# Concept for uncertainty estimation in AVHRR Aerosol Optical Depth CDR

The AVHRR instrument, which was designed for cloud and land observations, is a weak instrument if it is used for the retrieval of a Climate Data Record (CDR) of Aerosol Optical Depth (AOD) over land, because of its small information content with practically only one useful channel with uncertain calibration.

However, the AVHRR offers a long historic record back to 1978 from the series of instruments flown on NOAA and METOP platforms. AOD can be inverted from top-of-atmosphere (TOA) reflectance measurements in the red band $R_{TOA}$ based on a statistical correlation of the underlying surface reflectance with the mid-infrared channel at 1.67 µm and on assuming optical properties of atmospheric aerosol.

In doing so, estimating uncertainties of the retrieved AOD on pixel level $σ_{AOD}$ is crucial to understand the reliability of the results. This is particularly important, since the sensitivity of the retrieved AOD to the measured signal varies largely with retrieval conditions (AOD itself, surface brightness, aerosol optical properties / aerosol type, observing geometry).

Figure 1 shows two examples with different geometry, where AOD sensitivity to measured TOA reflectance behaves well (left case, where AOD is well determined for a given reflectance even with a small uncertainty) or is critical (right case, where even a tiny reflectance uncertainty may lead to a large uncertainty in retrieved AOD). Uncertainties are proportional to the derivatives of those graphs. In both plots AOD uncertainty values at AOD=0 which are due to a 2% reflectance uncertainty are plotted with vertical red bars – obviously uncertainties are highest for low AOD and grow with increasing surface reflectance.

Fig. 1: look-up-tables of $AOD_{670}$ as function of top-of-atmosphere reflectance $R_{670}$ for a pure weakly absorbing fine mode aerosol component under two typical retrieval conditions: good case (solar zenith angle $Ɵ_{0}$ = 47.5°, satellite zenith angle Ɵ = 0°, relative azimuth angle ϕ =180° - left plot), weak case ($Ɵ_{0}$ = 47.5°, Ɵ = 27.5°, ϕ =180° - right plot). Lines from left to right in each plot are for growing surface reflectance from 0.005 to 0.095. Red vertical lines indicate AOD uncertainty at AOD = 0. due to a 2% reflectance uncertainty.

Given the weak retrieval from AVHRR, we choose a pragmatic approach for the estimation of pixel-level AOD uncertainties, which is based on lessons learned during the ESA Aerosol_cci project (Popp, et al., 2016) and focuses on the dominant terms (also called effects):

1. TOA reflectance uncertainties $σ R_{TOA}$ due to uncertain calibration,
2. Surface reflectance uncertainties $σ Alb_{surf}$ due to significant noise in the correlation with the mid-infrared channel, and
3. Aerosol type uncertainties $σ_{AOD}^{ensemble}$ due to lacking knowledge of the true aerosol optical properties.

Equation (1) below summarizes, how these dominant uncertainties determine the retrieved AOD uncertainty. $$\sigma_{AOD} = \sqrt{ \left( \frac{∂_{AOD}}{∂ R_{TOA}} \sigma R_{TOA}\right)^{2} + \left(\frac{∂_{AOD}}{∂ Alb_{surf}} \sigma Alb_{surf}\right)^{2} + \left(\sigma_{AOD}^{ensemble}\right)^{2} + \sigma^{2}(0) }$$

This equation has been derived from the retrieval operator, which inverts measured TOA reflectance into retrieved AOD. Since this inversion cannot be solved analytically, the retrieval operator is depicted by a block diagram and is solved numerically rather than being able to provide an analytical measurement equation. Further uncertainties (summarized in the term $σ^{2} (0)$) are neglected to limit calculation time for uncertainties in the retrieval algorithm. This is justified, as long as these are considered significantly smaller or can be assumed to be fully independent, so that they average out on larger spatial / temporal scales.

In equation (1) sensitivities $\frac{∂AOD}{∂E}$ determine the strength of each uncertainty contribution due to effect E (e.g. the first sensitivity $\frac{∂AOD}{∂R_{TOA}})$ are the gradients / derivatives in figure 1).

Neglected uncertainties summarized in the term $σ^{2}(0)$ do contain trace gas absorption correction (small due to setup of window channels), altitude dependent Rayleigh scattering correction, vertical layering of AOD (both small in the red band), look-up table errors versus full radiative transfer calculations, including interpolation errors between distinct angular values (both proven with full radiative transfer calculations), and interpolation values between distinct aerosol types (at need a finer granularity could be used). Of these, there is evidence, that water vapour trace gas absorption could have structured behavior (i.e. trends over a 30 year record), which would then introduce systematic biases and even in (spatially) gridded or temporally aggregated (e.g. monthly) level3 products would then not cancel out. It needs further studies to determine, how large such uncertainties and consequential false trends in the AOD record could become and whether they need to be added explicitly to equation 1.

Equation (1) shows that the uncertainties in the AOD CDR do not only originate from propagation of uncertainties in measured reflectances, but also assumptions, simplifications, and lacking knowledge in the retrieval do add major contributions. Currently a demonstration is implemented, where a regional AVHRR AOD CDR of 10 years over land covering Europe and Northern Africa will be produced, which contains pixel-level AOD uncertainties according to the approach of dominant contributions as defined in equation (1).

The first term in equation (1) deals with propagating uncertainties of the measured input reflectances to retrieved AOD results. In the FIDUCEO project an easy-FCDR will be created for AVHRR which provides sophisticated but practical uncertainty information by aggregating all effects of signal processing into two uncertainty quantities: one integrated value of independent (i.e. completely uncorrelated) uncertainty and a second; average of all structured effects with a covariance matrix, that contains averaged spatial and temporal ranges of correlations. In creating level2 AOD uncertainties each of these two shall be propagated separately, so that for creating level3 aggregated uncertainties, their different correlation structures can be exploited.

The second term in equation (1) calculates the impact of uncertain estimation of the surface brightness (technically described as albedo) on resulting AOD uncertainties. By analysing a training dataset of atmospherically corrected radiometer data, a simple linear regression has been derived, which is used to estimate surface albedo in the red band from TOA reflectance in the mid-infrared channel as a function of normalized differential vegetation index NDVI ($NDVI = \frac{R_{TOA}^{870} - R_{TOA}^{670}} {R_{TOA}^{870} + R_{TOA}^{670}}$). This linear regression is provided in equation 2, below (Holzer-Popp, et al., 2008):

$$Alb_{surf}^{670} = a R_{TOA}^{1670} + b$$ with a = -1.5 NDVI + 1.5, and b = 0.1 NDVI - 0.01

Based on equation (2) uncertainties of surface albedo are calculated by propagating independent uncertainties of $R_{TOA}^{1670}$ while assuming the albedo uncertainties as completely independent between pixels. Since the linear regression of equation (2) has been determined once and is applied to the entire AVHRR dataset, its global uncertainty (as determined by a validation exercise with the training dataset) is used as global surface albedo uncertainty of 0.01. This second value is propagated separately to AOD uncertainty according to equation (1) and for gridded products is treated as a fully structured uncertainty (i.e. with global correlation, so that no averaging out occurs in aggregating).

For the third term of equation (1), the retrieval algorithm repeats the inversion for a discrete set of pre-defined aerosol types (mixtures of four basic components spanning the range of realistic optical aerosol properties in the atmosphere; where their optical properties are based on aggregated in-situ measurements; Holzer-Popp, et al., 2013 and de Leeuw, et al, 2015).

With this, an ensemble of AOD solutions is created, which describes the uncertainty of AOD results due to lacking knowledge of optical aerosol properties. This uncertainty grows with increasing AOD value (i.e. for AOD near zero it will be typically very small).

In the FIDUCEO project, a climatological prescription of the mixing factors (monthly mean on 1° spatial grid) will be used to identify the AOD of the most realistic aerosol type (in climatological sense), where the mixing factors have been determined from atmospheric modelling (Kinne, et al., 2006). A standard deviation of all AOD solutions will be used as the AOD uncertainty due to lacking knowledge in the ensemble of optical aerosol properties.

If additional information becomes available (e. g. a probability distribution of different aerosol types for a season and region), this can be used to better constrain the ensemble and its associated uncertainty. As for water vapour any systematic trend changing aerosol type (in particular aerosol absorption) over a 30-year record could introduce an apparent AOD trend – the size of such an effect and its impact on the uncertainty estimation needs further analysis.

In 2018 this approach shall be demonstrated in the FIDUCEO project and the AVHRR AOD CDR uncertainties will be validated statistically against true error estimates which are available with high accuracy in the differences to ground-based sun photometer AOD measurements.