Choosing Frequencies for Multifrequency Animal Abundance Estimation

Paul Roberts

AOS Seminar March 3, 2005

38 kHz?120 kHz?200 kHz?

Outline

• Background: Multifrequency Animal Abundance Estimation

• Frequency selection problem statement

• Condition Number as a selection metric

• Application to frequency selection in a multifrequency system

• Examples of condition number space

• “Minimization” results, interpretations, and summary

Biological Oceanography

Fundamentally concerned with:

•What (who) is out there (identification)

•How many of them are there (global abundance)

•How are they distributed (local abundance)

•What are the underlying processes that control the first three. (ecosystem model)

Biological Oceanography

In an effort to answer those questions, it is necessary to collect samples. There are three primary methods:

• (direct) Net sampling (avoidance. net-interaction)

• (indirect) Optical Sampling (limited range)

• (indirect) Acoustical Sampling (non-specificity)

Most scenarios will require some combination, preferably all three.

Multifrequency systems for Zooplankton

MAPS (Mulifrequency Acoustic Profiling System):

21 frequencies from 100 kHz to 10 MHz in log spacing

TAPS (Tracor Acoustic Profiling System):

265, 450, 700, 1100, 1850, 3000 kHz

HTI 244:

42, 120, 200, 420, 1000 kHz

Simrad EK500:

38, 120, 200 kHz

Important Definitions in Bioaoustics

Target Strength:

)(log10 10 bsTS

bs is the backscattering

cross-section. A measure of the reflectivity of the scatterer.

ka (or kL):

c

afaka

22

ka is a measure of how large the scatterer is compared to the wavelength of sound

Multifrequency Bioacoustic Inversion

Proposed in 1968 by McNaught (from Greenlaw 1983): Echosounders operating at different frequencies will be maximally sensitive to zooplaknton of different sizes.

1977 Van Holliday: “Extracting Bio-Physical Information from the Acoustic Signatures of Marine Organisms”

Van developed a mathematical procedure (linear inversion) to estimate fairly general bio-physical parameters belonging to a particular class of scatterers.

In most applications, the parameter is chosen to be abundance, and the class identifier is chosen to be size.

Multifrequency Abundance Estimation

f 1

...

f m

N S 1

N S p

X f 1

X f m

N S1

N SnTrue Abundance

Echo Vector

R s f , S1

Abundance Estimate

InversionData Processing

Multifrequency Abundance Estimation

In the framework proposed by Van, the estimation problem can be expressed as a linear inverse problem:

Given find

X(f) is a vector a processed echoes

R(f,S) is the reflectivity matrix

N(S) is the vector of abundance

)(),()( SNSfRfX )(SN

Multifrequency Abundance Estimation

The linear inverse framework has be the standard for the multifrequency abundance estimation problem. In many cases, the SNR could be low enough that estimates will be negative. To avoid this problem, a Non-Negative Least Squares method is used almost exclusively.

The NNLS method adds a single constraint of positivity to the standard 2-norm least squares problem.

It is possible to add additional constraints based on biological information from, for example, net tows.

Choosing A Model

In the past, the applicability of the technique to zooplankton was limited primarily by available models. In recent times, the models have become much more accurate. One good model is based on the Distorted Wave Born Approximation.

DWBA Model for E. Superba

• Expected value of Reduced Target Strength (RTS) over 10,000 uniformly random orientations.

• RTS = TS - 20log10(L) were L is the length of the scatterer.

• Derived from a single 38.35mm E. Superba that is digitized from an image.

Problem Statement

Assuming that we are free to select the transmit frequencies of the echosounder. On what basis should the selection be made.

Some Constraints:

SNR: Depends on range, density, size and reflectivity of the scatterers. In general will require that the wavelengths are larger than the size of the scatterer.

Need Additional Constraints

There are a wide range of possible frequencies that satisfy the constraints, but may be better or worse for the abundance estimation.

The quality of the estimate will depend on the frequency and size dependent reflectivity in the model.

The relationship between scattering, frequency and size is non-monotonic. It is therefore possible that some combinations of frequencies will yield linearly dependent vectors in the reflectivity matrix. In this case, it would not be possible to estimate abundance for each size class.

Optimal Frequencies

Do optimal frequencies exist?

For fixed size classes, species, they might, but would appear to be difficult to find. For example how do we define optimality in this case?

One idea: find f to minimize:

This however will require knowledge of the statistics of the population AND the estimator. These will likely be unavailable or difficult to derive. In addition, they will likely depend on many factors like the inversion method.

]))(ˆ))((ˆ[( TfNNfNNE

A Simple Metric

In order to reduce the complexity required to truly find optimal frequencies, we could base our selection on a simple metric which could be more more applicable in general.

To this end, we select the condition number of the reflectivity matrix as it is intimately related to the quality of the inversion.

The Condition Number

Fundamentally, the condition number bounds the potential magnification of relative error in computing the solution to a perturbed linear inverse problem:

It is commonly defined in terms of the matrix 2-norm in which case it is the ratio of the largest to the smallest singular value of the matrix:

B

BR

A

A

)(

Trrr

TTT vuvuvuVUR 222111

min

max)( R

BBAAR )(

Trr

r

TTT uvuvuvUVR111

222

111

11

The Condition Number

The singular values of the matrix define the amount of relative stretching that is applied to the semi-major axes of the hypersphere defined by the basis vectors of the matrix V:

If the singular values are equal, there is no relative stretching. However, if they are very unequal, there is a correspondingly large relative stretching (ellipse -> line)

v1

v2

R1u1

2u2

Application to a simple noise model

Consider a simple linear model using a square invertible matrix R with an additive zero mean Gaussian noise vector w. Define the data Y as:

wRNwRRNRYRN

RNXwXY1111ˆ

where

Our estimate is then the true abundance N perturbed by a noise term which is also Gaussian with zero mean. Applying the perturbed linear inverse model:

X

wR

wYR

wR

N

NN-)(

)(

ˆError Relative

1

1

Effect of the condition number on the model

The direction associated with both the signal and noise vector will dictate how much relative error exists in the solution.

X

wR

wYR

wR

N

NN-)(

)(

ˆError Relative

1

1

v1

v2

R1u1

2u2

Simulation Setup

•Consider exclusively the DWBA model for E. superba (Antarctic Krill).

•Restrict frequencies to the range: 30 - 1000 kHz

•Use fixed size classes in the range: 18 - 55 mm

Much of the following analysis amounts to picking points on the ka curve, building a matrix from those points, and then computing the condition number via SVD.

R f 1, S1

R f 2, S1

R f 3, S1

R f 4, S1

R f 1, S 2 R f 1, S 4

R f 2, S 2

R f 3, S 2

R f 4, S 2

R f 2, S 3

R f 3, S 3

R f 4, S 3

R f 2, S 4

R f 3, S 4

R f 4, S 4

R f 1, S3f 1

f 2

f 3

f 4

Building the Reflectivity Matrix

Reflectivity Matrix

• Select a frequency and size

• Compute corresponding ka value

• Find RTS value from model

• Convert RTS to TS

• Convert TS to

•Set R(f,S) = bsbs

Model: E. superba (DWBA)

R: 4x4 Sizes: 18, 30, 43, 55 mm10,000 simulations of estimateusing NNLS

Frequencies used:

Cond(R) = 17 @ 70, 170, 307, 366 kHz Cond(R) = 131 @ 70, 105, 200, 400 kHzCond(R) = 1410 @ 38, 70, 170, 200 kHz

Effect of condition number on relative error:

Order of magnitude increase in condition number could dramatically increase the expected relative error in the estimate.

This has important implications for applying the condition number metric to non-square estimation problems.

Effect of matrix shape on condition number:

Square matrix has much greater variance than both tall/thin and short/fat matrices.

The case with two frequencies and two sizes: 18, and 55mm. Even for frequencies which are very different, there is still potential for large condition number.

3 frequencies, 3 sizes: 18, 36, 55mm

4 frequencies, 4 sizes: 18, 30, 43, 55 mm

“Minimum” solution:

Condition Number = 17

Frequencies:

70, 170, 307, 366 kHz

Shows common bracketing of the transition region in scattering curve.

Many of the coefficients occupy the geometric region.

Comparison:

Using 70,120,200,420 kHz, the condition

number is 80

Summary

• The problem of selecting frequencies for a multifrequency system has been examined.

• The condition number has been shown to be important in the quality of the estimate and as such could provide an additional constraint in selecting frequencies.

• The condition number for square reflectivity matrices is shown to depend on size and frequency in a complicated and interesting way.

• It may be possible to chose frequencies that yield a low condition number while at the same time yielding stronger (geometric) scattering.