Uncertainty Propagation

Uncertainties are propagated from product to product using Monte Carlo (MC) approach (see Supplement 1 to the “Guide to the expression of uncertainty in measurement”). This MC approach is implemented using the punpy module (see also the punpy ATBD) from the open-source CoMet toolkit. A metrological approach is followed, where for each processing stage, a measurement function is defined, as well as the input quantities and the measurand.

Here we summarise the main steps and detail how these were implemented for HYPERNETS. The main stages consist of:

Formulation: Defining the measurand (output quantity Y), the input quantities \(X = (X_{i},\ldots,\ X_{N})\), and the measurement function (as a model relating Y and X). One also needs to asign Probability Density Functions (PDF) of each of the input quantities, as well as define the correlation between them (through joint PDF). In our method, the joint PDF’s are made by first generating uncorrelated PDF, which are then correlated using the Cholesky decomposition method.
Propagation: propagate the PDFs for the \(X_i\) through the model to obtain the PDF for Y.
Summarizing: Use the PDF for Y to obtain the expectation of Y, the standard uncertainty u(Y) associated with Y (from the standard deviation), and the covariance between the different values in Y.

For the HYPERNETS processing, the first propagation of uncertainties is applied when calibrating the L1A scans. The measurement function for this step is given in Calibrate - process to L1A. The input quantities are the measured signal (in digital numbers), the dark signal, the calibration coefficients, the non-linearity coefficients, and the integration time of the measurement. All that remains to be done in order to complete the `Formulation’ stage of MC is to define the uncertainties and error-correlations for each of the input quantities as well as their PDF.

For the measured signal and dark signal, the random uncertainties can be determined from calculating the standard deviation between the different scans. In addition, systematic uncertainties in the measured signal will cancel with systematic uncertainties in the dark signal. We can thus treat these two input quantities as having entirely random uncertainties. The calibration coefficients and non-linearity coefficients were characterised in the lab by Tartu University. Uncertainties were quantified for 16 different uncertainty contributions, and preliminary error correlation structures were defined. The uncertainties on the integration times were assumed to be neglibible. All PDF are assumed to be Gaussian with the input quantities as the mean and their uncertainties as the width.

Once all the uncertainties and their correlation have been quantified, punpy can be used to perform the Propagation and summarizing stage of MC. Punpy returns the measurand (calibrated (ir)radiances for L1A) as well as the uncertainties and correlation matrix w.r.t. wavelength. All the HYPERNETS products are stored as digital effects tables (see https://www.comet-toolkit.org/tools/obsarray/), meaning that all the uncertainties and error correlation information have been stored in a machine-readable format using a structured metadata standard.

For the processing to L1B, the input quantities are simply the measurands from L1A, for which the uncertainties and error-correlation are already defined. The measurement function is a simple averaging function. Uncertainties are again propagated with punpy.

Similarly, the input quantities and their uncertainties for the L1C processing are defined by the previous processing levels. The main steps in the L1C processing are the 2 interpolations. Here the measurement function is implemented within the interpolation tool in the CoMet toolkit. This tool also handles uncertainty propagation (which internally uses punpy) in an entirely analogous way. The measurand (interpolated irradiances) and asoociated uncertainties and error-correlations are returned by the tool.

Finally, uncertainty are propagated to L2A with punpy using the measurement functions in Calibrate - process to L1A and the input quantities from the L1C products. Some basic information on how to interface with the uncertainty information in the HYPERNETS products is given in the Using HYPERNETS data page. Further information and examples can be found on the CoMet website (https://www.comet-toolkit.org/).

Uncertainty contributions

Three uncertainty contributions are tracked throughout the processing:

Random uncertainty: Uncertainty component arising from the noise in the measurements, which does not have any error-correlation between different wavelengths or different repeated measurements (scans/series/sequences). The random uncertainties on the L0 data are taken to be the standard deviation between the scans that passed the quality checks. These uncertainties are then propagated all the way up to L2A.
Systematic independent uncertainty: Uncertainty component combining a range of different uncertainty contributions in the calibration. Only the components for which the errors are not correlated between radiance and irradiance are included. These include contributions from the uncertainties on the distance, alignment, non-linearity, wavelength, lamp (power, alignment, interpolation) and panel (calibration, alignment, interpolation, back reflectance) used during the calibration. Since the same lab calibration is used within the HYPERNETS PROCESSOR for repeated measurements (scans/series/sequences), the errors in the systematic independent uncertainty are assumed to be fully systematic (error-correlation of one) with respect to different scans/series/sequences. With respect to wavelength, we combine the different error-correlations of the different contributions and calculate a custom error-correlation matrix between the different wavelengths. These uncertainties are included in the L1A-L2A data products.
Systematic uncertainty correlated between radiance and irradiance: Uncertainty component combining a range of different uncertainty contributions in the calibration. Only the components for which the errors are correlated between radiance and irradiance are included. This error-correlation means this component will become negligible when taking the ratio of radiance and irradiance (i.e. in the L2A reflectance products), which is why we separate it from the systematic independent uncertainty. The systematic uncertainty correlated between radiance and irradiance includes contributions from the uncertainties on the lamp (calibration, age). Since the same lab calibration is used within the HYPERNETS PROCESSOR for repeated measurements (scans/series/sequences), the errors in the systematic independent uncertainty are assumed to be fully systematic (error-correlation made up of ones) with respect to different scans/series/sequences. With respect to wavelength, we combine the different error-correlations of the different contributions and calculate a custom error-correlation matrix between the different wavelengths. These uncertainties are present in the L1A-L1C products.

The temperature and spectral straylight uncertainties will be improved in future versions. Additionally, there is an uncertainty to be added on the HYPSTAR responsivity change since calibration (drift/ageing of spectrometer and optics). More post-deployment calibrations are necessary before we can quantify this contribution. Other uncertainty contributions not yet included in the uncertainty budget will also be considered in the future, such as uncertainties on the sensitivity to polarisation, uncertainties in the cosine response of the irradiance optics, the effects of the platform/mast on the observed upwelling radiances (e.g. Talone and Zibordi, 2018), or on the air-water interface reflectance corrections. Uncertainties on the Spectral Response Functions (SRF) of the radiance and irradiance sensors (particularly the difference between the two is important when calculating reflectance) should also be considered (see also Ruddick et al., 2023). To account for these missing uncertainty contributions, a placeholder uncertainty of 2% is added to the systematic independent uncertainty, assuming systematic spectral correlation. In the strong atmospheric absorption features (i.e. 757.5-767.5 nm and 1350-1390 nm), an additional placeholder uncertainty of 50% (assuming random spectral error correlation) is added to account for the difference in SRF becoming dominant.

Storing uncertainty information as digital effects tables

As previously mentioned, detailed error-correlation information is calculated as part of the uncertainty propagation. Storing this information in a space-efficient way is not trivial. To do this we use the obsarray module of the CoMet toolkit. obsarray uses a concept called ‘digital effects tables’ to store the error-correlation information. This concept takes the parameterised error-correlation forms defined in the Quality Assurance Framework for Earth Observation (QA4EO) and stores them in a standardised metadata format. By using these parameterised error-correlation forms, it is not necessary to explicitely store the error-correlation along all dimensions. Instead only the error-correlation with wavelength is explicitly stored, and error-correlation with scans/series is captured as the ‘random’ or ‘systematic’ error-correlation forms.

Another benefit to using obsarray, is that it allows for straightforward encoding of the uncertainty and error-correlation variables. The error-correlation (with respect to wavelength) does not need to be known at a very high precision. It can be saved as an 8-bit integer (leading to about a 0.01 precision in the error-correlation coefficient). Similarly, the uncertainties can be encoded using a 16-bit integer to a precision of 0.01%. Together, these encodings significantly reduce the amount of space required to store the uncertainty information.

Finally, having the HYPERNETS products saved as ‘digital effects tables’ means they can easily be used in further uncertainty propagation where all the error-correlation information is automatically taken into account. See De Vis & Hunt (in prep.) and the CoMet toolkit examples for further information (note there is one example specific to HYPERNETS).