Data format details

Working with angular data

Eradiate notably manipulates and produces what we refer to as angular data, which represent variables dependent on one or more directional parameters. Typical examples are BRDFs (\(f_\mathrm{r} (\theta_\mathrm{i}, \varphi_\mathrm{i}, \theta_\mathrm{o}, \varphi_\mathrm{o})\)) or top-of-atmosphere BRFs (\(\mathit{BRF}_\mathrm{TOA} (\theta_\mathrm{sun}, \varphi_\mathrm{sun}, \theta_\mathrm{view}, \varphi_\mathrm{view})\)): a xarray data array representing them has at least one angular dimension (and corresponding coordinates). Eradiate has specific functionality to deal more easily with this sort of data.

Angular dependencies and coordinate variable names

Angular variable naming in Earth observation and radiative transfer modelling may sometimes clash or be confusing. Eradiate clearly distinguishes between two types of angular dependencies for its variables:

• Physical properties such as BRDFs and phase functions have intrinsic bidirectional dependencies which are referred to as incoming and outgoing directions. Data sets representing such quantities use coordinate variables phi_i, theta_i for the incoming direction’s azimuth and zenith angles, and phi_o, theta_o for their outgoing counterparts.

• Observations are usually parametrised by illumination (or solar) and viewing (or sensor) directions. For data sets representing such results, Eradiate uses coordinate variables sza, saa for solar zenith/azimuth angle and vza, vaa for viewing zenith/azimuth angle. A typical example of such variable is the top-of-atmosphere bidirectional reflectance factor (TOA BRF).

Under specific circumstances, one can directly convert an observation dataset to a physical property dataset. This, for instance, applies to top-of-atmosphere BRF data, but also any BRF computed or measured in a vacuum. In such cases, incoming/outgoing directions can be directly converted to illumination/viewing directions. But in general, this does not work.

Angular data set types

While one should clearly distinguish intrinsic and observation angular dependencies for correct physical interpretation of radiative data, both share an asymmetry between ‘incoming’ and ‘outgoing’ directions. Eradiate uses similar semantics to handle both angular data types, and the table below clarifies the nomenclature for the two types:

Type	Incoming	Outgoing
Intrinsic	\(\varphi_\mathrm{i}\), \(\theta_\mathrm{i}\)	\(\varphi_\mathrm{o}\), \(\theta_\mathrm{o}\)
Observation	\(\varphi_\mathrm{s}\), \(\theta_\mathrm{s}\)	\(\varphi_\mathrm{v}\), \(\theta_\mathrm{v}\)

Eradiate’s xarray containers do not explicitly keep track of the angular data set type. However, when relevant, coordinate naming is used to determine whether an angular data set is of intrinsic or observation type.

Angular data sets with a pair of angular dimensions \((\theta, \varphi)\) are called hemispherical. If they have two pairs of angular dimensions (incoming and outgoing), they are then called bi-hemispherical.

Measure data formats

Most measures in Earth observation radiative transfer modelling have angular dependencies. However, Eradiate uses storage data structures inherited from computer graphics technology and measure results are usually mapped against film coordinates \((x, y) \in [0, 1]^2\). When those data represent hemispherical quantities, a mapping transformation associate angles to film coordinates. For convenience, Eradiate ships helpers to convert data from film coordinates to angular coordinates.