Geodesic source grid

The two previous sections demonstrated that we can neglect the dimensions of time and depth in the forward model. This means that the forward model parameters m only need to describe the distribution of vocalizing whales over latitude and longitude. The whales are replaced with a set of sound sources. We assume that these sound sources emit continuous signals at the frequency of 20 Hz. The sound sources can not be allowed to take arbitrary positions, since this would require an extreme computational effort. Instead the possible locations of the sound sources are reduced to nodes on a grid. This grid is termed the simulated source grid.

Using a rectangular latitude-longitude grid will result in uneven distribution of grid nodes across ocean basin scales. Even distribution of grid nodes across these scales can be achieved using a geodesic grid. Teanby (2006) described an algorithm to develop such grids based on approximating the shape of a sphere using an icosahedron. It is available as Matlab code and was implemented into the forward model. Figure26shows the loca- tion of the grid nodes for 3 resolution steps in the North Atlantic.

Figure 26: Position of simulated source grid nodes for 3 resolution steps are displayed as blue dots. Geodesic grid nodes span a regular icosahedron around the earth based onTeanby(2006).

After discretizing the search space, there are many ways to formulate the forward prob- lem. A first approach was to place one source on each grid node and allow each to take one of 10 preset sound pressure levels. However, the parameter estimation algorithm could not find the maximum likelihood parameters. There were too many degrees of free- dom in the system to find the global maximum of the likelihood function within realizable computation time. It was necessary to find a way to reduce the degrees of freedom and render suboptimal solutions less attractive for the parameter estimation algorithm.

After experimenting with different formulations of the forward model and different pa- rameter estimation algorithms, an efficient formulation of the inverse problem was found. Instead of searching directly for the sound pressure value of each grid node, a fixed num- ber of simulated sources is moved across the grid nodes. This has the effect of reducing the number of possible combinations to nnsimulated sources

nodes (in the benchmark scenario it is

195195). The number of simulated sources is the same as the number of grid nodes, to allow all source location combinations. These range from one simulated source at each node to all simulated sources being at one node.

nsimulated sources= nnodes (24)

The parameter vector m is then a vector containing the node index describing where each simulated source is located. If there is one simulated source on each node, the parameter vector will have the following values:

m = [1, 2, 3, ... nnodes] (25)

This can be interpreted as: "Source one is located at node one, source two located at node node two,..." until "source nsimulated sources is located at node nnodes". To further

reduce the degrees of freedom, the total sound pressure of all simulated sources is fixed to the value ptotal. Each simulated source is then assigned the same sound pressure,

based on equation26.

psource=

ptotal

nsimulated sources

5. Methods

Since the true ptotal is usually unknown, we assume it is somewhere between a mini-

mum and maximum value pminand pmax. The minimum and maximum total pressure are

part of the a priori probability distribution ρ(m). Appropriate values for pminand pmaxcan

be obtained from published values on fin whale acoustics.

The source level of each node is determined by the number of simulated sources posi- tioned at it:

nsources on node(inode) =

X

(inode ∈ m) (27)

Here inodedescribes the index of each node. The source pressure and source level of

each node are determined by the following equations:

p(inode) = nsources on node(inode) psource (28)

SL(inode) = 20 log10(p(inode)) (29)

The source pressure at each grid node describes, where on the grid sound energy is emitted and how much. It is an approximation of the distribution of vocalizing fin whales. In regions where the grid nodes have non-zero sound pressure values, vocalizing fin whales must be present. In regions where the grid nodes have zeros as sound pressure values, no vocalizing fin whales are present. Nodes with high source pressure indicate in- creased fin whale calling activity, compared to nodes with low source pressure. However, increased calling activity does not automatically imply an increased whale abundance. Increased calling activity could either be caused by an increasing number of vocalizing whales or an increase in the whales source levels and call rates.

Compared to the simulated sources (that are confined to take positions on a geodesic grid) the recorders can take any position within the ocean. According to the sonar equa- tion (Eq. 4), the simulated received levels depend on the transmission loss between the grid node and recorder locations. For a given set of recorders, equation30 determines the simulated received levels:

RLsimulated(irecoder) = nnodes

X

inode=1

SL(inode) − T L(inode, irecoder) (30)

This is a simplified version of the sonar equation, RL stands for received level, SL for source level and TL for transmission loss. The indices inode and irecoder describe the

respective elements of the source grid and recorder array. The transmission loss can be determined using an arbitrary sound propagation model, as long as the TL can be given as a matrix connecting each node and recorder. Two ways of calculating the TL matrix were tested in this thesis: Geometrical spreading and ray trace modeling using BELLHOP. They will be described in the next section.