## table of contents

t.rast.algebra(1grass) | GRASS GIS User's Manual | t.rast.algebra(1grass) |

# NAME¶

**t.rast.algebra** - Apply temporal and spatial
operations on space time raster datasets using temporal raster algebra.

# KEYWORDS¶

temporal, algebra, raster, time

# SYNOPSIS¶

**t.rast.algebra**

**t.rast.algebra --help**

**t.rast.algebra** [-**sngd**] **expression**=*string*
**basename**=*string* [**suffix**=*string*]
[**nprocs**=*integer*] [--**help**] [--**verbose**]
[--**quiet**] [--**ui**]

## Flags:¶

**-s**-

Check the spatial topology of temporally related maps and process only spatially related maps **-n**-

Register Null maps **-g**-

Use granularity sampling instead of the temporal topology approach **-d**-

Perform a dry run, compute all dependencies and module calls but don’t run them **--help**-

Print usage summary **--verbose**-

Verbose module output **--quiet**-

Quiet module output **--ui**-

Force launching GUI dialog

## Parameters:¶

**expression**=*string***[required]**-

r.mapcalc expression for temporal and spatial analysis of space time raster datasets **basename**=*string***[required]**-

Basename of the new generated output maps

A numerical suffix separated by an underscore will be attached to create a unique identifier **suffix**=*string*-

Suffix to add at basename: set ’gran’ for granularity, ’time’ for the full time format, ’num’ for numerical suffix with a specific number of digits (default %05)

Default:*num* **nprocs**=*integer*-

Number of r.mapcalc processes to run in parallel

Default:*1*

# DESCRIPTION¶

*t.rast.algebra* performs temporal and spatial map algebra
operations on space time raster datasets (STRDS) using the temporal raster
algebra.

## PROGRAM USE¶

The module expects an **expression** as input parameter in the
following form:

**"result = expression"**

The statement structure is similar to that of r.mapcalc. In this
statement, **result** represents the name of the space time raster
dataset (STRDS) that will contain the result of the calculation that is
given as **expression** on the right side of the equality sign. These
expressions can be any valid or nested combination of temporal operations
and spatial overlay or buffer functions that are provided by the temporal
algebra.

The temporal raster algebra works only with space time raster
datasets (STRDS). The algebra provides methods for map selection based on
their temporal relations. It is also possible to temporally shift maps, to
create temporal buffer and to snap time instances to create a valid temporal
topology. Furthermore, expressions can be nested and evaluated in
conditional statements (if, else statements). Within if-statements, the
algebra provides temporal variables like start time, end time, day of year,
time differences or number of maps per time interval to build up conditions.

In addition the algebra provides a subset of the spatial operations from
r.mapcalc. All these operations can be assigned to STRDS or to the map lists
resulting of operations between STRDS.

By default, only temporal topological relations among space time
datasets (STDS) are evaluated. The **-s** flag can be used to
additionally activate the evaluation of the spatial topology based on the
spatial extent of maps.

The expression option must be passed as **quoted** expression,
for example:

t.rast.algebra expression="C = A + B" basename=resultWhere

**C**is the new space time raster dataset that will contain maps with the basename "result" and a numerical suffix separated by an underscore that represent the sum of maps from the STRDS

**A**and temporally equal maps (i.e., maps with equal temporal topology relation) from the STRDS

**B**.

The map **basename** for the result STRDS must always be
specified.

# TEMPORAL RASTER ALGEBRA¶

The temporal algebra provides a wide range of temporal operators and functions that will be presented in the following section.

## TEMPORAL RELATIONS¶

Several temporal topology relations are supported between maps
registered in space time datasets:

equals A ------

B ------ during A ----

B ------ contains A ------

B ---- starts A ----

B ------ started A ------

B ---- finishs A ----

B ------ finished A ------

B ---- precedes A ----

B ---- follows A ----

B ---- overlapped A ------

B ------ overlaps A ------

B ------ over both overlaps and overlapped

The relations must be read as: A is related to B, like - A equals B - A is during B - A contains B.

Topological relations must be specified with curly brackets {}.

## TEMPORAL OPERATORS¶

The temporal algebra defines temporal operators that can be
combined with other operators to perform spatio-temporal operations. The
temporal operators process the time instances and intervals of two
temporally related maps and calculate the resulting temporal extent in five
possible different ways.

LEFT REFERENCE l Use the time stamp of the left space time dataset INTERSECTION i Intersection DISJOINT UNION d Disjoint union UNION u Union RIGHT REFERENCE r Use the time stamp of the right space time dataset

## TEMPORAL SELECTION¶

The temporal selection simply selects parts of a space time
dataset without processing any raster or vector data. The algebra provides a
selection operator **:** that by default selects parts of a space time
dataset that are temporally equal to parts of a second space time dataset.
The following expression

C = A : B

means: select all parts of space time dataset A that are equal to B and store them in space time dataset C. These parts are time stamped maps.

In addition, the inverse selection operator **!:** is defined
as the complement of the selection operator, hence the following expression

C = A !: Bmeans: select all parts of space time time dataset A that are not equal to B and store them in space time dataset C.

To select parts of a STRDS using different topological relations
regarding to other STRDS, the temporal topology selection operator can be
used. This operator consists of the temporal selection operator, the
topological relations that must be separated by the logical OR operator
**|** and, the temporal extent operator. All three parts are separated by
comma and surrounded by curly brackets as follows: {"temporal selection
operator", "topological relations", "temporal
operator"}.

**Examples:**

C = A {:,equals} B C = A {!:,equals} BWe can now define arbitrary topological relations using the OR operator "|" to connect them:

C = A {:,equals|during|overlaps} BSelect all parts of A that are equal to B, during B or overlaps B.

In addition, we can define the temporal extent of the resulting STRDS by adding the temporal operator.

C = A {:,during,r} BSelect all parts of A that are during B and use the temporal extents from B for C.

The selection operator is implicitly contained in the temporal topology selection operator, so that the following statements are exactly the same:

C = A : B C = A {:} B C = A {:,equal} B C = A {:,equal,l} BSame for the complementary selection:

C = A !: B C = A {!:} B C = A {!:,equal} B C = A {!:,equal,l} B

## CONDITIONAL STATEMENTS¶

Selection operations can be evaluated within conditional
statements as showed below. Note that A and B can be either space time
datasets or expressions. The temporal relationship between the conditions
and the conclusions can be defined at the beginning of the if statement
(third and fourth examples below). The relationship between then and else
conclusion must be always equal.

if statement decision option temporal relations

if(if, then, else)

if(conditions, A) A if conditions are True; temporal topological relation between if and then is equal.

if(conditions, A, B) A if conditions are True, B otherwise; temporal topological relation between if, then and else is equal.

if(topologies, conditions, A) A if conditions are True; temporal topological relation between if and then is explicitly specified by topologies.

if(topologies, conditions, A, B) A if conditions are True, B otherwise; temporal topological relation between if, then and else is explicitly specified by topologies.

The conditions are comparison expressions that are used to
evaluate space time datasets. Specific values of temporal variables are
compared by logical operators and evaluated for each map of the STRDS.

**Important:** The conditions are evaluated from left to right.

## Logical operators¶

Symbol description

== equal

!= not equal

> greater than

>= greater than or equal

< less than

<= less than or equal

&& and

|| or

## Temporal functions¶

The following temporal functions are evaluated only for the STDS
that must be given in parenthesis.

td(A) Returns a list of time intervals of STDS A start_time(A) Start time as HH::MM:SS start_date(A) Start date as yyyy-mm-DD start_datetime(A) Start datetime as yyyy-mm-DD HH:MM:SS end_time(A) End time as HH:MM:SS end_date(A) End date as yyyy-mm-DD end_datetime(A) End datetime as yyyy-mm-DD HH:MM start_doy(A) Day of year (doy) from the start time [1 - 366] start_dow(A) Day of week (dow) from the start time [1 - 7], the start of the week is Monday == 1 start_year(A) The year of the start time [0 - 9999] start_month(A) The month of the start time [1 - 12] start_week(A) Week of year of the start time [1 - 54] start_day(A) Day of month from the start time [1 - 31] start_hour(A) The hour of the start time [0 - 23] start_minute(A) The minute of the start time [0 - 59] start_second(A) The second of the start time [0 - 59] end_doy(A) Day of year (doy) from the end time [1 - 366] end_dow(A) Day of week (dow) from the end time [1 - 7], the start of the week is Monday == 1 end_year(A) The year of the end time [0 - 9999] end_month(A) The month of the end time [1 - 12] end_week(A) Week of year of the end time [1 - 54] end_day(A) Day of month from the start time [1 - 31] end_hour(A) The hour of the end time [0 - 23] end_minute(A) The minute of the end time [0 - 59] end_second(A) The second of the end time [0 - 59]

## Comparison operator¶

As mentioned above, the conditions are comparison expressions that
are used to evaluate space time datasets. Specific values of temporal
variables are compared by logical operators and evaluated for each map of
the STDS and (optionally) related maps. For complex relations, the
comparison operator can be used to combine conditions.

The structure is similar to the select operator with the addition of an
aggregation operator: {"comparison operator", "topological
relations", aggregation operator, "temporal operator"}

This aggregation operator (| or &) defines the behaviour when a map is
related to more than one map, e.g. for the topological relation
’contains’. Should all (&) conditions for the related maps
be true or is it sufficient to have any (|) condition that is true. The
resulting boolean value is then compared to the first condition by the
comparison operator (|| or &&). By default, the aggregation operator
is related to the comparison operator:

comparison operator -> aggregation operator:

|| -> | and && -> &

**Examples:**

Condition 1 {||, equal, r} Condition 2 Condition 1 {&&, equal|during, l} Condition 2 Condition 1 {&&, equal|contains, |, l} Condition 2 Condition 1 {&&, equal|during, l} Condition 2 && Condition 3 Condition 1 {&&, equal|during, l} Condition 2 {&&,contains, |, r} Condition 3

## Hash operator¶

Additionally, the number of maps in intervals can be computed and
used in conditional statements with the hash (#) operator.

A {#, contains} B

This expression computes the number of maps from space time
dataset B which are during the time intervals of maps from space time
dataset A.

A list of integers (scalars) corresponding to the maps of A that contain maps
from B will be returned.

C = if({equal}, A {#, contains} B > 2, A {:, contains} B)

This expression selects all maps from A that temporally contain at least 2 maps from B and stores them in space time dataset C. The leading equal statement in the if condition specifies the temporal relation between the if and then part of the if expression. This is very important, so we do not need to specify a global time reference (a space time dataset) for temporal processing.

Furthermore, the temporal algebra allows temporal buffering,
shifting and snapping with the functions buff_t(), tshift() and tsnap(),
respectively.

buff_t(A, size) Buffer STDS A with granule ("1 month" or 5) tshift(A, size) Shift STDS A with granule ("1 month" or 5) tsnap(A) Snap time instances and intervals of STDS A

## Single map with temporal extent¶

The temporal algebra can also handle single maps with time stamps
in the tmap() function.

tmap()

For example:

C = A {:, during} tmap(event)

This statement selects all maps from space time data set A that are during the temporal extent of the single map ’event’

## Spatial raster operators¶

The module supports the following raster operations:

Symbol description precedence

% modulus 1

/ division 1

* multiplication 1

+ addition 2

- subtraction 2

And raster functions:

abs(x) return absolute value of x float(x) convert x to foating point int(x) convert x to integer [ truncates ] log(x) natural log of x sqrt(x) square root of x tan(x) tangent of x (x is in degrees) round(x) round x to nearest integer sin(x) sine of x (x is in degrees) isnull(x) check if x = NULL isntnull(x) check if x is not NULL null set null value exist(x) Check if x is in the current mapset

## Single raster map¶

The temporal raster algebra features also a function to integrate
single raster maps without time stamps into the expressions.

map()

For example:

C = A * map(constant_value)

This statement multiplies all raster maps from space time raster data set A with the raster map ’constant_value’

## Combinations of temporal, raster and select operators¶

The user can combine the temporal topology relations, the temporal
operators and the spatial/select operators to create spatio-temporal
operators as follows:

{"spatial or select operator", "list of temporal relations", "temporal operator"}

For multiple topological relations or several related maps the
spatio-temporal operators feature implicit aggregation. The algebra
evaluates the stated STDS by their temporal topologies and apply the given
spatio-temporal operators in a aggregated form. If we have two STDS A and B,
B has three maps: b1, b2, b3 that are all during the temporal extent of the
single map a1 of A, then the following arithmetic calculations would
implicitly aggregate all maps of B into one result map for a1 of A:

C = A {+, contains} B --> c1 = a1 + b1 + b2 + b3

**Important**: the aggregation behaviour is not symmetric

C = B {+, during} A --> c1 = b1 + a1

c2 = b2 + a1

c3 = b3 + a1

## Temporal neighbourhood modifier¶

The neighbourhood modifier of *r.mapcalc* is extended for the
temporal raster algebra with the temporal dimension. The format is
strds[t,r,c], where t is the temporal offset, r is the row offset and c is
the column offset.

strds[2]

refers to the second successor of the current map.

strds[1,2]refers to the cell one row below and two columns to the right of the current cell in the current map.

strds[1,-2,-1]refers to the cell two rows above and one column to the left of the current cell of the first successor map.

strds[-2,0,1]refers to the cell one column to the right of the current cell in the second predecessor map.

# EXAMPLES¶

## Computation of NDVI¶

# Sentinel-2 bands are stored separately in two STDRS "S2_b4" and "S2_b8" g.region raster=sentinel2_B04_10m -p t.rast.list S2_b4 t.rast.list S2_b8 t.rast.algebra basename=ndvi expression="ndvi = float(S2_b8 - S2_b4) / ( S2_b8 + S2_b4 )" t.rast.colors input=ndvi color=ndvi

## Sum of space-time raster datasets¶

Sum maps from STRDS A with maps from STRDS B which have equal time
stamps and are temporally before Jan. 1. 2005 and store them in STRDS D:

D = if(start_date(A) < "2005-01-01", A + B)

Create the sum of all maps from STRDS A and B that have equal time
stamps and store the new maps in STRDS C:

C = A + B

## Sum of space-time raster datasets with temporal topology relation¶

Same expression with explicit definition of the temporal topology
relation and temporal operators:

C = A {+,equal,l} B

## Selection of raster cells¶

Select all cells from STRDS B with equal temporal relations to
STRDS A, if the cells of A are in the range [100.0, 1600] of time intervals
that have more than 30 days (Jan, Mar, May, Jul, Aug, Oct, Dec):

C = if(A > 100 && A < 1600 && td(A) > 30, B)

## Selection of raster cells with temporal topology relation¶

Same expression with explicit definition of the temporal topology
relation and temporal operators:

C = if({equal}, A > 100 && A < 1600 {&&,equal} td(A) > 30, B)

## Conditional computation¶

Compute the recharge in meters per second for all cells of
precipitation STRDS "Prec" if the mean temperature specified in
STRDS "Temp" is higher than 10 degrees. Computation is performed
if STRDS "Prec" and "Temp" have equal time stamps. The
number of days or fraction of days per interval is computed using the td()
function that has as argument the STRDS "Prec":

C = if(Temp > 10.0, Prec / 3600.0 / 24.0 / td(Prec))

## Conditional computation with temporal topology relation¶

Same expression with explicit definition of the temporal topology
relation and temporal operators:

C = if({equal}, Temp > 10.0, Prec / 3600.0 / 24.0 {/,equal,l} td(Prec))

## Computation with time intervals¶

Compute the mean value of all maps from STRDS A that are located
during time intervals of STRDS B if more than one map of A is contained in
an interval of B, use A otherwise. The resulting time intervals are either
from B or A:

C = if(B {#,contain} A > 1, (B {+,contain,l} A - B) / (B {#,contain} A), A)

## Computation with time intervals with temporal topology relation¶

Same expression with explicit definition of the temporal topology
relation and temporal operators:

C = if({equal}, B {#,contain} A > 1, (B {+,contain,l} A {-,equal,l} B) {equal,=/} (B {#,contain} A), A)

# SEE ALSO¶

*r.mapcalc,* *t.vect.algebra,*
*t.rast3d.algebra,* *t.select,* *t.rast3d.mapcalc,*
*t.rast.mapcalc*

Temporal data processing Wiki

# REFERENCES¶

The use of this module requires the following software to be installed: PLY(Python-Lex-Yacc)

# Ubuntu/Debian sudo apt-get install python3-ply # Fedora sudo dnf install python3-ply # MS-Windows (OSGeo4W: requires "python3-pip" package to be installed) python3-pip install ply

Related publications:

- Gebbert, S., Pebesma, E. 2014.
*TGRASS: A temporal GIS for field based environmental modeling*. Environmental Modelling & Software 53, 1-12 (DOI) - preprint PDF - Gebbert, S., Pebesma, E. 2017.
*The GRASS GIS temporal framework*. International Journal of Geographical Information Science 31, 1273-1292 (DOI) - Gebbert, S., Leppelt, T., Pebesma, E., 2019.
*A topology based spatio-temporal map algebra for big data analysis*. Data 4, 86. (DOI)

# SEE ALSO¶

*v.overlay,* *v.buffer,* *v.patch,*
*r.mapcalc*

# AUTHORS¶

Thomas Leppelt, Sören Gebbert, Thünen Institute of Climate-Smart Agriculture

# SOURCE CODE¶

Available at: t.rast.algebra source code (history)

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