects, starting with the following example:

```{r 02-spatial-data-52, message=FALSE, results='hide'}

Our new object, `new_vector`, is a an `sf` data frame representing countries worldwide (see the help page `?spData::world` for details).

The CRS can be retrieved with the **sf** function `st_crs()`.

```{r 02-spatial-data-53, eval=FALSE}

```{r, echo=FALSE, eval=FALSE}

The output is a list containing two main components:

1. `User input` (in this case `WGS 84`, a synonym for `EPSG:4326` which in this case was taken from the input file), corresponding to CRS identifiers described above

1. `wkt`, containing the full WKT string with all relevant information about the CRS.

The `input` element is flexible, and depending on the input file or user input, can contain the `AUTHORITY:CODE` representation (e.g., `EPSG:4326`), the CRS's name (e.g., `WGS 84`), or even the `proj4string` definition.

The `wkt` element stores the WKT representation, which is used when saving the object to a file or doing any coordinate operations.

Above, we can see that the `new_vector` object has the WGS84 ellipsoid, uses the Greenwich prime meridian, and the latitude and longitude axis order.

In this case, we also have some additional elements, such as `USAGE` explaining the area suitable for the use of this CRS, and `ID` pointing to the CRS's identifier: `EPSG:4326`.

The `st_crs` function also has one helpful feature -- we can retrieve some additional information about the used CRS.

For example, try to run:

- `st_crs(new_vector)$IsGeographic` to check is the CRS is geographic or not

- `st_crs(new_vector)$units_gdal` to find out the CRS units

- `st_crs(new_vector)$srid` extracts its 'SRID' identifier (when available)

- `st_crs(new_vector)$proj4string` extracts the `proj4string` representation

In cases when a coordinate reference system (CRS) is missing or the wrong CRS is set, the `st_set_crs()` function can be used (in this case the WKT string remains unchanged because the CRS was already set correctly when the file was read-in):

```{r 02-spatial-data-54}

```{r, echo=FALSE, eval=FALSE}

Getting and setting CRSs works in a similar way for **raster** geographic data objects.

The `crs()` function in the `terra` package accesses CRS information from a `SpatRaster` object (note the use of the `cat()` function to print it nicely):

```{r 02-spatial-data-55}

The output is the WKT representation of CRS.

The same function, `crs()`, is can be also used to set a CRS for raster objects.

```{r 02-spatial-data-56}

Here, we can use either the identifier (recommended in most cases) or complete WKT representation.

Alternative methods to set `crs` include `proj4string` strings or CRSs extracted from other existing object with `crs()`, although these approaches may be less future proof.

Importantly, the `st_crs()` and `crs()` functions do not alter coordinates' values or geometries.

Their role is only to set a metadata information about the object CRS.

In some cases the CRS of a geographic object is unknown, as is the case in the `london` dataset created in the code chunk below, building on the example of London introduced in Section \@ref(vector-data):

```{r 06-reproj-2}

The output `NA` shows that **sf** does not know what the CRS is and is unwilling to guess (`NA` literally means 'not available').

Unless a CRS is manually specified or is loaded from a source that has CRS metadata, **sf** does not make any explicit assumptions about which coordinate systems, other than to say "I don't know".

This behavior makes sense given the diversity of available CRSs but differs from some approaches, such as the GeoJSON file format specification, which makes the simplifying assumption that all coordinates have a lon/lat CRS: `EPSG:4326`.

A CRS can be added to `sf` objects in three main ways:

- By assigning the CRS to a pre-existing object, e.g. with `st_crs(london) = "EPSG:4326"`

- By passing a CRS to the `crs` argument in **sf** functions that create geometry objects such as `st_as_sf(... crs = "EPSG:4326")`. The same argument can also be used to set the CRS when creating raster datasets (e.g., `rast(crs = "EPSG:4326")`)

- With the `st_set_crs()`, which returns a version of the data that has a new CRS, an approach that is demonstrated in the following code chunk

```{r 06-reproj-3}

Datasets without a specified CRS can cause problems: all geographic coordinates have a coordinate system and software can only make good decisions around plotting and and geometry operations if it knows what type of CRS it is working with.

If no CRS has been set, **sf** uses the GEOS geometry library for many operations.

GEOS is not well suited to lon/lat CRSs, as we will see later in this chapter.

If a CRS has been set, **sf** will use either GEOS or the S2 *spherical geometry engine* depending on the type of CRS.

Since **sf** version 1.0.0, R's ability to work with geographic vector datasets that have lon/lat CRSs has improved substantially, thanks to its integration with S2 introduced in Section \@ref(s2).

To demonstrate the importance of CRSs, we will in this section create a buffer of 100 km around the `london` object created in the previous section.

We will also create a deliberately faulty buffer with a 'distance' of 1 degree, which is roughly equivalent to 100 km (1 degree is about 111 km at the equator).

Before diving into the code, it may be worth skipping briefly ahead to peek at Figure \@ref(fig:crs-buf) to get a visual handle on the outputs that you should be able to reproduce by following the code chunks below.

The first stage is to create three buffers around the `london` and `london_geo` objects created above with boundary distances of 1 degree and 100 km (or 100,000 m, which can be expressed as `1e5` in scientific notation) from central London:

```{r 06-reproj-4-1}

In the first line above, **sf** assumes that the input is projected and generates a result that has a buffer in units of degrees, which is problematic, as we will see.

In the second line, **sf** silently uses the spherical geometry engine S2, introduced in Chapter \@ref(spatial-class), to calculate the extent of the buffer using the default value of `max_cells = 1000` --- set to `100` in line three --- the consequences which will become apparent shortly (see `?s2::s2_buffer_cells` for details).

To highlight the impact of **sf**'s use of the S2 geometry engine for unprojected (geographic) coordinate systems, we will temporarily disable it with the command `sf_use_s2()` (which is on, `TRUE`, by default), in the code chunk below.

Like `london_buff_no_crs`, the new `london_geo` object is a geographic abomination: it has units of degrees, which makes no sense in the vast majority of cases:

```{r 06-reproj-4-2}

The warning message above hints at issues with performing planar geometry operations on lon/lat data.

When spherical geometry operations are turned off, with the command `sf::sf_use_s2(FALSE)`, buffers (and other geometric operations) may result in worthless outputs because they use units of latitude and longitude, a poor substitute for proper units of distances such as meters.

```{block2 06-reproj-5, type="rmdnote"}

Do not interpret the warning about the geographic (`longitude/latitude`) CRS as "the CRS should not be set": it almost always should be!

It is better understood as a suggestion to *reproject* the data onto a projected CRS.

This suggestion does not always need to be heeded: performing spatial and geometric operations makes little or no difference in some cases (e.g., spatial subsetting).

But for operations involving distances such as buffering, the only way to ensure a good result (without using spherical geometry engines) is to create a projected copy of the data and run the operation on that.

This is done in the code chunk below:

```{r 06-reproj-6}

The result is a new object that is identical to `london`, but reprojected onto a suitable CRS (the British National Grid, which has an EPSG code of 27700 in this case) that has units of meters.

We can verify that the CRS has changed using `st_crs()` as follows (some of the output has been replaced by `...`):

```{r 06-reproj-7, eval=FALSE}

Notable components of this CRS description include the EPSG code (`EPSG: 27700`), the projection ([transverse Mercator](https://en.wikipedia.org/wiki/Transverse_Mercator_projection), `+proj=tmerc`), the origin (`+lat_0=49 +lon_0=-2`) and units (`+units=m`).^[

For a short description of the most relevant projection parameters and related concepts, see the fourth lecture by Jochen Albrecht hosted at

http://www.geography.hunter.cuny.edu/~jochen/GTECH361/lectures/ and information at https://proj.org/usage/projections.html.

Other great resources on projections are spatialreference.org and progonos.com/furuti/MapProj.

The fact that the units of the CRS are meters (rather than degrees) tells us that this is a projected CRS: `st_is_longlat(london_proj)` now returns `FALSE` and geometry operations on `london_proj` will work without a warning, meaning buffers can be produced from it using proper units of distance.

The following line of code creates a buffer around *projected* data of exactly 100 km:

```{r 06-reproj-8}

The geometries of the three `london_buff*` objects that *have* a specified CRS created above (`london_buff_s2`, `london_buff_lonlat` and `london_buff_projected`) created in the preceding code chunks are illustrated in Figure \@ref(fig:crs-buf).

```{r crs-buf-old, include=FALSE, eval=FALSE}

```{r crs-buf, fig.cap="Buffers around London showing results created with the S2 spherical geometry engine on lon/lat data (left), projected data (middle) and lon/lat data without using spherical geometry (right). The left plot illustrates the result of buffering unprojected data with sf, which calls Google's S2 spherical geometry engine by default with `max_cells = 1000` (thin line). The thick 'blocky' line illustrates the result of the same operation with `max_cells = 100`.", fig.scap="Buffers around London with a geographic and projected CRS.", echo=FALSE, fig.asp=0.39, fig.width = 8}

uk = rnaturalearth::ne_countries(scale = 50) %>%

st_as_sf() %>%

filter(grepl(pattern = "United Kingdom|Ire", x = name_long))

library(tmap)

tm1 = tm_shape(london_buff_s2, bbox = st_bbox(london_buff_s2_100_cells)) +

tm_graticules(lwd = 0.2) +

tm_borders(col = "black", lwd = 0.5) +

tm_shape(london_buff_s2_100_cells) +

tm_borders(col = "black", lwd = 1.5) +

tm_shape(uk) +

tm_polygons(lty = 3, alpha = 0.2, col = "#567D46") +

tm_shape(london_proj) +

tm_symbols()

tm2 = tm_shape(london_buff_projected, bbox = st_bbox(london_buff_s2_100_cells)) +

tm_grid(lwd = 0.2) +

tm_borders(col = "black", lwd = 0.5) +

tm_shape(uk) +

tm_polygons(lty = 3, alpha = 0.2, col = "#567D46") +

tm_shape(london_proj) +

tm_symbols()

tm3 = tm_shape(london_buff_lonlat, bbox = st_bbox(london_buff_s2_100_cells)) +

tm_graticules(lwd = 0.2) +

tm_borders(col = "black", lwd = 0.5) +

tm_shape(uk) +

tm_polygons(lty = 3, alpha = 0.2, col = "#567D46") +

tm_shape(london_proj) +

tm_symbols()

tmap_arrange(tm1, tm2, tm3, nrow = 1)

To make the `london` and `london_proj` objects geographically comparable one of them must be transformed into the CRS of the other.

But which CRS to use?

The answer depends on context: many projects, especially those involving web mapping, require outputs in EPSG:4326, in which case it is worth transforming the projected object.

If, however, the project requires planar geometry operations rather than spherical geometry operations engine (e.g. to create buffers with smooth edges), it may be worth transforming data with a geographic CRS into an equivalent object with a projected CRS, such as the British National Grid (EPSG:27700).

That is the subject of Section \@ref(reproj-vec-geom).

\index{CRS!reprojection}

\index{projection!World Geodetic System}

The question of *which CRS* is tricky, and there is rarely a 'right' answer:

"There exist no all-purpose projections, all involve distortion when far from the center of the specified frame" [@bivand_applied_2013].

Additionally, you should not be attached just to one projection for every task.

It is possible to use one projection for some part of the analysis, another projection for a different part, and even some other for visualization.

Always try to pick the CRS that serves your goal best!

When selecting **geographic CRSs**, the answer is often [WGS84](https://en.wikipedia.org/wiki/World_Geodetic_System#A_new_World_Geodetic_System:_WGS_84).

It is used not only for web mapping, but also because GPS datasets and thousands of raster and vector datasets are provided in this CRS by default.

WGS84 is the most common CRS in the world, so it is worth knowing its EPSG code: 4326.

This 'magic number' can be used to convert objects with unusual projected CRSs into something that is widely understood.

What about when a **projected CRS** is required?

In some cases, it is not something that we are free to decide:

"often the choice of projection is made by a public mapping agency" [@bivand_applied_2013].

This means that when working with local data sources, it is likely preferable to work with the CRS in which the data was provided, to ensure compatibility, even if the official CRS is not the most accurate.

The example of London was easy to answer because (a) the British National Grid (with its associated EPSG code 27700) is well known and (b) the original dataset (`london`) already had that CRS.

\index{UTM}

A commonly used default is Universal Transverse Mercator ([UTM](https://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_system)), a set of CRSs that divides the Earth into 60 longitudinal wedges and 20 latitudinal segments.

The transverse Mercator projection used by UTM CRSs is conformal but distorts areas and distances with increasing severity with distance from the center of the UTM zone.

Documentation from the GIS software Manifold therefore suggests restricting the longitudinal extent of projects using UTM zones to 6 degrees from the central meridian (source: [manifold.net](http://www.manifold.net/doc/mfd9/universal_transverse_mercator_projection.htm)).

Therefore, we recommend using UTM only when your focus is on preserving angles for relatively small area!

Almost every place on Earth has a UTM code, such as "60H" which refers to northern New Zealand where R was invented.

UTM EPSG codes run sequentially from 32601 to 32660 for northern hemisphere locations and from 32701 to 32760 for southern hemisphere locations.

```{r 06-reproj-12, eval=FALSE, echo=FALSE}

To show how the system works, let's create a function, `lonlat2UTM()` to calculate the EPSG code associated with any point on the planet as [follows](https://stackoverflow.com/a/9188972/):

```{r 06-reproj-13}

The following command uses this function to identify the UTM zone and associated EPSG code for Auckland and London:

```{r 06-reproj-14, echo=FALSE, eval=FALSE}

```{r 06-reproj-15}

Currently, we also have tools helping us to select a proper CRS, which includes the **crssuggest** package <!--add ref or docs-->.

The main function in this package, `suggest_crs()`, takes a spatial object with geographic CRS and returns a list of possible projected CRSs that could be used for the given area.^[This package also allows to figure out the true CRS of the data without any CRS information attached.]

Another helpful tool is a webpage https://jjimenezshaw.github.io/crs-explorer/ that lists CRSs based on selected location and type.

Important note: while these tools is helpful in many situations, you need to be aware of the properties of the recommended CRS before you apply it.

\index{CRS!custom}

In cases where an appropriate CRS is not immediately clear, the choice of CRS should depend on the properties that are most important to preserve in the subsequent maps and analysis.

All CRSs are either equal-area, equidistant, conformal (with shapes remaining unchanged), or some combination of compromises of those (section \@ref(projected-coordinate-reference-systems)).

Custom CRSs with local parameters can be created for a region of interest and multiple CRSs can be used in projects when no single CRS suits all tasks.

'Geodesic calculations' can provide a fall-back if no CRSs are appropriate (see [proj.org/geodesic.html](https://proj.org/geodesic.html)).

Regardless of the projected CRS used, the results may not be accurate for geometries covering hundreds of kilometers.

\index{CRS!custom}

When deciding on a custom CRS, we recommend the following:^[

Many thanks to an anonymous reviewer whose comments formed the basis of this advice.

\index{projection!Lambert azimuthal equal-area}

\index{projection!Azimuthal equidistant}

\index{projection!Lambert conformal conic}

\index{projection!Stereographic}

\index{projection!Universal Transverse Mercator}

- A Lambert azimuthal equal-area ([LAEA](https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection)) projection for a custom local projection (set latitude and longitude of origin to the center of the study area), which is an equal-area projection at all locations but distorts shapes beyond thousands of kilometers

- Azimuthal equidistant ([AEQD](https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection)) projections for a specifically accurate straight-line distance between a point and the center point of the local projection

- Lambert conformal conic ([LCC](https://en.wikipedia.org/wiki/Lambert_conformal_conic_projection)) projections for regions covering thousands of kilometers, with the cone set to keep distance and area properties reasonable between the secant lines

- Stereographic ([STERE](https://en.wikipedia.org/wiki/Stereographic_projection)) projections for polar regions, but taking care not to rely on area and distance calculations thousands of kilometers from the center

One possible approach to automatically select a projected CRS specific to a local dataset is to create an azimuthal equidistant ([AEQD](https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection)) projection for the center-point of the study area.

This involves creating a custom CRS (with no EPSG code) with units of meters based on the center point of a dataset.

Note that this approach should be used with caution: no other datasets will be compatible with the custom CRS created and results may not be accurate when used on extensive datasets covering hundreds of kilometers.

The principles outlined in this section apply equally to vector and raster datasets.

Some features of CRS transformation however are unique to each geographic data model.

We will cover the particularities of vector data transformation in Section \@ref(reproj-vec-geom) and those of raster transformation in Section \@ref(reproj-ras).

Next, the last section, shows how to create custom map projections (Section \@ref(mapproj)).

\index{CRS!reprojection}

\index{vector!reprojection}

Chapter \@ref(spatial-class) demonstrated how vector geometries are made-up of points, and how points form the basis of more complex objects such as lines and polygons.

Reprojecting vectors thus consists of transforming the coordinates of these points, which form the vertices of lines and polygons.

Section \@ref(whenproject) contains an example in which at least one `sf` object must be transformed into an equivalent object with a different CRS to calculate the distance between two objects.

```{r 06-reproj-10}

Now that a transformed version of `london` has been created, using the **sf** function `st_transform()`, the distance between the two representations of London can be found.

It may come as a surprise that `london` and `london2` are just over 2 km apart!^[

The difference in location between the two points is not due to imperfections in the transforming operation (which is in fact very accurate) but the low precision of the manually-created coordinates that created `london` and `london_proj`.

Also surprising may be that the result is provided in a matrix with units of meters.

This is because `st_distance()` can provide distances between many features and because the CRS has units of meters.

Use `as.numeric()` to coerce the result into a regular number.

```{r 06-reproj-11}

This is demonstrated below with reference to `cycle_hire_osm`, an `sf` object from **spData** that represents 'docking stations' where you can hire bicycles in London.

The CRS of `sf` objects can be queried --- and as we learned in Section \@ref(reproj-intro) set --- with the function `st_crs()`.

The output is printed as multiple lines of text containing information about the coordinate system:

As we saw in Section \@ref(crs-setting), the main CRS components, `User input` and `wkt`, are printed as a single entity, the output of `st_crs()` is in fact a named list of class `crs` with two elements, single character strings named `input` and `wkt`:

```{r 06-reproj-16, echo=FALSE}

Additional elements can be retrieved with the `$` operator, including `Name`, `proj4string` and `epsg` (see [`?st_crs`](https://r-spatial.github.io/sf/reference/st_crs.html) and the CRS and tranformation tutorial on the GDAL [website](https://gdal.org/tutorials/osr_api_tut.html#querying-coordinate-reference-system) for details):

As mentioned in Section \@ref(crs-in-r), WKT representation, stored in the `$wkt` element of the `crs_lnd` object is the ultimate source of truth.

This means that the outputs of the previous code chunk are queries from the `wkt` representation provided by PROJ, rather than inherent attributes of the object and its CRS.

Both `wkt` and `User Input` elements of the CRS are changed when the object's CRS is transformed.

In the code chunk below, we create a new version of `cycle_hire_osm` with a projected CRS (only the first 4 lines of the CRS output are shown for brevity):

```{r 06-reproj-18, eval=FALSE}

The resulting object has a new CRS with an EPSG code 27700.

But how to find out more details about this EPSG code, or any code?

One option is to search for it online,

```{r 06-reproj-19}

The result shows that the EPSG code 27700 represents the British National Grid, a result that could have been found by searching online for "[EPSG 27700](https://www.google.com/search?q=CRS+27700)".

```{block2 06-reproj-21, type='rmdnote'}

\index{raster!reprojection}

\index{raster!warping}

\index{raster!transformation}

\index{raster!resampling}

The projection concepts described in the previous section apply equally to rasters.

However, there are important differences in reprojection of vectors and rasters:

transforming a vector object involves changing the coordinates of every vertex but this does not apply to raster data.

Rasters are composed of rectangular cells of the same size (expressed by map units, such as degrees or meters), so it is usually impracticable to transform coordinates of pixels separately.

Raster reprojection involves creating a new raster object, often with a different number of columns and rows than the original.

The attributes must subsequently be re-estimated, allowing the new pixels to be 'filled' with appropriate values.

In other words, raster reprojection can be thought of as two separate spatial operations: a vector reprojection of the raster extent to another CRS (Section \@ref(reproj-vec-geom)), and computation of new pixel values through resampling (Section \@ref(resampling)).

Thus in most cases when both raster and vector data are used, it is better to avoid reprojecting rasters and reproject vectors instead.

```{block2 06-reproj-35a, type='rmdnote'}

```{r, include=FALSE}

The raster reprojection process is done with `project()` from the **terra** package.

Like the `st_transform()` function demonstrated in the previous section, `project()` takes a geographic object (a raster dataset in this case) and some CRS representation as the second argument.

On a side note -- the second argument can also be an existing raster object with a different CRS.

Let's take a look at two examples of raster transformation: using categorical and continuous data.

Land cover data are usually represented by categorical maps.

The `nlcd.tif` file provides information for a small area in Utah, USA obtained from [National Land Cover Database 2011](https://www.mrlc.gov/data/nlcd-2011-land-cover-conus) in the NAD83 / UTM zone 12N CRS, as shown in the output of the code chunk below (only first line of output shown):

```{r 06-reproj-29, results='hide'}

In this region, 8 land cover classes were distinguished (a full list of NLCD2011 land cover classes can be found at [mrlc.gov](https://www.mrlc.gov/data/legends/national-land-cover-database-2011-nlcd2011-legend)):

```{r 06-reproj-30}

When reprojecting categorical rasters, the estimated values must be the same as those of the original.

This could be done using the nearest neighbor method (`near`), which sets each new cell value to the value of the nearest cell (center) of the input raster.

An example is reprojecting `cat_raster` to WGS84, a geographic CRS well suited for web mapping.

The first step is to obtain the PROJ definition of this CRS, which can be done, for example using the [http://spatialreference.org](http://spatialreference.org/ref/epsg/wgs-84/) webpage.

The final step is to reproject the raster with the `project()` function which, in the case of categorical data, uses the nearest neighbor method (`near`):

```{r 06-reproj-31}

Many properties of the new object differ from the previous one, including the number of columns and rows (and therefore number of cells), resolution (transformed from meters into degrees), and extent, as illustrated in Table \@ref(tab:catraster) (note that the number of categories increases from 8 to 9 because of the addition of `NA` values, not because a new category has been created --- the land cover classes are preserved).

```{r catraster, echo=FALSE}

Reprojecting numeric rasters (with `numeric` or in this case `integer` values) follows an almost identical procedure.

This is demonstrated below with `srtm.tif` in **spDataLarge** from [the Shuttle Radar Topography Mission (SRTM)](https://www2.jpl.nasa.gov/srtm/), which represents height in meters above sea level (elevation) with the WGS84 CRS:

```{r 06-reproj-32}

We will reproject this dataset into a projected CRS, but *not* with the nearest neighbor method which is appropriate for categorical data.

Instead, we will use the bilinear method which computes the output cell value based on the four nearest cells in the original raster.^[Other methods mentioned in Section \@ref(resampling) also can be used here.]

The values in the projected dataset are the distance-weighted average of the values from these four cells:

the closer the input cell is to the center of the output cell, the greater its weight.

The following commands create a text string representing WGS 84 / UTM zone 12N, and reproject the raster into this CRS, using the `bilinear` method:

```{r 06-reproj-34}

Raster reprojection on numeric variables also leads to small changes to values and spatial properties, such as the number of cells, resolution, and extent.

These changes are demonstrated in Table \@ref(tab:rastercrs)^[

Another minor change, that is not represented in Table \@ref(tab:rastercrs), is that the class of the values in the new projected raster dataset is `numeric`.

This is because the `bilinear` method works with continuous data and the results are rarely coerced into whole integer values.

This can have implications for file sizes when raster datasets are saved.

]:

```{r rastercrs, echo=FALSE}

```{block2 06-reproj-35, type='rmdnote'}

Established CRSs captured by `AUTHORITY:CODE` identifiers such as `EPSG:4326` are well suited for many applications.

However, it is desirable to create a new, custom CRS in some cases.

Section \@ref(which-crs) mentioned reasons for using custom CRSs, and provided several possible approaches.

Here, we show how to apply these ideas in R.

One possible approach to creating a custom CRS is to take an existing WKT definition of a CRS, modify some of its elements, and then use the new definition for reprojecting.

This can be done for spatial vectors with `st_crs()$wkt` and `st_transform()`, and for spatial rasters with `crs()` and `project()`.

Let's try it by transforming the `zion` object to a custom azimuthal equidistant (AEQD) CRS.

Using a custom AEQD CRS requires knowing the coordinates of the center point of a dataset in degrees (geographic CRS).

In our case, this information can be extracted by calculating a centroid of the `zion` area and transforming it into WGS84.

```{r, warning=FALSE}

Next, we can use the newly obtained values to update the WKT definition of the azimuthal equidistant (AEQD) CRS seen below.

Notice that we modified just two values below -- `"Central_Meridian"` to the longitude and `"Latitude_Of_Origin"` to the latitude of our centroid.

This approach's last step is to transform our original object (`zion`) to our new custom CRS (`zion_aeqd`).

Custom projections can also be made interactively, for example, using the [Projection Wizard](https://projectionwizard.org/#) web application [@savric_projection_2016].

This website allows you to select a spatial extent of your data and a distortion property, and returns a list of possible projections.

The list also contains WKT definitions of the projections that you can copy and use for reprojections.

See @opengeospatialconsortium_wellknown_2019 for details on creating custom CRS definitions using WKT strings.

\index{CRS!proj4string}

A `proj4string` definition can also be used to create custom projections, as long we accept its limitations mentioned in Section \@ref(crs-in-r).

For example, many projections have been developed and can be set with the `+proj=` element of `proj4string`s.^[

The Wikipedia page 'List of map projections' has 70+ projections and illustrations.

When mapping the world while preserving area relationships, the Mollweide projection is a good choice [@jenny_guide_2017] (Figure \@ref(fig:mollproj)).

To use this projection, we need to specify it using the `proj4string` element, `"+proj=moll"`, in the `st_transform` function:

```{r 06-reproj-22}

```{r mollproj, fig.cap="Mollweide projection of the world.", warning=FALSE, message=FALSE, echo=FALSE}

On the other hand, it is often desirable to have as little distortion as possible for all spatial properties (area, direction, distance) when mapping the world.

One of the most popular projections to achieve this is the Winkel tripel projection (Figure \@ref(fig:wintriproj)).^[

This projection is used, among others, by the National Geographic Society.

`st_transform_proj()` from the **lwgeom** package allows for coordinate transformations to a wide range of CRSs, including the Winkel tripel projection:

```{r 06-reproj-23}

```{r wintriproj, fig.cap="Winkel tripel projection of the world.", echo=FALSE}

```{block2 06-reproj-24, type='rmdnote', echo=FALSE}

```{r 06-reproj-25, eval=FALSE, echo=FALSE}

Moreover, `proj4string` parameters can be modified in most CRS definitions, for example the center of the projection can be adjusted using the `+lon_0` and `+lat_0` parameters.

The below code transforms the coordinates to the Lambert azimuthal equal-area projection centered on the longitude and latitude of New York City (Figure \@ref(fig:laeaproj2)).

```{r 06-reproj-27}

```{r laeaproj2, fig.cap="Lambert azimuthal equal-area projection of the world centered on New York City.", fig.scap="Lambert azimuthal equal-area projection centered on New York City.", warning=FALSE, echo=FALSE}

More information on CRS modifications can be found in the [Using PROJ](https://proj.org/usage/index.html) documentation.

```{r, echo=FALSE, results='asis'}