Observation Error Model equations

The Observation Error Model (OEM) caricatures the observation processes that produce the complete set of data required for the assessment of each managed stock.

Catch observations

MixME calculates the observed catches $C^{obs}$ for fleet $f$ catching stock $s$ at age $a$ in year $y$ by applying pre-sampled fleet-and-stock-specific catch observation noise $\epsilon^c$ to the catch time-series.

$$C_{f,s,a,y}^{obs} = C_{f,s,a,y} \cdot \epsilon_{f,s,a,y}^c$$

Under single-fleet management, these are then aggregated into an overall commercial catch time-series.

$$C_{s,a,y}^{obs} = \sum_f(C_{f,s,a,y}^{obs})$$

Survey observations

MixME recognises two types of survey index: age-resolved indices and biomass indices.

The age-resolved index $I$ for survey $i$ is calculated from the operating model stock numbers $N$ available at the time of the survey given a survey-specific age-resolved catchability $q^*$:

$$ I_{i,s,a,y} = q_{i,s,a,y}^* \cdot N_{s,a,y} \cdot exp^{-(F_{s,a,y}+M_{s,a,y})t} $$

where $t$ is the time of the year, expressed as a proportion, that survey $i$ takes place. The biomass index $I$ for survey $i$ is similarly, but the catchability is applied to the stock biomass available at the time of the survey:

$$ I_{i,s,y} = q_{i,s,y}^* \cdot \sum_a(N_{s,a,y} \cdot exp^{-(F_{s,a,y}+M_{s,a,y})t} \cdot w_{s,a,y}) $$

where $w$ is the individual mean weight at age for the stock. For both types of survey index, the observed index is then obtained by applying pre-sampled index-specific observation noise $\epsilon^o$:

$$ I_{i,s,a,y}^{obs} = I_{i,s,a,y} + \epsilon_{i,s,a,y}^o $$