# Motion correction (Dunford method)

This operator compensates sample values for attenuation due to changes in the orientation of the transducer, which occur between the time of the pulse transmission and the time of sample reception.

It accepts operands with the following data types:

Operand 1:

• Linear
• Power dB
• Sv
• TS
• Unspecified dB

Operand 2:

• Roll

Operand 3:

• Pitch

Note: If you plan on using the Towed body operator, it is recommended you use the Motion correction (Dunford method) operator before the Towed body operator.

## Settings

The Motion Correction (Dunford method) Variable Properties dialog box pages include (common) Variable Properties pages and these operator pages:

### Motion correction page

 Setting Description 3dB beam angle factor A multiplying factor for the 3dB beam angle used to threshold the motion correction output. This setting enables you to specify the degree to which the Dunford algorithm is applied to your data. Values in the Information section of the Motion correction page show the impact of the threshold on your data.

## Algorithm

The algorithms for the Motion correction (Dunford method) operator are based on those developed by Dunford.

Refer to:

The Dunford method algorithms are valid for circular transducers and apply a motion correction factor k1 to each sample of each ping.

If s is the original linear backscatter value of a sample, its adjusted value s', after applying the operator becomes:

s’ = k × s.

This correction factor (k) must be applied in the linear domain. Hence, the formula used within Echoview for Sv, TS and dB variables becomes:

s' = s + 10 log k

## Calculation of k

k = 0.17083 x 5 - 0.39660 x 4 + 0.53851 x 3 + 0.13764 x 2 + 0.039645 x + 1

where x = sin(γ) / sin(α/2),

 α is the full-width half-power beam width of the transducer. This is specified as the Minor axis 3dB angle or Major axis 3dB angle on the Calibration page of the Variable Properties dialog box. Note: For a circular transducer the Minor axis 3dB angle equals the Major axis 3dB angle. γ is the separation angle between the transducer’s pointing direction at transmission and reception and is defined by the roll data and pitch data. Dunford specifies that the practical upper limit for γ should be, γ equal to the transducer Major axis 3dB angle or the Major axis 3dB angle. Samples where γ is greater than Major or Minor axis 3dB angle are set to no data. The algorithm for γ is given below.

## Derivation of γ

The angle γ is obtained from roll and pitch data. In order to obtain accurate roll and pitch values for each sample, we need precise timing information. We derive such times from the time stamp of the ping, the value for the range of the sample and the sound speed.

For a ping, let tt be the transmission time of the pulse.

If r is the range (m) of a sample, and c the sound speed (m/s) then t r (s) is the time of reception for that sample

tr = tt + 2r/c

We use tt and tr to find the corresponding roll and pitch values, by linear interpolation from the roll and pitch variables

Let:

 at, ar represent the roll angles at transmission (t) and reception (r) bt, br represent the pitch angles at transmission (t) and reception (r)

Dunford Figure 12, defines azimuth and zenith angles at transmission and reception.

 θ is the azimuth angle, measured from the positive x-axis towards the positive y-axis. φ is the zenith angle, measured from the positive z-axis.

### Azimuth and zenith angles expressed in terms of roll and pitch angles

Expressing Dunford defined azimuth and zenith angles in terms of roll and pitch angles:

φ t = atan { sqrt( tan2(at) + tan2(bt) ) },

θ t = atan { tan(at) / tan(bt) },

and similarly for the received versions of φ r and θ r.

### Unit vectors

We construct unit vectors along the pointing directions at transmission and reception:

pt = ( sin(φ t) cos(θ t), sin(φ t) sin(θ t), cos(φ t) )

pr = ( sin(φ r) cos(θ r), sin(φ r) sin(θ r), cos(φ r) )

The angle γ is now obtained from the identity: cos(γ) = p t . p r (vector dot product). This holds since ||p r|| = ||p t|| = 1.

### Expression for γ

Expanding this out we get following formula for γ :

γ = acos{ sin(φ t) cos(θ t) sin(φ r) cos(θ r) + sin(φ t) sin(θ t) sin(φ r) sin(θ r) + cos(φ t) cos(φ r) }

Notes:

• It is assumed that the timestamp given within Echoview is the precise time of the beginning of both transmission and logging of sample data.
• This algorithm is only valid for circular transducers.
• A value of k = 1.0 corresponds to no correction. Correction factors will always be greater than 1.0. It is assumed that targets that were on-axis at transmission, end up off-axis at reception. Dunford's paper specifies that the practical upper limit for γ (which in turn limits k) is the transducer Major axis 3dB angle or Minor axis 3dB angle.

## Threshold

By default, the Motion correction algorithm applies a threshold to data values outside the practical upper limit of γ (effectively the 3dB angle). See Calculation of k above.

To give you additional control over the application of the algorithm Echoview offers a setting on the Motion correction page of the Variable Properties dialog box that can threshold data values to a user defined limit of f γ. Where f (the 3dB beam angle factor) is a multiplier for the 3dB angle. In effect, you limit the correction factor k.

Values in the Information section of the Motion correction page display the separation angles and resultant correction factors:

 Current 3dB beam angle Minor axis 3dB angle or Major axis 3dB angle on the Calibration page of the Variable Properties dialog box. This is γ in the algorithm. Maximum separation angle fγ Maximum correction (factor) Maximum k is a function of α and fγ. Maximum correction Maximum k is a function of α and γ.