06-reproj.Rmd

# Reprojecting geographic data {#reproj-geo-data}

## Prerequisites {-}

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

```{r, message=FALSE}
library(sf)
library(raster)
library(tidyverse)
library(spData)
library(spDataLarge)
```

## Introduction

<!-- A vital type of geometry transformation is *reprojecting* from one coordinate reference system (CRS) to another. -->
<!-- Because of the importance of reprojection, introduced in Chapter \@ref(spatial-class), and the fact that it applies to raster and vector geometries alike, it is the topic of the first section in this chapter. -->

Section \@ref(crs-intro) introduced coordinate reference systems (CRSs) and demonstrated their importance.
This chapter goes further.
It highlights issues that can arise when using inappropriate CRSs and how to *transform* data from one CRS to another.

As illustrated in Figure \@ref(fig:vectorplots), there are two types of CRS: *geographic* ('lon/lat', with units in degrees longitude and latitude) and *projected* (typically in units of meters from a datum).
This has consequences because many geometric operations a *projected* CRS:
most geometric operations in **sf**, for example, assume a projected CRS and generate warnings if the data is *geographic*, using the function `st_is_longlat()` (this is because under the hood GEOS assumes projected data).
Unfortunately R does not always know the CRS of an object, as shown below using the example of London introduced in section \@ref(vector-data):

<!-- , which is created by *coercing* a `data.frame` into an `sf` object (the `coords` argument specifies the coordinates): -->

```{r}
london = data.frame(lon = -0.1, lat = 51.5) %>% 
  st_as_sf(coords = c("lon", "lat"))
st_is_longlat(london)
```

This shows that unless a CRS is manually specifified or is loaded from a source that has CRS metadata, the CRS is `NA`.
A CRS can be added to `sf` objects with `st_set_crs()` as follows:^[
The CRS can also be added when creating `sf` objects with the `crs` argument (e.g. `st_sf(geometry = st_sfc(st_point(c(-0.1, 51.5))), crs = 4326)`).
The same argument can also be used to set the CRS when creating raster datasets (e.g. `raster(crs = "+proj=longlat")`).
]

```{r}
london_geo = st_set_crs(london, 4326)
st_is_longlat(london_geo)
```

If the CRS of an object with lon/lat coordinates is not specified CRSs, this can lead to problems.
An example is provided below, which creates a buffer of one unit around `london` and `london_geo` objects:

```{r}
london_buff_no_crs = st_buffer(london, dist = 1)
london_buff = st_buffer(london_geo, dist = 1)
```

Only the second operation generates a warning.
The warning message it useful, telling us that result may be of limited use because it is in units of latitude and longitude, rather than meters or some other suitable measure of distance assumed by `st_buffer()`.
The consequences of a failure to work on projected data are illustrated in Figure \@ref(fig:crs-buf) (left panel):
the buffer is elongated in the north-south direction because lines of longitude converge towards the Earth's poles.

```{block2 type="rmdnote"}
The distance between two lines of longitude, called meridians, is around 111 km at the equator (execute `geosphere::distGeo(c(0, 0), c(1, 0))` to find the precise distance).
This shrinks to zero at the poles.
At the latitude of London, for example, meridians are less than 70 km apart (challenge: execute code that verifies this).
<!-- `geosphere::distGeo(c(0, 51.5), c(1, 51.5))` -->
Lines of latitude, by contrast, have constant distance from each other irrespective of latitude: they are always around 111 km apart, including at the equator and near the poles.
This is illustrated in Figures \@ref(fig:crs-buf) and \@ref(fig:wintriproj).  
```

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 is to create a projected copy of the data and run the operation on that.
This is done in the code chunk below:

```{r}
london_proj = data.frame(x = 530000, y = 180000) %>% 
  st_as_sf(coords = 1:2, crs = 27700)
```

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 replace by `...`):

```{r, eval=FALSE}
st_crs(london_proj)
#> Coordinate Reference System:
#>   EPSG: 27700 
#>   proj4string: "+proj=tmerc +lat_0=49 +lon_0=-2 ... +units=m +no_defs"
```

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: [geography.hunter.cuny.edu/~jochen/GTECH361/lectures/](http://www.geography.hunter.cuny.edu/~jochen/GTECH361/lectures/lecture04/concepts/Map%20coordinate%20systems/Projection%20parameters.htm) as well as [http://proj4.org/parameters.html](http://proj4.org/parameters.html). 
Another great resource on projection definitions is [http://spatialreference.org/](http://spatialreference.org/).
]
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.
<!-- 
1 degree distance (great circle distance) at the equator:
geosphere::alongTrackDistance(c(0, 0), c(0, 1), c(0, 1)) 
but 1 degree converted into m distance at the latitude of London:
coords = st_coordinates(london)
geosphere::alongTrackDistance(coords, coords + c(1, 0), coords + c(1, 0))
-->
As pointed out above, moving one degree means moving a bit more than 111 km at the equator (to be precise: 111,320 meters).
This is used as the new buffer distance:

```{r}
london_proj_buff = st_buffer(london_proj, 111320)
```

The result in Figure \@ref(fig:crs-buf) (right panel) shows that buffers based on a projected CRS are not distorted:
every part of the buffer's border is equidistant to London.

```{r crs-buf, fig.cap="Buffer on vector representations of London with a geographic (left) and projected (right) CRS. The circular point represents London and the grey outline represents the outline of the UK.", fig.asp=1, fig.show='hold', out.width="45%", echo=FALSE}
uk = rnaturalearth::ne_countries(scale = 50) %>% 
  st_as_sf() %>% 
  filter(grepl(pattern = "United Kingdom|Ire", x = name_long))
plot(london_buff, graticule = st_crs(4326), axes = TRUE)
plot(london_geo, add = TRUE)
plot(st_geometry(uk), add = TRUE, border = "grey", lwd = 3)
uk_proj = uk %>%
  st_transform(27700)
plot(london_proj_buff, graticule = st_crs(27700), axes = TRUE)
plot(london_proj, add = TRUE)
plot(st_geometry(uk_proj), add = TRUE, border = "grey", lwd = 3)
```

The importance of CRSs (primarily whether they are projected or geographic) has been demonstrated using the example of London.
The subsequent sections go into more depth, exploring which CRS to use and the details of reprojecting vector and raster objects.

## When to reproject?

The previous section showed how to set the CRS manually, with `st_set_crs(london, 4326)`.
In real world applications, however, CRSs are usually set automatically when data is read-in.
The main task involving CRSs is often to *transform* objects, from one CRS into another.
But when should data be transformed? And into which CRS?
There are no clear-cut answers to these questions and CRS selection always involves trade-offs [@maling_coordinate_1992].
However there are some general principles, provided in this section, that can help decide. 

First it's worth considering *when to transform*.
In some cases transformation to a projected CRS is essential, such as when using geometric functions such as `st_buffer()`, Figure \@ref(fig:crs-buf) shows.
Conversely, publishing data online with the **leaflet** package may require a geographic CRS.
<!-- If the visualization phase of a project involves publishing results using [leaflet](https://github.com/Leaflet/Leaflet) via the common format [GeoJSON](http://geojson.org/) (a common scenario) projected data should probably be transformed to WGS84.  -->
Another case is when two objects with different CRSs must be compared or combined, as shown when we try to find the distance between two objects with different CRSs:

```{r, eval=FALSE}
st_distance(london_geo, london_proj)
# > Error: st_crs(x) == st_crs(y) is not TRUE
```

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 is usually 'to the projected CRS', which in this case is the British National Grid (BNG, EPSG:27700):

```{r}
london2 = st_transform(london_geo, 27700)
```

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}
st_distance(london2, london_proj)
```

## Which CRS to use?

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].
For geographic CRSs the answer is often [WGS84](https://en.wikipedia.org/wiki/World_Geodetic_System#A_new_World_Geodetic_System:_WGS_84), not only for web mapping (covered in the previous paragraph) 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 CRS 'BNG' (with its associated EPSG code 27700) is well-known and b) the original dataset (`london`) already had that CRS.

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, equi-distant, conformal (with shapes remaining unchanged), or some combination of compromises of those.
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 [proj4.org/geodesic.html](https://proj4.org/geodesic.html)).
For any projected CRS the results may not be accurate when used on geometries covering hundreds of kilometers.

When deciding a custom CRS we recommend the following:^[
Many thanks to an anonymous reviewer whose comments formed the basis of this advice.
]

- A Lambert azimuthal equal-area ([LAEA](https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection)) projection for a custom local projection (set `lon_0` and `lat_0` to the center of the study area), which is an equal-area projection at all locations but distorts shapes beyond thousands of kilometres.
-  Azimuthal equidistant ([AEQD](https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection)) projections for a specifically accurate straight-line distance between a point and the centre point of the local projection.
- Lambert conformal conic ([LCC](https://en.wikipedia.org/wiki/Lambert_conformal_conic_projection)) projections for regions covering thousands of kilometres, 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 kilometres from the center.

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)).

Almost everywhere on Earth has a UTM code, such as "60H" which refers to northern New Zealand where R was invented.
All UTM projections have the same datum (WGS84) and their EPSG codes run sequentially from 32601 to 32660 (for northern hemisphere locations) and 32701 to 32760 (southern hemisphere locations).

```{r, eval=FALSE, echo=FALSE}
utm_nums_n = 32601:32660
utm_nums_s = 32701:32760
crs_data = rgdal::make_EPSG()
crs_data[grep(utm_nums_n[1], crs_data$code), ] # zone 1N
crs_data[grep(utm_nums_n[60], crs_data$code), ] # zone 60N
crs_data[grep(utm_nums_s[1], crs_data$code), ] 
crs_data[grep(utm_nums_s[60], crs_data$code), ] 
crs_data[grep("UTM zone 60N", crs_data$note), ] # many
crs_data[grep("UTM zone 60S", crs_data$note), ] # many
crs_data[grep("UTM zone 60S", crs_data$note), ] # many
crs_utm = crs_data[grepl("utm", crs_data$prj4), ] # 1066
crs_utm_zone = crs_utm[grepl("zone=", crs_utm$prj4), ]
crs_utm_south = crs_utm[grepl("south", crs_utm$prj4), ]
```

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:^[
Credit: Josh O'Brien's post on [stackoverflow](https://stackoverflow.com/a/9188972) inspired this function.
] 

<!-- Idea: create full function with message and flexibility in later chapter (RL) -->
<!-- I think this code needs a short description (JM)-->
```{r}
lonlat2UTM = function(lonlat) {
  utm = (floor((lonlat[1] + 180) / 6) %% 60) + 1
  if(lonlat[2] > 0) utm + 32600 else
    utm + 32700
}
```

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

```{r, echo=FALSE, eval=FALSE}
stplanr::geo_code("Auckland")
```


```{r}
epsg_utm_auk = lonlat2UTM(c(174.7, -36.9))
epsg_utm_lnd = lonlat2UTM(st_coordinates(london))
st_crs(epsg_utm_auk)$proj4string
st_crs(epsg_utm_lnd)$proj4string
```

Maps of UTM zones such as that provided by [dmap.co.uk](http://www.dmap.co.uk/utmworld.htm) confirm: the London is in UTM zone 30U.
<!-- London can be transformed into this CRS as follows (result not shown): -->
<!-- idea: create figure showing UTM zones -->

<!-- ```{r} -->
<!-- lnd_utm = st_transform(london, crs = epsg_utm) -->
<!-- ``` -->

Another approach to automatically selecting 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 centerpoint of a dataset.
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.

Although we used vector datasets to illustrate the points outlined in this section, the principles apply equally to raster datasets.
The subsequent sections explain features of CRS transformation that are unique to each geographic data model, continuing with vector data in the next section (section \@ref(reproj-vec-geom)) and moving-on to explain how raster transformation is different, in section \@ref(reprojecting-raster-geometries).

<!-- This approach is used in the **stplanr** function `geo_select_crs()` which returns a CRS object that can be used in other functions (see `?stplanr::geo_select_aeq` for further details): -->

<!-- ```{r} -->
<!-- stplanr::geo_select_aeq(london) -->
<!-- ``` -->

<!-- Another **stplanr** function, `geo_buffer()`, uses this behind the scenes to enable buffers to be created around objects with geographic CRSs with units of metres, and returns the result in the original CRS, as illustrated in the code chunk below: -->

<!-- ```{r} -->
<!-- london_proj_buff2 = stplanr::geo_buffer(london, dist = 111320) -->
<!-- ``` -->

<!-- ```{r, eval=FALSE, echo=FALSE} -->
<!-- library(tmap) -->
<!-- tmap_mode("view") -->
<!-- qtm(st_transform(london_proj_buff, 4326)) + -->
<!--   qtm(london_proj_buff2, "red") + -->
<!--   qtm(london_buff) -->
<!-- ``` -->


## Reprojecting vector geometries {#reproj-vec-geom}

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.
<!-- Depending on projections used, reprojection could be either lossy or lossless. -->
<!-- I don't understand the following sentence -->
<!-- For example, loss of spatial information could occur when the new CRS is only adequate for smaller area than input vector. -->
<!-- Do you have an example for the next sentence? -->
<!-- The precision could be also lost when transforming coordinate systems with different datums - in those situations approximations are used. -->
<!-- However, in most cases CRS vector transformation is lossless. -->
This is illustrated by `cycle_hire_osm`, an `sf` object from **spData** that represents cycle hire locations across London.
The previous section showed how the CRS of vector data can be queried with `st_crs()`.
Although the output of this function is printed as a single entity, the result is in fact a named list of class `crs`, with names `proj4string` (which contains full details of the CRS) and `epsg` for its code.
This is demonstrated below:

```{r}
crs_lnd = st_crs(cycle_hire_osm)
class(crs_lnd)
crs_lnd$epsg
```

This duality of CRS objects means that they can be set either using an EPSG code or a `proj4string`.
This means that `st_crs("+proj=longlat +datum=WGS84 +no_defs")` is equivalent to `st_crs(4326)`, although not all `proj4string`s have an associated EPSG code.
Both elements of the CRS are changed by transforming the object to a projected CRS:

```{r, eval=FALSE, echo=FALSE}
crs1 = st_crs("+proj=longlat +datum=WGS84")
crs2 = st_crs("+datum=WGS84 +proj=longlat")
crs3 = st_crs(4326)
crs1 == crs2
crs1 == crs3
```

```{r}
cycle_hire_osm_projected = st_transform(cycle_hire_osm, 27700)
```

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.
Another option is to use a function from the **rgdal** package to find the name of the CRS:

```{r}
crs_codes = rgdal::make_EPSG()[1:2]
dplyr::filter(crs_codes, code == 27700)
```

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)".
But what about the `proj4string` element?
`proj4string`s are text strings in a particular format the describe the CRS.
They can be seen as formulas for converting a projected point into a point on the surface of the Earth and can be accessed from `crs` objects as follows (see [proj4.org](http://proj4.org/) for further details of what the output means):

```{r, eval=FALSE}
st_crs(27700)$proj4string
#> [1] "+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 ...
```

```{block2 type='rmdnote'}
Printing a spatial object in the console, automatically returns its coordinate reference system.
To access and modify it explicitly, use the `st_crs` function, for example, `st_crs(cycle_hire_osm)`.
```

## Modifying map projections

Established CRSs captured by EPSG codes are well-suited for many applications.
However in some cases it is desirable to create a new CRS, using a custom `proj4string`.
This system allows a very wide range of projections to be created, as we'll see in some of the custom map projections in this section.
<!-- as we mentioned in section \@ref(crs-in-r). -->

A long and growing list of projections has been developed and many of these these can be set with the `+proj=` element of `proj4string`s.^[See the Wikipedia page '[List of map projections](https://en.wikipedia.org/wiki/List_of_map_projections)' for 70+ projections, and illustrations.]
When mapping the world while preserving areal 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}
world_mollweide = st_transform(world, crs = "+proj=moll")
```
<!-- plot(world_mollweide$geom) -->
<!-- plot(world_mollweide$geom, graticule = TRUE) -->

```{r mollproj, echo=FALSE, fig.cap="Mollweide projection of the world.", warning=FALSE}
par_old = par()
par(mar = c(0, 0, 1, 0))
plot(st_geometry(world_mollweide), graticule = TRUE, main = "the Mollweide projection")
par(par_old)
```

On the other hand, when mapping the world, it is often desirable to have as little distortion as possible for all spatial properties (area, direction, distance).
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}
world_wintri = lwgeom::st_transform_proj(world, crs = "+proj=wintri")
```
<!-- plot(world_wintri$geom) -->
<!-- plot(world_wintri$geom, graticule = TRUE) -->

```{r wintriproj, echo=FALSE, fig.cap="Winkel tripel projection of the world.", warning=FALSE}
world_wintri_gr = st_graticule(lat = c(-89.9, seq(-80, 80, 20), 89.9)) %>% 
  lwgeom::st_transform_proj(crs = "+proj=wintri")
par_old = par()
par(mar = c(0, 0, 1, 0))
plot(st_geometry(world_wintri_gr), main = "the Winkel tripel projection", col = "grey")
plot(st_geometry(world_wintri), add = TRUE)
par(par_old)
```

```{block2 type='rmdnote'}
The two main functions for transformation of simple features coordinates are `sf::st_transform()` and `sf::sf_project()`. 
The `st_transform` function uses the GDAL interface to PROJ.4, while `sf_project()` (which works with two-column numeric matrices, representing points) and `lwgeom::st_transform_proj()` use the PROJ.4 API directly.
The first one is appropriate for most situations, and provides a set of the most often used parameters and well defined transformations.
The second one allows for greater customization of a projection, which includes cases when some of the PROJ.4 parameters (e.g., `+over`) or projection (`+proj=wintri`) is not available in `st_transform()`.
```

```{r, eval=FALSE, echo=FALSE}
# demo of sf_project
mat_lonlat = as.matrix(data.frame(x = 0:20, y = 50:70))
plot(mat_lonlat)
mat_projected = sf_project(from = st_crs(4326)$proj4string, to = st_crs(27700)$proj4string, pts = mat_lonlat)
plot(mat_projected)
```

Moreover, PROJ.4 parameters can be modified in most CRS definitions.
The below code transforms the coordinates to the Lambert azimuthal equal-area projection centered on longitude and latitude of `0` (Figure \@ref(fig:laeaproj1)).

```{r}
world_laea1 = st_transform(world, crs = "+proj=laea +x_0=0 +y_0=0 +lon_0=0 +lat_0=0")
```
<!-- plot(world_laea1$geom) -->
<!-- plot(world_laea1$geom, graticule = TRUE) -->

```{r laeaproj1, echo=FALSE, fig.cap="Lambert azimuthal equal-area projection of the world centered on longitude and latitude of 0.", warning=FALSE}
par_old = par()
par(mar = c(0, 0, 1, 0))
plot(st_geometry(world_laea1), graticule = TRUE, main = "the Lambert azimuthal equal-area projection")
par(par_old)
```

We can change the PROJ.4 parameters, for example the center of the projection using the `+lon_0` and `+lat_0` parameters. 
The code below gives the map centered on New York City (Figure \@ref(fig:laeaproj2)).

```{r}
world_laea2 = st_transform(world, crs = "+proj=laea +x_0=0 +y_0=0 +lon_0=-74 +lat_0=40")
```
<!-- plot(world_laea2$geom) -->
<!-- plot(world_laea2$geom, graticule = TRUE) -->

```{r laeaproj2, echo=FALSE, fig.cap="Lambert azimuthal equal-area projection of the world centered on New York City.", warning=FALSE}
world_laea2_g = st_graticule(ndiscr = 10000) %>%
  st_transform("+proj=laea +x_0=0 +y_0=0 +lon_0=-74 +lat_0=40.1 +ellps=WGS84 +no_defs") %>% 
  st_geometry()
par_old = par()
par(mar = c(0, 0, 1, 0))
plot(world_laea2_g, main = "the Lambert azimuthal equal-area projection", col = "grey")
plot(st_geometry(world_laea2), add = TRUE)
par(par_old)
```

More information on CRS modifications can be found in the [Using PROJ.4](http://proj4.org/usage/index.html) documentation.

<!-- https://github.com/r-spatial/lwgeom/issues/6 -->
<!-- ```{r} -->
<!-- # devtools::install_github("r-spatial/lwgeom") -->
<!-- library(lwgeom) -->
<!-- world_3 = lwgeom::st_transform_proj(world, crs = "+proj=wintri") -->
<!-- plot(world_3$geom) -->
<!-- ``` -->
<!-- http://bl.ocks.org/vlandham/raw/9216751/ -->

## Reprojecting raster geometries

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

The raster reprojection process is done with `projectRaster()` from the **raster** package.
Like the `st_transform()` function demonstrated in the previous section, `projectRaster()` takes a geographic object (a raster dataset in this case) and a `crs` argument.
However, `projectRaster()` only accepts the lengthy `proj4string` definitions of a CRS rather than concise EPSG codes.

```{block2 type='rmdnote'}
It is possible to use a EPSG code in a `proj4string` definition with `"+init=epsg:MY_NUMBER"`.
For example, one can use the `"+init=epsg:4326"` definition to set CRS to WGS84 (EPSG code of 4326).
The PROJ.4 library automatically adds the rest of parameters and converts it into `"+init=epsg:4326 +proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0"`,
```

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 `nlcd2011.tif` file provides information for a small area in Utah, USA obtained from [National Land Cover Database 2011](https://www.mrlc.gov/nlcd2011.php) in the NAD83 / UTM zone 12N CRS.

```{r}
cat_raster = raster(system.file("raster/nlcd2011.tif", package = "spDataLarge"))
crs(cat_raster)
```

In this region, 14 land cover classes were distinguished^[Full list of NLCD2011 land cover classes can be found at https://www.mrlc.gov/nlcd11_leg.php]:

```{r}
unique(cat_raster)
```

When reprojecting categorical raster, we need to ensure that our new estimated values would still have values of our original classes.
This could be done using the nearest neighbor method (`ngb`).
In this method, value of the output cell is calculated based on the nearest cell center of the input raster.

For example, we want to change the CRS to WGS 84. 
It can be desired when we want to visualize a raster data on top of a web basemap, such as the Google or OpenStreetMap map tiles.
The first step is to obtain the proj4 definition of this CRS, which can be done using the [http://spatialreference.org](http://spatialreference.org/ref/epsg/wgs-84/) webpage. 
The second and last step is to define the reprojection method in the `projectRaster()` function, which in case of categorical data is the nearest neighbor method (`ngb`):

```{r}
wgs84 = "+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs"
cat_raster_wgs84 = projectRaster(cat_raster, crs = wgs84, method = "ngb")
```

Many properties of the new object differ from the previous one, which include 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 14 to 15 because of the addition of `NA` values, not because a new category has been created --- the land cover classes are preserved).
<!-- freq(cat_raster_wgs84) -->
<!-- freq(cat_raster) -->

```{r catraster, echo=FALSE}
data_frame(
  CRS = c("NAD83", "WGS84"),
  nrow = c(nrow(cat_raster), nrow(cat_raster_wgs84)),
  ncol = c(ncol(cat_raster), ncol(cat_raster_wgs84)),
  ncell = c(ncell(cat_raster), ncell(cat_raster_wgs84)),
  resolution = c(mean(res(cat_raster)), mean(res(cat_raster_wgs84), na.rm = TRUE)),
  unique_categories = c(length(unique(values(cat_raster))), length(unique(values(cat_raster_wgs84))))
) %>% knitr::kable(caption = "Key attributes in the original ('cat_raster') and projected ('cat_raster_wgs84') categorical raster datasets.", digits = 4)
```

Reprojecting raster data with continuous (`numeric` or in this case `integer`) values follows an almost identical procedure.
`srtm.tif` in **spDataLarge** contains a digital elevation model from [the Shuttle Radar Topography Mission (SRTM)](https://www2.jpl.nasa.gov/srtm/) representing height above sea level (elevation) in meters.
Its CRS is WGS84:

```{r}
con_raster = raster(system.file("raster/srtm.tif", package = "spDataLarge"))
crs(con_raster)
```

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.
<!--
"Quadric and cubic polynomials are also popular interpolation functions for resampling with more complexity and improved accuracy" [@liu_essential_2009].
However, these interpolation methods are still unavailable in the **raster** package.
-->
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 the Oblique Lambert azimuthal equal-area projection, and reproject the raster into this CRS, using the `bilinear` method:

<!-- nice link, but does not fit into the text here in my opinion
First, we need to obtain the proj4 definition of the existing projected CRS appropriate for this area or create a new one using the [Projection Wizard](http://projectionwizard.org/) online tool [@savric_projection_2016].
-->

```{r}
equalarea = "+proj=laea +lat_0=37.32 +lon_0=-113.04"
con_raster_ea = projectRaster(con_raster, crs = equalarea, method = "bilinear")
crs(con_raster_ea)
```

Raster reprojection on numeric variables also leads to small changes 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}
data_frame(
  CRS = c("WGS84", "Equal-area"),
  nrow = c(nrow(con_raster), nrow(con_raster_ea)),
  ncol = c(ncol(con_raster), ncol(con_raster_ea)),
  ncell = c(ncell(con_raster), ncell(con_raster_ea)),
  resolution = c(mean(res(cat_raster)), mean(res(cat_raster_wgs84), na.rm = TRUE)),
  mean = c(mean(values(con_raster)), mean(values(con_raster_ea), na.rm = TRUE))
) %>% knitr::kable(caption = "Key attributes original ('con_raster') and projected ('con_raster') continuous raster datasets.", digits = 4)
```


```{block2 type='rmdnote'}
Of course, the limitations of 2D Earth projections apply as much to vector as to raster data.
At best we can comply with two out of three spatial properties (distance, area, direction).
Therefore, the task at hand determines which projection to choose. 
For instance, if we are interested in a density (points per grid cell or inhabitants per grid cell) we should use an equal-area projection (see also chapter \@ref(location)).
```

There is more to learn about CRSs.
An excellent resource in this area, also implemented in R, is the website R Spatial.
Chapter 6 for this free online book is recommended reading --- see [rspatial.org/spatial/rst/6-crs.html](http://rspatial.org/spatial/rst/6-crs.html)

<!-- why new na? -->

<!-- res option in projectRaster? -->
<!-- note1: in most of the cases reproject vector, not raster-->
<!-- note2: equal area projections are the best for raster calculations -->
<!-- q: should we mentioned gdal_transform? -->

## Exercises

<!-- CRS CONVERSION -->
<!-- 1. vector reprojection exercise (e.g. modification of proj4) -->
1. Create a new object called `nz_wgs` by transforming `nz` object into the WGS84 CRS.
    - Create an object of class `crs` for both and use this to query their CRSs.
    - With reference to the bounding box of each object, what units does each CRS use?
    - Remove the CRS from `nz_wgs` and plot the result: what is wrong with this map of New Zealand and why?
```{r, eval=FALSE, echo=FALSE}
st_crs(nz)
nz_wgs = st_transform(nz, 4326)
nz_crs = st_crs(nz)
nz_wgs_crs = st_crs(nz_wgs)
nz_crs$epsg
nz_wgs_crs$epsg
st_bbox(nz)
st_bbox(nz_wgs)
nz_wgs_NULL_crs = st_set_crs(nz_wgs, NA)
nz_27700 = st_transform(nz_wgs, 27700)
par(mfrow = c(1, 3))
plot(st_geometry(nz))
plot(st_geometry(nz_wgs))
plot(st_geometry(nz_wgs_NULL_crs))
# answer: it is fatter in the East-West direction
# because New Zealand is close to the South Pole and meridians converge there
plot(st_geometry(nz_27700))
par(mfrow = c(1, 1))
```
1. Transform the `world` dataset to the transverse Mercator projection (`"+proj=tmerc"`) and plot the result.
What has changed and why?
Try to transform it back into WGS 84 and plot the new object.
Why does the new object differ from the original one?
```{r, echo=FALSE, eval=FALSE}
# see https://github.com/r-spatial/sf/issues/509
world_tmerc = st_transform(world, "+proj=tmerc")
plot(st_geometry(world_tmerc))
world_4326 = st_transform(world_tmerc, 4326)
plot(st_geometry(world_4326))
```
1. Transform the continuous raster (`cat_raster`) into WGS 84 using the nearest neighbor interpolation method. 
What has changed?
How does it influence the results?
```{r, echo=FALSE, eval=FALSE}
con_raster = raster(system.file("raster/srtm.tif", package="spDataLarge"))
con_raster_wgs84 = projectRaster(con_raster, crs = wgs84, method = "ngb")
con_raster_wgs84
```
1. Transform the categorical raster (`cat_raster`) into WGS 84 using the bilinear interpolation method.
What has changed?
How does it influence the results?
```{r, echo=FALSE, eval=FALSE}
wgs84 = "+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs"
cat_raster_wgs84 = projectRaster(cat_raster, crs = wgs84, method = "bilinear")
cat_raster_wgs84
```