1. Coho Model Structure

1.1. Stocks

Currently, 123 stock groups are represented in Coho FRAM. Each of these stock groups have both marked (adipose fin clipped) and unmarked components to permit assessment of mark-selective fishery regulations; therefore, the current version of FRAM has a total of 246 stock units (Appendix 1). Model stocks may represent an individual salmon stock or an aggregate of stocks from the same region. Stock units represented in FRAM were chosen based on the level of management interest, their contribution to PFMC fisheries, and the availability of representative coded-wire tag (CWT) recoveries in the Regional Mark Information System (RMIS). Coho FRAM includes a comprehensive set of stocks originating from Central California to Southeast Alaska and represents total West Coast production. For Coho, only age three fish are included in the model.

1.2. Fisheries

The FRAM includes pre-terminal and terminal fisheries in southeast Alaska, Canada, Puget Sound, and off the coasts of Washington, Oregon, and California. There are 198 fisheries in Coho FRAM. A unique fishery identification number and fishery name for each of the FRAM fisheries are listed in Appendix 3. Terminal fisheries in Coho FRAM include individual freshwater fisheries. FRAM can model directed fisheries, mark-selective fisheries, and non-retention fisheries, and also calculates fisheries-related incidental mortalities.

1.3. Time Steps

The time step structure used in FRAM represents a level of resolution that corresponds to data availability, fishery management structure, and species-specific migration and maturation schedules. Recoveries of CWTs for each stock unit during years with widespread tagging and fisheries (the reference base period, explained further in Section 3) are used to estimate model parameters such as exploitation rates. The amount of recovery data available by stock unit and fishery in the historical CWT database limit the time-step resolution of the model. Decreasing the number of CWT recoveries in a time step by making the time step shorter can increase the variance of the estimated parameters in those strata. In recognition of these data limitations, efforts were made to restrict the level of time step resolution to that necessary for life history fishery management purposes.

At each time step, a stock experiences natural mortality, and can be subjected to pre-terminal fisheries, and terminal fisheries.

FRAM contains five time periods for Coho, covering a single calendar year (Appendix 5).

Table 1. Time step structure used in Coho FRAM.

Time Step	Months
1	January - June
2	July
3	August
4	September
5	October - December

2. Assumptions and Limitations

Major assumptions and limitations of the model are briefly described below.

CWT fish accurately represent the modeled stock. Many FRAM stocks are represented by CWTs from only one production type, usually hatchery origin. For example, in nearly all cases wild/natural stocks were aggregated in the base period with hatchery stocks and both are represented by the hatchery stock’s CWT data. Therefore it is assumed that the tagged and untagged fish are distributed and exploited at the same rates.

Each stock unit and age class is exploited as a single pool. All pre-terminal fisheries operate on the entire cohort simultaneously in each time step, and all terminal fisheries operate on the mature run.

Natural mortality only varies by time step and is constant between stocks and from year to year.

Release mortality and drop-off rates are fishery and time step specific and constant between stocks and from year to year. Note: Mortality rates can be updated, but this is done very infrequently, i.e., only a few rates have been updated in the past 30 years.

Stock distribution and migration is constant from year to year and is represented by the average distribution of CWT recoveries during the reference base period. We currently lack data on the annual variability in distribution and migration patterns of Chinook and coho salmon stocks. In the absence of such estimates, fishery-specific exploitation rates are computed relative to the entire cohort.

3. Model Parameterization

FRAM data are stored as tables in a Microsoft Access database. All the information needed for a “simple model run” (User Manual) is stored in these tables, containing input as well as output data. Input data can be grouped into two main categories: static base period data and annual run data. Main FRAM outputs are mortalities (landed, mark-selective, non-retention, drop-off) and abundances (starting cohorts at each time step; abundances after natural mortalities, pre-terminal, and terminal fisheries).

Table 2. Coho FRAM Microsoft Access Database Tables.

3.1. FRAM Input Data

FRAM input data consist of two main input types: base period reference data and annual data. Base period data remain constant until a new base period replaces the existing one. The principal base period inputs are stock-fishery-time step specific exploitation rates. Other base period parameters include base period catches and cohort sizes. The second input type consists of annual data that can vary with each model run, such as stock specific abundance estimates, fishery catches, size limits, etc.

3.1.1. Base Period Data

CWT recoveries are used to estimate model parameters such as time/area/fishery exploitation rates. The years from which CWT recoveries are used to estimate these parameters are referred to as the reference “base period.” These parameter estimates are derived through stock-specific cohort analyses. Each cohort analysis is a series of procedures that use CWT recoveries and base period catch and escapement data to “back-calculate,” or reconstruct, a pre-fishing cohort size for each stock using assumed natural mortality and incidental mortality rates. See (Model Evaluation Workgroup, 2008) for a more detailed description of the cohort analysis procedures.

Model base period data for the Coho FRAM is derived from fishery and escapement recoveries of CWTs and terminal area run size estimates for return years 1986-1992 (Packer et al. 2007). In 2006, the previously used base period data from 1986-1991 were re-analyzed, changes were made to the algorithm, and 1992-1997 catch year data were compiled. A decision was made to only incorporate the 1992 catch year with the previous base period data, and thus the Coho base period for return years 1986-1992 was generated.

If modeled fishery impacts for a stock deviate significantly from regulatory expectations or an analysis of CWT data is re-examined, a stock fishery rate scaler can be used for individual stocks within a fishery and time step to deviate from the reference base period. The stock fishery rate scaler is optional, must be agreed-to, and can be tied to an individual annual fishery input or remain as a static input to adjust the reference base period exploitation rate for a stock-fishery-time step.

3.1.2. Annual and Semi-static Data

These data are comprised of five general types of input. Three of these input types (a, b, d) are submitted annually to reflect projected stock abundances and proposed fishery regulations for the current model year. The remaining two types of input (c, e) are specifications for fishery-related mortalities that can change as more information becomes available through additional data collection or studies, but typically do not change annually.

a. Cohort Abundance: For each stock unit, an annual abundance is obtained from a regional expert, typically in the form of an ocean age-3 run size (pre-fishing age-3 abundance in the ocean after natural mortality has been subtracted). In a pre-season context these abundances come from annual forecasts, whereas in a post-season context the abundances are derived from estimates of actual returns. For Coho, an initial stock abundance is needed for adult fish (age-3) by mark status.

b. Fishery Landed Catch: The model provides three options for setting the catch in a fishery: a quota, a fishery scaler, and a harvest rate (for Puget Sound and Washington coastal terminal fisheries).

Quota: Catch in the fishery is set equal to a numeric value input by the user.
Fishery Scaler: The fishery is scaled relative to the effort during the reference base period using a scaler value input by the user. The catch resulting from a scaler is a function of the base period exploitation rates and stock abundances.
Harvest Rate: Using the Puget Sound TAMM, a terminal area harvest rate can be applied to terminal area fisheries (time step 4 and 5). In the Coho TAMM, the terminal run size definitions can be customized (using the TAAETERSList table in the Microsoft Access database) and must be aligned with calculations used for each model input. For a list of terminal fisheries see Appendix 3.

FRAM inputs for quota and fishery scaler can be identified as either a conventional retention fishery or a mark-selective fishery and modeled accordingly. Modeling as a mark-selective fishery initiates additional calculations to estimate catches, encounters, and mortalities differently for marked and unmarked groups.

c. Release Mortality Rates: This is the mortality associated with the release of landed fish from hook-and-line and other gear types. Release mortality rates are designated by species, geographic area, fishery type and gear type (Appendix 6). Release mortality is assessed when Coho are not retained (“non-retention” or CNR fisheries) as well as in mark-selective fisheries. A number of studies have estimated release mortality for hook-and-line fisheries, and release mortality rates for troll and recreational fisheries in the ocean have been formally adopted by the PFMC.

Release mortality in the majority of net fisheries with non-retention is estimated externally to FRAM using agreed-to rates.

d. Mark-selective fisheries have two additional variations of “release” mortality that are described as either the inappropriate retention of an unmarked fish or the release of a marked fish that consequently experiences release mortality. The failure to release an unmarked fish is a user input to the model called “Unmarked Retention Error” (ure, or Retention Error Rate) and is the proportion of the unmarked fish encountered that are retained. The release of marked fish is a user input to the model called “Marked Recognition Error” (mre) and it is the proportion of the marked fish encountered that are released; these released marked fish are then subject to release mortality. These rates are updated annually based on fishery monitoring data.

e. Other Non-landed Mortality Rates (Incidental Rates): This includes fishing-induced mortality not associated with directly handling fish. Drop-off mortality can occur when fish in sport and troll hook-and-line fisheries drop off the hook before they are brought to the vessel. Drop-out mortality occurs when fish in commercial net fisheries are not brought on board but die from injury as a result of encountering the net. For simplicity, both types are referred to as drop-off mortality in FRAM. Net drop-out mortality rates vary depending on species, net type, or timing (pre-terminal or terminal) of the fishery. In general, a 5% drop-off mortality rate is applied to the landed catch to account for “other non-landed mortalities” in hook-and-line fisheries (Appendix 6).

4. Model Calculations

4.1. Overview

FRAM processes information through a time step loop, beginning with time step 1 and ending with time step 5 for Coho. Within each time step, a series of four computational processes occur for each stock as depicted in Figure 1: (1) calculate starting cohort size, (2) remove natural mortality, (3) remove pre-terminal mortalities, (4) remove terminal fishery mortalities (time 5 only).

Figure 1. Conceptual flow chart for the Coho FRAM model.

4.2. Computational Processes

In the following equations, variables are presented with origin specific formatting:

Variables estimated in FRAM are shown in regular italics.
Variables that are input by a user, including externally estimated values, are shown as bold.
Variables that are estimated during the base period are shown with an underline.
Variables that are estimated using a BkFRAM run are shown as blue.

Cohort Abundance

Process 1: Cohort Abundance at the Start of the Time Step

The starting cohort size in time step 1 is a product of two parameters: (1) the base period cohort size for stock s at age 3 in the first time period (BPCohorts_{s, a = 3}) and (2) a stock and age-specific scaler (StockScalers_{s, a = 3}).

Pre-season, the starting cohort is generally calculated from forecasts of ocean age-3 (OA3) abundances. These forecasts are expanded for natural mortalities by multiplying OA3 * 1.2315 to arrive at January age-3 (JA3) starting cohorts.

Post-season, the starting cohort is generally calculated by expanding escapement estimates for fisheries mortalities and natural mortalities during BkFRAM (see Backwards FRAM chapter).

(1)

Cohort_{s, a = 3, t = 1} = BPCohort_{s, a = 3} × StockScaler_{s, a = 3}

where Cohort_{s, a = 3, t = 1} is the initial cohort size for stock s, age 3, during time step t=1.

Coho FRAM only models age-3 fish (a=3). The starting cohort size is the projected number of age-3 fish in January (JA3) of the fishing year for each stock. All Coho escape in the final time step (t=5) and thus are not aged in the model.

Natural Mortality

Process 2: Natural mortality during each time step

During each time step, the stock-age cohort size at the start of the time step is decreased to account for natural mortality:

(2)

Cohort_{s, a = 3, t} = Cohort_{s, a = 3, t} × (1−M_{a = 3, t})

where M_{a = 3, t} is the discrete natural mortality rate for age 3 fish during time step t (Appendix 8).

Pre-terminal Fishery Mortality

Process 3: Pre-terminal Fishery Mortality

The remaining cohort is then subjected to removals by pre-terminal fisheries; both landed catch and non-landed mortalities associated with each fishery are calculated. FRAM simulates fishery mortalities using different processes depending upon the type of fishery: retention fishery (non-selective), non-retention fishery, or mark-selective fishery.

3a. In regular retention fisheries (non-selective), landed catch is estimated by:

(3)

Catch_{s, a = 3, f, t} = Cohort_{s, a = 3, t} × BPER_{s, a = 3, f, t} × FisheryScaler_f, t × PV_{s, a = 3, f, t} × SHRF_s, f, t

where:

BPER_{s, a = 3, f, t} is the base period exploitation rate for stock s, at age 3, in fishery f, during time step t.
FisheryScaler_f, t is an annual model input that relates effort in the model year back to the effort during the base period.
PV_{s, a = 3, f, t} is the portion of the stock cohort s, at age 3, during time step t that is of legal size in fishery f (i.e. portion vulnerable; PV). For coho, PV is always 1 and can be eliminated from the equation.
SHRF_s, f, t is the stock fishery rate scaler for stock s in fishery f during time step t. The SHRF is optional and can be used to adjust impacts for individual stocks in a fishery and time step; i.e., if FRAM output for an individual stock significantly deviates from expectations, then the user can make adjustments by providing a SHRF scaler. The stock fishery rate scaler was conceived to adjust the base period exploitation rate without the need to conduct a full calibration.

The FisheryScaler_f, t is the foundation for the fishery simulation algorithms. FRAM can evaluate two general types of fisheries: effort-based or catch-based. For effort-based fisheries, the FisheryScaler_f, t is specified by the user to reflect expected effort during the model year relative to the average effort observed during the base period. For catch-based fisheries, the FisheryScaler_f, t is computed by FRAM to obtain a user-specified catch level (i.e. a quota).

3b. Drop-off mortalities are estimated by multiplying either a) landed catch in a non-selective retention fishery, or b) total encounters in a mark-selective fishery, by a user-specified drop-off mortality rate (DropRate_f see Appendix 6)

a) Non-selective retention fishery f:

(4a)

Dropoff_{s, a = 3, f, t} = Catch_{s, a = 3, f, t} × DropRate_f

b) Mark-selective fishery f:

(4b)

Dropoff_{s, a = 3, f, t} = Encounters_{s, a = 3, f, t} × DropRate_f

3c. Coho non-retention (CNR) mortalities are estimated when fishing is allowed, but the retention of Coho is prohibited.

Non-retention mortalities are calculated external to the model, often derived using historical observations, and are model fishery inputs by time step in units of dead age-3 fish (EstCNRMorts_f, t). Non-retention mortalities by stock are then calculated in FRAM by multiplying the total non-retention mortality by the stock proportion within each fishery and time step (PropTempCatch_{s, a = 3, f, t}).

The stock-specific CNR mortality in a fishery and time step is computed in four steps.

Step 1: Calculate the stock specific catch in a retention fishery (TempCatch_s, f, t) with a FisheryScaler of 1 (i.e., at base period effort).

(5)

TempCatch_{s, a = 3, f, t} = Cohort_{s, a = 3, t}× BPER_{s, a = 3, f, t} × FisheryScaler_f, t× SHRF_s, f, t

Step 2: Sum over stocks for a fishery and time step to get the total catch.

(6)

TotalTempCatch_f, t = ∑_{s, a = 3}TempCatch_{s, a = 3, f, t}

Step 3: Calculate the proportion of the total catch for each stock.

(7)

PropTempCatch_{s, a = 3, f, t} = $\frac{TempCatch_{s,a \! = \! 3,f,t}}{TotalTempCatch_{f,t}}$

Step 4: Calculate the stock-specific mortality estimate for the provided non-retention fishery input (EstCNRMorts_f, t).

(8)

CNR_{s, a = 3, f, t} = EstCNRMorts_f, t × PropTempCatch_{s, a = 3, f, t}

where:

stock s, age a = 3, fishery f, time step t
CNR_{s, a = 3, f, t} is the Coho age = 3 non-retention mortality
PropTempCatch_{s, a = 3, f, t} is the proportion of encounters of stock s, age a = 3, in fishery f, at time step t using FishScaler = 1.0
TotalTempCatch_f, t is the total catch in fishery f at time step t summed over all stocks s using FisheryScaler = 1

3d. Mark-selective fisheries (MSF) require additional computations to calculate both the landed catch and the mortalities due to the release of fish.

To simplify calculations, the SHRS is excluded from the below calculations:

For marked stock units, the landed catch is calculated using an additional term to account for marked-recognition error (the release of a marked fish) and is fishery and time step specific (mre_f, t):

(9a)

MSFCatch_{s, a = 3, f, t} = Cohort_{s, a = 3, t} × BPER_{s, a = 3, f, t} × FisheryScaler_f, t × (1 − mre_f, t)

For unmarked stock units, the landed catch is calculated using an additional term to account for unmarked retention error (retaining an unmarked fish in a mark-selective fishery) and is fishery and time step specific (ure_f, t):

(9b)

MSFCatch_{s, a = 3, f, t} = Cohort_{s, a = 3, t} × BPER_{s, a = 3, f, t} × FisheryScaler_f, t × ure_f, t

Equations used to calculate adult release mortalities (AdultRelMort) in mark-selective fisheries must account for marked recognition (mre) and unmarked retention error (ure) for marked and unmarked stock units and utilize fishery and time step-specific release mortality rates (sfm_f, t) (Appendix 6).

(10a) Marked:

AdultRelMort_{s, a = 3, f, t} = Cohort_{s, a = 3, t} × BPER_{s, a = 3, f, t} × FisheryScaler_f, t × mre_f × sfm_f, t

(10b) UnMarked:

AdultRelMort_{s, a = 3, f, t} = Cohort_{s, a = 3, t} × BPER_{s, a = 3, f, t} × FisheryScaler_f, t × (1 − ure_f) × sfm_f, t

Drop-off mortalities in mark-selective fisheries are calculated by multiplying the drop-off mortality rate with the number of total encounters, as described in process 3b above.

3e. All pre-terminal fishery mortalities in time step t for stock s at age 3 are totaled (TotMort) and the size of the cohort is reduced accordingly. Since the Coho model contains only age-3 mature fish, the summing of fishery mortalities occurs only once in each time step.

(11)

TotMort_{s, a = 3, t} = ∑_f (Catch_{s, a = 3, f, t} + Dropoff_{s, a = 3, f, t} + AdultRelMort_{s, a, f, t} + CNR_{s, a = 3, f, t})

The remaining cohort is then calculated as:

(12)

Cohort_{s, a = 3, t} = Cohort_{s, a = 3, t} − TotMort_{s, a = 3, t}

Fisheries during time steps 1 through 4 are considered to be on immature fish and, by default, all Coho fisheries in time step 5 are on mature fish. Thus,

MatureCohort_{s, a = 3, t = 5} = Cohort_{s, a = 3, t = 5}.

Process 4: Terminal Fishery Mortality

Terminal fishery mortality is calculated for the mature cohort only, using the same equations as for pre-terminal fishery mortality. Fishery mortalities are summed, and the remainder of the mature cohort constitutes the escapement from FRAM fisheries. Since only age-3 adult Coho are modeled and the mature time step is always 5, age and time step parameters can be eliminated from the equations.

(13)

TotTermMort_{s, a = 3, t = 5} = ∑_{f ∈ terminal} (Catch_{s, a = 3, f, t = 5} + Dropoff_{s, a, f, t} + AdultRelMort_{s, a, f, t} + CNR_{s, a = 3, f, t = 5})

(14)

Escapement_s, a, t = MatureCohort_s, a, t − TotTermMort_s, a, t

Escapement is defined as the number of fish that remain in the mature cohort after removal of all terminal-area fishery related mortality. For coho modeling, which includes freshwater fisheries, escapement represents the adult spawning abundance if mortality during “pre-spawning” holding time is negligible.

Process 5: TAMM Iterations

For Coho, traditionally, the word “Terminal” has two meanings. It can either denote fisheries occurring on the mature cohort(see previous paragraph) or it can describe fisheries that occur in terminal marine areas (see Appendix 3). These fisheries occur in time step 4 and 5, are reported from TAMM to FRAM, and involve iterative solutions to calculate stock specific impacts. During a first pass, FRAM completes a run with the terminal fishery inputs stored in the FRAM database. These “seed” inputs usually stem from a previous model run. After this pass is completed, FRAM estimates terminal fishery impacts using values from TAMM and compares them to the original FRAM results. If these numbers are equal (less than 0.1% or fewer than 4 fish difference), the model is done. If these numbers are not equal, an iterative process is initiated during which TAMM terminal fishery impacts are used to adjust FRAM terminal fishery impacts until there is convergence (see Chapter 5).

Finally, FRAM sends terminal marine and freshwater run sizes and fishing mortality to the TAMM file for use in additional calculations and final reporting.

4.3. Processing Schematic

To facilitate understanding of the sequencing of FRAM processing steps, Table 3 provides an example of how a stock’s cohort is processed through time step from starting cohort to escapement.

The starting cohort in time step 1 is first reduced by natural mortality. The remaining cohort is vulnerable to pre-terminal fisheries. Coho not caught become the starting cohort of the next time step. This cycle repeats until time step 5, when 100% of Coho mature after pre-terminal fishing. The mature cohort is susceptible to terminal fisheries. Coho not caught after pre-terminal and terminal fisheries in time step 5, go to escapement.

Table 3. Example values of Coho FRAM processing steps for a stock by time step.

4.4. Exploitation Rate Calculation

FRAM calculates stock-specific mortalities and escapements that are ultimately used to compute exploitation rates (ER). Exploitation rate calculations are not part of the main FRAM algorithms and are not calculated by FRAM. Exploitation rates are computed from FRAM or TAMM output as the sum of all fishery mortalities divided by fishery mortalities plus escapement.

(15) $$ER_s = \frac{\displaystyle\sum\limits_{t} TotMort_{s,a \! = \! 3,t}}{\displaystyle\sum\limits_{t} TotMort_{s,a \! = \! 3,t} + Escapement_{s}}$$

4.5. Bias Corrected Mark-Selective Fishery Equations for Coho

Mark-selective fisheries (MSF) allow anglers to keep legal-size, adipose-fin clipped hatchery fish (hereafter, “marked”) and require the release of any fish with an adipose fin (hereafter, “unmarked”). Originally, both Chinook and Coho FRAM used estimates of release mortality (δ) and time-period specific exploitation rates (ER) from non-selective fisheries as a surrogate for encounter rates of unmarked stocks to estimate the mortality of unmarked fish (equations 9a and 9b).

(16)

SimpleExploitationRate_Unmarked = ER × δ

Chinook FRAM continues to rely on this method. However, the release of salmon causes bias in these linear exploitation rate calculations. The true mortality for unmarked fish is underestimated because it is an increasing function of time-period specific encounter rates. The unmarked-to-marked ratio of all fish in the pool increases over time as the result of selective removal of marked fish in MSFs and released unmarked fish that survive may encounter the fishing gear more than once during the time period (Figure 2). The original linear functions do not capture this process. Additionally, because all fisheries during a modeled time period are assumed to operate simultaneously on a single pool of fish, this bias also occurs in any modeled non-selective fisheries (NSF) that take place during the same model time period as the MSF.

Figure 2: During MSF, the selective removal of marked fish (black) may increase the unmarked-to-marked ratio. In this example, the mark rate is 50% at the start of the time-period and 25% at the end. Additionally, released unmarked fish that survive encountering the fishing gear (light blue) may encounter the fishing gear again. In this example, two fish (dark blue), encountered gear again and were thus subjected to release mortality more than once, plus two additional unmarked fish encounters (light blue) occurred.

Bias Corrected Mark-Selective Fishery Calculations for Coho

The biased exploitation rate for a stock in a mark-selective fishery, shown in red, is simply the time-period-specific average exploitation rate from the base period multiplied by a scalar that relates the expected current year effort to the base period effort and a release mortality rate for the fishery.

To simplify calculations, the SHRF, mre, and ure were excluded from the calculations below:

(17)

ER_Marked= FisheryScaler × BPER × 1

(18)

ER_Unmarked= FisheryScaler × BPER × δ

In the original biased equations, the exploitation rates of non-selective fisheries were the same as the marked exploitation rate (ER_Marked) in mark-selective fisheries. Subsequent unbiased equations were developed with a marked frame of reference. In the following equations, ER_Marked is synonymous with the original non-selective exploitation rates.

The equations above are biased as they approximate a nonlinear Baranov equation with a linear model. Conrad, Hagen-Breaux, and Yuen (2013) developed unbiased calculations for the number of unmarked mortalities when there are multiple MSF and NSF acting on a stock in a time period. This paper is summarized briefly here, see the publication for more detailed documentation.

When the only fishery acting on a stock in a time period is a MSF, the unbiased method for calculating the total unmarked exploitation rate is:

(19)

$\widehat{ER}$_Unmarked= 1 − (1−ER_Marked) ^δ,

and δ is the release mortality rate of the fishery.

This becomes more complicated when there are multiple fisheries acting on an unmarked cohort during the same time period. Even when assuming a constant release mortality across all fisheries, because of nonlinearity it is not correct to perform bias corrections separately on each fishery and then sum over all fisheries. This incorrect method results in the underestimation of the true exploitation rate on the unmarked cohort. When there are multiple fisheries acting on a stock in a single pool model, and the release mortality is constant across all fisheries, the total bias-corrected exploitation rate on the unmarked cohort across all fisheries is calculated as:

(20)

$\widehat{ER}$_{Unmarked, F}= 1 − (1− $\sum_f^F$ER_Marked, f) ^δ,

where f is an individual fishery and F is the set of all fisheries.

In reality, in any given time period many stocks are subject to a mixture of NSF and MSF with varying release mortality rates, and mark-recognition is not perfect. Recognition errors for marked fish (mre, probability that a marked fish is released) reduce the exploitation rate on the marked cohort and introduce bias in the exploitation rate calculation for the marked portion of the stock. Conversely, recognition errors for unmarked fish in MSF (ure, unmarked fish are kept) increase the exploitation rate on an unmarked cohort.

An overall release mortality rate in fishery f that accounts for mark-recognition error for a marked cohort (δ_Marked, f) is defined by:

(21)

δ_Marked, f = mre_f × δ_f + (1 − mre_f)

Similarly, for the unmarked cohort:

(22)

δ_{Unmarked, f} = (1 − ure_f) × δ_f + ure_f

Because of mark recognition error, weighted release mortality rates (δ^W) across all fisheries in the modeled time period are calculated for marked and unmarked cohorts. These weighted release mortality rates are the ratio of the sum of the biased exploitation rate calculations, which account for release mortalities and the sum of the marked exploitation rates.

Weighted release mortalities for marked and unmarked are calculated as:

(23) $$\delta^{W}_{Marked} = \sum^{F}_{f}(\frac{ER_{Marked,f} \times \delta_{Marked,f}}{\sum^{F}_{f} ER_{Marked, f}}) = \sum^{F}_{f}(\frac{ER_{Marked,f} \times \{ mre_f \times \boldsymbol{\delta}_f + (1 - mre_f)\}}{\sum^{F}_{f} ER_{Marked,f}})$$

(24) $$\delta^{W}_{Unmarked} = \sum^{F}_{f}(\frac{ER_{Marked,f} \times \delta_{Unmarked,f}}{\sum^{F}_{f} ER_{Marked, f}}) = \sum^{F}_{f}(\frac{ER_{Marked,f} \times \{ (1 - ure_f) \times \boldsymbol{\delta}_f + ure_f ) \}}{\sum^{F}_{f} ER_{Marked,f}})$$

If there is no mark recognition error, bias-adjusted exploitation rates for marked and unmarked stock units are calculated as:

(25)

$\widehat{ER}_{Marked, F} = \sum_{f}^{F} ER_{Marked,f}$

(26)

$\widehat{ER}_{Unmarked,F} = 1 - (1 - \sum_{f}^{F} ER_{Marked,f})^{\delta^{W}_{Unmarked}}$

If there is mark recognition error, bias-adjusted exploitation rates for marked and unmarked stock units are calculated as:

(27)

$\widehat{ER}_{Marked,F} = 1 - (1 - \sum_{f}^{F} ER_{Marked,f})^{\delta^{W}_{Marked}}$

(28)

$\widehat{ER}_{Unmarked,F} = 1 - (1 - \widehat{ER}_{Marked,F})^{\delta^{W}_{Marked} / \delta^{W}_{Unmarked}}$

Once unbiased exploitation rate calculations are completed, the total number of unmarked and marked mortalities that are projected to occur in all fisheries are allocated to each fishery in each time period. The unbiased total mortalities (D̂) are allocated proportional to the simple, biased exploitation rate of each fishery. This total is calculated as:

(29)

$\widehat{D}_{Unmarked,f} = N_{Unmarked} \times \widehat{ER}_{Unmarked,F} \times P_f$

Here, N_Unmarked is the initial number of unmarked fish after natural mortality and P_f is the proportion of total unmarked fish mortalities in all fisheries that occurred in fishery f.

(30)

$P_f = \frac{ER_{Marked} \times \delta_{Unmarked,f}}{\sum_{f}^{F} ER_{Marked} \times \delta_{Unmarked,f}}$

A similar process can be used to allocate marked mortalities to fisheries.

After total mortalities for a fishery have been calculated, they are assigned as catch (L) and non-landed mortalities (N) using recognition rates and release mortalities.

For an unmarked cohort, landed catch for fishery i (D̂_{L, Unmarked, f}) is calculated:

(31)

$\widehat{D}_{L, Unmarked, f} = \widehat D_{Unmarked,f} \times \frac{ure_f}{(1-ure_f) \times \boldsymbol{\delta}_f + ure_f}$

and non-landed mortality (D̂_{N, Unmarked, f}) is calculated as:

(32)

$\widehat{D}_{N, Unmarked, f} = \widehat D_{Unmarked,f} \times \frac{(1 - ure_f) \times \boldsymbol{\delta}_f}{(1-ure_f) \times \boldsymbol{\delta}_f + ure_f}$

Similarly, for a marked cohort:

(33)

$\widehat D_{L, Marked, f} = \widehat D_{Marked,f} \times \frac{(1 - mre_f)}{mre_f \times \boldsymbol{\delta}_f + (1 - mre_f) }$

and non-landed mortality (D̂_{N, Marked, f}) is calculated as:

(34)

$\widehat D_{N,Marked,f} = \widehat D_{Marked,f} \times \frac{mre_f \times \boldsymbol{\delta}_f}{mre_f \times \boldsymbol{\delta}_f + (1 - mre_f)}$

Applying MSF Bias Calculations to FRAM

If MSF are modeled as rates in FRAM, the equations presented above work well. These equations are rate based and incorporate fishery effort scalers to model the magnitude of fisheries. However, many fisheries are modeled as quotas with a catch input; in essence supplying the results, expected mortalities, of the unbiased equations. The model has to find the fishery effort scaler that produces these results utilizing the unbiased equations. This problem cannot be corrected at a stock level, since the catch of a fishery is made up of many stocks. The unbiased exploitation rates can be found using an empirical, iterative solution. First, the unbiased exploitation rates are calculated using base period exploitation rates (FisheryScaler = 1), then the landed catch of marked and unmarked fish is calculated (Catch). This landed catch is compared to the target catch. A new fishery effort scaler is computed as FisheryScaler = TargetCatch/Catch. Iterations continue until the calculated FRAM landed catch matches the quota.

Figure 3: Flowchart illustrates the iterative process of calculating unbiased exploitation rates when a MSF is modeled as a quota.

If you would like to see the exact steps completed in FRAM (as shown in Figure 3), read on. Otherwise, stop here. All steps are completed for each stock.

Step 1: Calculate unbiased marked ($\widehat{ER}_{Marked,s,f,t}$) and unmarked ($\widehat{ER}_{Marked,s,f,t}$) exploitation rate for stock s in time period t using equations 27 and 28 (FisheryScaler = 1). During the first iteration, ER_{Marked, s, f, t} = BPER_s, f, t.

Step 2: Calculate total landed catch of marked (D_{Marked, L, s, f, t}) and unmarked (D_{Unmarked, L, s, f, t}) cohorts of stock s using equations 31 and 33.

Step 3: Calculate total landed catch across all stocks encountered in a fishery.

Catch_f = ∑_sD_{Marked, L, s, f, t} + ∑_sD_{Unmarked, L, s, f, t}

Step 4: Calculate fishery scaler.

$$FisheryScaler_f = \frac{TargetCatch_f}{Catch_f}$$

Step 5: Scale base period exploitation rate.

ER_{Marked, s, f} = FisheryScaler_f× BPER_s, f

These steps are repeated until Catch_f is equal to the target quota.

5. Terminal Area Management Module (TAMM)

The FRAM program interacts with the TAMM, a species-specific Microsoft Excel file, which allows users to specify alternative terminal fishery inputs and to calculate impacts on a finer level of resolution than FRAM. The TAMM has separate sections for each of the six Puget Sound terminal regions defined in the Puget Sound Salmon Management Plan (PSSP 1985, Table 3) for the State of Washington and the Washington Treaty Indian Tribes of Puget Sound, as well as Washington coastal terminal regions. This structure has supported development of unique regional management goals and allows managers the flexibility to analyze and report FRAM model output according to their needs.

TAMM is best understood in its historic context as a terminal-area-centric fisheries model. Prior to FRAM development, information of stock-specific fishery impacts was not readily available. At that time, management was based on terminal run reconstructions that assigned 100% of terminal catches to only local terminal stocks. However, in the current model framework, FRAM takes TAMM fishery inputs and accounts for non-local and local stock impacts before reporting stock-specific impacts back to TAMM. Additionally, the TAMM also conducts alternative modeling for Washington coastal Coho stocks in terminal area fisheries, as well as generating inputs for Columbia River models.

Table 4. Puget Sound Terminal Management Regions.

Management Regions
Nooksack-Samish
Skagit
Stillaguamish-Snohomish
South Sound
Hood Canal
Strait of Juan de Fuca

Common TAMM features:

Receive user inputs for TAMM terminal fisheries
Receive user input for TAMM stock management criteria (updated annually)
Provide fishery inputs to FRAM (time step 4 and 5) during FRAM-TAMM iterations
Receive FRAM output of fishery impacts and terminal abundances
Use FRAM output to complete TAMM fishery impact modeling for Puget Sound and the Washington Coast
Generate TAMM reports of combined FRAM and TAMM fishery impacts

FRAM iteratively finds the fishery scalers that produce the same catches (Catch_FRAM) as those supplied by TAMM (Catch_TAMM) in Puget Sound and Washington coastal fisheries (Figure 4). TAMM catches are either input into the TAMM as catch values or harvest rates. Harvest rates are multiplied by a unique terminal run size definition (TRS). The TRS for a TAMM region/stock represents ‘run-to-the-river’ of a stock plus catches of all stocks in the terminal fisheries designated for that region. In Coho FRAM-TAMM, terminal run size definitions can be customized (using the TAAETRSList table in the Microsoft Access database) and must be aligned with calculations used for each model input.

FRAM calculations for each terminal area:

During a first pass, FRAM completes a run with the terminal fishery inputs stored in the FRAM database. After the pass is completed:

Sum escapements over appropriate time steps
Add terminal catches (Catch_FRAM) to calculate TRS (includes non-local stocks)
Calculate the FRAM fishery scaler that produces the Catch_TAMM:
FisheryScaler(area,timestep) = Catch_TAMM(area,timestep) ÷ Catch_FRAM(area,timestep)
Rerun FRAM and repeat these steps until Catch_TAMM is within 0.1% (or fewer than 4 fish) of Catch_FRAM.

Figure 4. Conceptual flow chart for FRAM-TAMM calculations.

6. Backwards FRAM

Backwards FRAM (BkFRAM) is a utility that determines FRAM starting cohorts when estimates of escapements and fishery catches (landed and non-retention) are provided. The program iteratively adjusts stock recruit scalers (a surrogate for starting cohorts, see equation below) and runs FRAM forward until the resulting escapements match the target escapements. Starting cohorts are in units of January-age-3 adult fish.

Post-season BkFRAM runs are generally conducted to create the starting cohorts that result in observed escapements given known fishery catches. These cohorts are needed to calculate post-season exploitation rates.

Working backwards from time step 5 to time step 1, for each time step, fishery and natural mortalities of a stock are added to the escapement target. Total mortalities are calculated by summing landed, drop-off, and non-retention mortalities over all fisheries for a stock and time step.

Mark-selective fishery bias correction calculations used for Coho produce an error when the exploitation rate exceeds 100%. This frequently occurs during early iterations, because the abundances from the seed run (usually the pre-season run) are not related to the post-season fishery catches (e.g., low pre-season forecasts but high post-season terminal catches). To avoid errors, the first iteration initiates starting cohorts scaled 1000 times greater than the base period abundance. Additionally, the first 7 iterations are run without bias correction to get sufficiently close to the target escapement, before adding mark-selective fishing bias calculations for all remaining iterations.

(35)

StockScaler_{s, a = 3} = ((((((BkTarget_s + (TotMort_{s, a = 3, t = 5} + TotTermMort_{s, a = 3, t = 5}) ÷ (1− M_{a = 3, t = 5}) + TotMort_{s, a = 3, t = 4}) ÷ (1− M_{a = 3, t = 4}) + TotMort_{s, a = 3, t = 3}) ÷ (1− M_{a = 3, t = 3}) + TotMort_{s, a = 3, t = 2}) ÷ (1− M_{a = 3, t = 2}) + TotMort_{s, a = 3, t = 1}) ÷ (1− M_{a = 3, t = 1})) ÷ BPCohort_{s, a = 3}

where:

BkTarget is the BkFRAM escapement target
BPCohort is the base period starting cohort abundance
StockScaler is the stock recruit scaler (abundance)

The program iterates until the FRAM escapement is within one fish of the target stock escapement and then terminates the iteration process.

(36)

| FRAMescapement_{s, a = 3} − BkTarget_{s, a = 3} | ≤ 1

where:

FRAMescapement is the escapement resulting from a forward FRAM run.

7. Output Reports

Model results are available in FRAM screen reports viewed within the software, FRAM reports, TAMM Microsoft Excel files, or can be extracted from the species-specific Microsoft Access database associated with the FRAM run. FRAM reports include summaries of projected catch and mortalities by fishery, stock, and age. The TAMM files provide comprehensive summaries of fishery mortalities, exploitation rates, run sizes, and escapements for key stocks in the PFMC and North of Falcon (NOF) annual pre-season planning process, as well as Canadian Coho stocks relevant to the Pacific Salmon Treaty (PST). For a full scope of FRAM report generating functions, refer to the online FRAM User Manual.

To cite this page:
Salmon modeling and analysis workgroup. 2023. Coho Model Detail in FRAM Documentation. https://framverse.github.io/fram_doc/ built September 21, 2023.