NECESIDAD DE RECONOCER LO TRASCENDENTE

The models are defined in three stages, firstly the probability density function (pdf) of the availability of krill swarms within the survey site that can be detected by the MBE is

defined. Secondly the filter through which this underlyingpdf of krill swarm availability is observed using the MBE. The filter is known as the detectability of krill swarms. Finally the modified 2-D distance sampling function as derived by Marques (2007) is given.

Krill swarm availability

Thepdf of krill swarm centre coordinates, their availability, is dependent on the horizontal,

h(x), and vertical,v(z), distributions of distances to the geometric centre of krill swarms an is defined by:

π(x, z) = h(x)v(z) (6.3)

This assumes that the horizontal and vertical krill density distances are independent. Also, h(x) is assumed to be uniform giving:

π(x, z) = 1

w sin(π/3)v(z) (6.4)

where π/3 is half the swath width (Figure 6.2). Equation 6.4 also assumes a constant seabed depth, although seabed depth throughout the survey region varied from 46 to 120 m. This effectively varied the survey effort with respect to depth: krill swarms cannot occupy depths greater the the seabed depth and not considering this attenuation effect may give a biased estimate of the krill density gradient. For example, consider a survey site that has a uniform distribution of krill with respect to depth, the site has two depths that occur equally across the site, 50 and 100 m. In this instance using the observed vertical distribution of krill (z) to estimate the krill pdf, v∗(z), will lead to a depth gradient that estimates double the abundance of krill in the upper 50 m of the water column, which is biased. To correct for this bias the attenuation effect, a(z), was estimated, allowing an estimate of the true krill density gradient unbiased by seabed depth, v(z).

v∗(z)∝ˆa(z)v(z) (6.5)

The attenuation effect was estimated by fitting a logistic function to the proportion of depths (z) available over the survey area. The study site depths were measured using the mean observed seabed depth from the three centre beams of the SM20 MBE within 100 m line transect lengths. The seabed attenuation effect was incorporated into the krill swarm location pdf (equation 6.4), giving:

π(x, z) = 1

Figure 6.2: 2D swath geometry shown in the cross-track distance (x) and depth (z) dimensions, showing the maximum swath width z = wcos(π/3), where w is the seabed depth. Swarmi is located atri =

p

x2

i +z2i and θi =atan(xi/zi). The thick line denotes the external swath boundary. The MBE swath has been folded about its axis of symmetry (a vertical line at the centre of the swath).

Krill swarm detectability

The detectability of krill swarms was determined using two independent detection func- tions. The first is the range detection function g(r), where r is the radial distance from the MBE head to the krill swarm geometric centre. The second detection function is the angular detection function, p(θ), where θ is the angle from the centre of the MBE swath. These functions act as a filter on the available krill swarms (equation 6.6), conse- quently the krill location pdf cannot be observed directly. The range detection function was assumed to be half normal:

g(r|σd) = 1 p (2π)σ2 d e −r2 2σ2 d (6.7) where σd is the scale parameter.

Krill swarms do not have equal detectability across the MBE swath. As with many MBE systems it is less likely to detected areference swarm (a swarm with uniform char- acteristics) in the outer beams than in the centre beams whilst using the SM20 MBE. The variation in krill swarm detectability with θ was caused by changes in the across-swath MBE sensitivity, which is in turn caused by the beam pattern of the MBE transmitted acoustic pulse and the MBE head geometry (see Cochrane et al. 2003). The beam-by- beam sensitivity was estimated using observations of MBE beam-by-beam noise. Noise was assumed to be isotropic and estimated using mean volume backscatter (Sv) data that were collected 07/02/2007 in the Cape Shirreff study site. For this experiment the SM20 MBE was set to passive mode (no sound transmitted), with a 200 m observation range (r) and gain at 20log10(r), which allowed the observation of MBE system noise. Krill

were considered to have a lower detectability in beams with a higher Sv value, because a higher Sv meant there was more background acoustic noise over which to detect krill.

The beam-by-beam Sv observations were transformed to the linear domain (sv = 10{Sv/10}_{) and inverted, giving a relative measure of krill swarm detection probability,}

where it was less likely to detect krill swarms in the noisier beams. These data were rescaled so that the detection probability was one at the swath centre,q(0) = 1. Since the SM20 MBE does not have a nadir beam, minimum detectability of the three centre beams (Melvin et al., 2003) was used to rescale the inverted sv observations. To accommodate the MBE observations in a distance sampling framework the sensitivity of the MBE was considered to be symmetrical about the swath centre, so only the angle from the swath centre was used (θ), rather than beam number was used. This approach meant that the data were folded, meaning the variation in across-swath sensitivity with respect to detection angle was 0≤θ ≤π/3. The across swath sensitivity was assumed to affect the detectability of a krill swarm and was modelled using a modified hazard rate detection

function with form:

q(θ) =γh1−e−(θa)

−bi

(6.8) The hazard rate model parameters were estimated using a non-linear least squares algorithm implemented in R v2.4.0 (Vienna, Austria). Given that the two detection functionsg(r) and q(θ) are independent the probability of observing a krill swarm, when it is present at (r, θ) is given by:

P(observe|r, θ) = ˆg(r)ˆq(θ)

= d(r, θ) (6.9)

Where d(r, θ) is the combined range and angular detection function.

Likelihood: MBE observations

The pdf of the radial distance (r) and the sighting angle (θ) of detected krill swarms as derived by Marques (2007) is:

f(r, θ) = π(r, θ)g(r) Rw 0 Rθmax 0 π(r, θ)g(r)dθdr (6.10)

whereθmax =π/3, which is the limit of the MBE swath (see Figure 6.1(a))

Incorporating both the joint angular and range detection function d(r, θ) and the attenuation effecta(z) gives a joint pdf for radial distances and sighting angles of detected swarms: f(r, θ) = π(r, θ)d(r, θ)a(rsinθ) Rw 0 Rθmax 0 π(r, θ)d(r, θ)a(rsinθ)dθdr (6.11) This is the basis of a likelihood that can be maximised to estimate the unknown parameters of the vertical locationpdf φ1, and the range detection functionφ2, given the

seabed attenuation detection function a(r,sinθ):

L(φ1, φ2 |r, θ) = n Y i=1 π(ri, θi)d(ri, θi)a(risinθi) Rrmax 0 Rθmax 0 θ(r, θ)d(r, θ)a(rsinθ)dθdr (6.12) Where the maximum observation range w is the seabed depth so rmax = w and n is the number of detected krill swarms.

For the krill swarm analysis the vertical locationpdf was considered to have either a beta form, normal, or a log-normal form each of which have two parameters to estimate

in φ1. A uniform model, with no unknown parameters was also considered.