analysis of cross-sections of otolith of icefish

1. Measurement and analysis of optical density profiles of otolith from
Ps. georgianus and Ch. gunnarii.
Ryszard Traczyk
Introduction.
Analysis of cross-sections and whole otoliths, for their growth poses many difficulties, it is very tedious and time consuming. Results of analyzes are very different, largely depends on the method used, the individual researcher's experience, as well as his physical and mental abilities: visual acuity of vision - the distinction of color, degree of fatigue, or memory of experiments in the large number of observations and measurements derived under the microscope. There are only stored information which are research purpose, additional are lost.
In this paper, a variety of investigation and measurement of otolith increments were performed by computer measurement and harmonic analysis of the optical density - daily and other otoliths increments are repeated several times at intervals of time and space.
The peaks of optical density of daily increments in otoliths mackerel icefish were more suppressed by noise records than South Georgia icefish. Ch. gunnarii is a pelagic fish, where there are more noises and Ps. georgianus is more demersal, where its gets less noises. Statistical significance for power of daily increments peaks was tested.
For nuclei of larval otoliths of Pseudochaenichthys georgianus average width of daily increments was 9.41 * 10-4 mm. Width of daily otolith increments for juvenile of Champsocephalus gunnarii were 0.0024 mm. Cyclical increments in larval otoliths of Ps. georgianus proceeded by a sine wave: y=3,42sin( x[mm] -0,08)+235,1, and for juvenile of Ch. gunnarii by a sine wave: y=3,82sin( x[mm] -24,83)+90.
Materials and Methods.
Measurements of optical density of otolith increments were carried out from digital photos of otolith slices, obtained by the excision of otoliths, two-sided grinding, polishing and etching with EDTA (1). Optical density measurements were performed in the program "Fiji image" developed for visualization and measurements of microscope images (2). The otolith material were collected during research cruises on the RV. "Professor Siedlecki" (3; 1).
The images were processed to measurements: contrast was optimized, background was cleared, the SI mm unit was setup. For each change its optical density profile were recorded with 1231 measurements along otoliths radius and stored. Further study were on pictures with background evenly to dark.
Exploration and study of cyclic variation of optical density of increments otoliths were carried out three ways:
1) Spectral analysis and REDFIT procedures of statistical package PAST (4; 5), utilizing Fourier transformation algorithm of Lomb’s periodogram (6; 7; 8; 9);
2) the procedure for the fitting sinusoid to the empirical data using program PAST.
3) minimizing the sum of squares of model deviations from the empirical data using Excel Solver tool.

2. Fig. 1. Image processing, contrast optimization, extraction and reset the background.
In the model REDFIT the procedure to increase the number of points on the frequency axis was used (5) - after passing the series test fitting empirical data to red noise model (5).
Average cycle of daily increments was determined by moving a series of measurements (713 for Ps. georgianus and 503 for Ch. gunnarii) of the optical density along whole density profile of the same otolith radius (which retains a fixed number for subtracted value) and determining the first minimum in sums of squared differences of optical density in the adjacent of each measure of series. The distance in moving from the initial difference of none (0 - subtract the same value) to the first minimum was adopted for the period: 24 h periodicity unit.
A series of 713 measurements were selected from the optical density profile of otolith of Ps. georgianus starting from measure No 7 from the edge nucleus to the 720 measure along the radius – because measurements began not exactly from the edge, but before it and additionally the initial marginal, usually unclear measurements should be excluded from the analysis of the pattern.
Fig. 2. Measuring the density of the image having background aligned. Determination of the period between troughs. In column D is the subtraction of series after its first shifts (subtraction without any move produces a sum of squared difference equal to 0). Subtraction series after its 16 time shift resulted in second column of S giving minimum sum of squared differences. In further columns that sums grow. The blue line - series of 1231 optical density measurements, green - 713 initial measurements from 7 to 720 measurement that were subject to move with one unit measurement: every 0.000059 mm.
A series of 503 measurements of the otolith optical density of Ch. gunnarii, were selected from its profile on radius from 507 measure from the edge to the 1009 measure.

3. Fig. 3. Profile of the optical density of otolith nuclei of Ch. gunnarii (r = 0.04 mm), with reset the background.
Fig. 4. Profile of the optical density extended outside of otolith nucleus of Ch. gunnarii (r = 0.22 mm), with reset the background.

4. Fig. 5. Profile of the optical density for cross section of otolith of Ch. gunnarii (r = 0.44 mm), with reset the background.
Fig. 6. Measurement of the optical density of the images having reduced background. Determination of the period between troughs. Column D contain results of the subtraction of series after its first shifts. Subtraction series after the moving of one of two with about six units resulted in second minimum sum of squared differences placed in column I. Shift by next six unit gave third minimum Sum in column N. Shift by next six unit gave minimum No 4. The blue line - series of 1615 optical density measurements, green - middle 503 measurements from 507 to 1009, that were moved with unit: the distance between measurement equal to 0.0004 mm.
Shifts of large series of measurements form a representative unit of daily increment.

5. Results
Pseudochaenichthys georgianus.
The average period of diurnal cycling in the increments specified by relative displacement of 713 measurements became 9.41 * 10-4 mm = 941 nm. Harmonic analysis Lomb-Scargl in the program PAST indicated the presence in the otolith increments two statistically significant periodicity: daily and 8-daily. A large portion of the variability of the analyzed record still have the transformations above daily periodicity in the echo effects: peak No 3 of 5 daily cycle, and peaks No 4 and 5 of circadian cycles.
Fig. 7. Participation sinusoids in the variability of density record. The dotted and dashed lines define the probability levels for which the harmonics included there in the part of the noise generated by it. Peaks 1-5 with a power above the level of 11.87, with a frequency of 0.0075 cycles / 60 nm, have a probability close to 0, which is equal to and less 0.01, which means that in the analyzed series of space - time there are exists cyclicality with such frequencies.
Other harmonics have a smaller share of variability and can be a large percentage of noise - are generated randomly by the noise.
However, in the procedure fit profile to tested the red noise model (used to analyze the signals common in the the marine environment), MonteCarlo simulation test 99% show that in the analyzed variation can not be identified significantly strong periodicity, which would not have been generated by incidental noise, beyond the peak four-hour cycle. The use this model here is appropriate, as indicated by series test.
Fig. 8. Spectral analysis of the optical density of otolith increments for fitting spectrum data to the red noise model (5). Because at test, fitting sample to the model, the number of runs 171 has been set at 5% confidence interval of: 161-198, the above noise model is appropriate for the tested series, which contains not a strong periodicities. Plot of curves of χ² and Monte Carlo for studies of statistical significance of power of peaks. Calculate the critical level indicator (False-al.) the probability of random creation of a sine wave: here in 99.86% a sine wave is generated by noise -enters into its composition.
As it was small sample for above procedure, the number of points on the frequency axis
0
2
4
6
8
10
12
14
16
18
20
22
24
26
0
0,1
0,2
0,3
0,4
0,5
p(random) = 0,000000009961
moc~A2
poziom 0,01 = 11,87
poziom 0,05 = 10,24
frequency
Peak1 f= 0,007528 cycles per 59 nm, or per 1,5 h; 0,0078 mm/cycle, or 8 days/cycle
Peak3 f = 0,013305 cycles per 59 nm, or per1,5 h; 0,0044 mm/cycle, or 5 days/cycle
Peak5 f = 0,05007 cycles per 59 nm, or per 1,5 h; 0,001 mm/cycle, or 1,25 day/cycle
Peak4 f = 0,051996 cycles per 59 nm, or per 1,5 h; 0,001 mm/cycle, or 1,2 day/cycle
Peak2 f = 0,0649 cycles per 59 nm, or per 1,5 h; 0,001 mm/cycle, or 1 day/cycle
power ~A²
level 0.01=11.87
level 0.05=10.24
0
1000
2000
3000
4000
5000
6000
7000
0
0,1
0,2
0,3
0,4
0,5
τ: 2,2696, Bnwith: 0,001692, False-al.: 99,86
moc
chi2 99%
chi2 95%
Teoret.AR(1)
Monte Carlo 95%
Monte Carlo 99%
krytycz. Chi2
frequency
Peak1 f = 0,00699 cycle per 59 nm, or per 1,5 h.; 0,0084 mm/cycle, or 9 days/cycle
Runs Test: k=171 is in 5% aceptance interval: 161-198.
Peak3 f =0,039 cycle per 59 nm, or per 1,5 h.; 0,002 mm/cycle, or 1,6 day/cycle
Pea4 f=0.05175 cycle per 59 nm, or per 1.5 h.; 0.001 mm/cycle, or 1.2 day/cycle
Peak5 f=0.064336 cycle per 59 nm, or per1.5 h.; 0.0009 mm/cycle, or 1 day/cycle
Peak2 f=0.2112 cycle per 59 nm, or per 1.5 h.; 0.0003 mm/cycle, or 7 h./cycle
Peak6 f=0.35524 cycle per 59 nm, or per 1,5 h
0.00017 mm/cycle, or 4 h./cykl
power
99%χ²
95%χ²
Theoret. AR(1)
99%Monte Carlo
95%Monte Carlo
Crit.level χ²

6. was doubled that give periodicity considerably strong.
0
1000
2000
3000
4000
5000
6000
7000
0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1
Moc χ² 99%
χ² 95% Teoret.AR(1)
frekw
Pik1 f =0,0073427 cykli na 59 nm, lub w 1,5 godz., a to jest 0,008 mm/cykl, lub 8,5 doby/cykl
Pik3 f =0,013287 cykli na 59 nm, lub w 1,5 godz., a to jest 0,0044 mm/cykl, lub 4,7 doby/cykl
Pik5 f =0,05 cykli na 59 nm, lub w 1,5 godz., a to jest 0,001 mm/cykl, lub 1,3 doby/cykl
Pik4 f =0,052098 cykli na 59 nm, lub w 1,5 godz., a to jest 0,001 mm/cykl, lub 1,2
Pik2 f =0,065035 cykli na 59 nm, lub w 1,5 godz., a to jest 0,001
mm/cykl, lub 1 doba/cykl
Pik7 f =0,01958 cykli na 59 nm, lub w 1,5 godz., a to jest 0,003 mm/cykl, lub 3,2 doby/cykl
Pik6 f =0,039161 cykli na 59 nm, lub w 1,5 godz., a to jest 0,0015 mm/cykl, lub 1,6
Fig. 9. By increasing the number of points on the frequency axis, two considerably periodicity in otolith increments, were
underlined (by 99%χ²): 8.5 diurnal cycle and 1 daily. Less significant (under 99%, overgrove 95% Monte Carlo) were
circadian cycles. The variability of otolith increments have a large share of periodicity of 5, 3 and 2 daily cycles -
transformations from one to more daily cycles in echo effects (7). Symbols as in above figure.
Sinusoidal curve fitting. The first seven sinusoids (y=Ansin(
x + φn)+const.) matched
to empirical measurement of optical density well explains the variation of the optical density of
otolith increments.
Two sine waves in the first look dominate: one daily and 8 daily cycle, as in the spectral
analysis. Parameters derived: amplitude - A period - T phase - φ, were similar to those
measured with known periods in spectral analysis, Tab. 1.
Fig. 103 Fitting empirical 713 measurements by model composed of seven sinusoids..
Most closely fitted course of the 7 main sinusoids, were determined by optimizing all 22
parameters to minimizing the sum of squared deviations of the model, yi from empirical data,
ye. This sum amounted to min|Σ(yi-ye)2|=72257,5. Such estimates for previous methods were
higher.
As a result calculation of the third method it was obtained that the strength of periodicity
(proportional to A2) comes from two main cycles: the first 8 daily cycle A1 = 4.52 (T1 = 0.00789
mm, φ1 = -2.64) and the second daily cycle, A2 = 3.42 (T2 = 0.00091 mm, φ2 = -0.08. Constans
= 235.1.
Variability in optical density of the otolith, still explain the circadian sinusoid: 1.2 of daily
180
190
200
210
220
230
240
250
260
0 0,01 0,02 0,03 0,04
nucleus edge
gęst. optyczna fragmentu promienia jądra, N = 713 pomiarów y dopasowane, nieznane A, T, φ:
mm
unknown A, T, φ: A 1=4,51, φ1=-2,41, T1=0,007955; A2=3,49, φ2=-0,566, T2=0,000908; A3=3,19, φ3=-3,11, T3=0,004395; A4=3,28, φ4=1,1,
T4=0,001124; A5=2,96, φ5=-2,98, T5=0,001495; A6=2,77, φ6=-1,68, T6=0,002959; A7=2,7, φ7=2,8, T7=0,002111; stała =235,1
χ² = 72951; R² = 0,27408; p = 1,7577E-42; Akaike IC = 72979; Σ(yₑ-yₓ)²=72951; Σ(yₓ-yₓ)²=5837
optical density of nucleus radius, N=713 measurements y fitted, unknown A, T, φ

7. cycle A4 = 6.07 (T4 = 0.00114 mm, φ4 = 0.81) and 1.3 of daily cycle: A5 = 5.33 (T5 = 0.00115 mm, φ5 = -2.87).
3 cycles of other incorporated into the model are likely to be interpreted as transformation of the first – daily cycle in echo effect: 5 daily with A3 = 3.31 (T3 = 0.00446 mm, φ3 = 3.11), 3 daily with A7 = 2.79 (T7 = 0 , 00299 mm, φ7 = -3.21) and 2 daily A6 = 3.11 (T6 = 0.0015 mm, φ6 = 3.89). Existence of harmonic component with a frequency of daily increment determines the frequencies of multiple components: the effects of jumps in periods of 2 *, 3 *, 4 * daily, etc.
Tab. 1. Sinusoids parameters (Ps. georgianus larvae), daily increments in bold.
From spectral analysis of PAST.
From fitting sinusoids of PAST.
From minimizing the sum of squared deviations
No
A
φ
T
Const.
A
φ
T
Const.
A
φ
T
Const.
1
Sin
4,47
-2,53
0,008011
235,1
4,51
-2,41
0,007955
235,1
4,5153
-2,6358
0,007892
235,1
2
Sin
3,39
-0,0911
0,000905
χ²
3,49
-0,566
0,000908
χ²
3,4242
-0,082
0,000906
3
Cos
3,29
2,95
0,004427
73979
3,19
-3,11
0,004395
72951
3,3089
3,1087
0,004461
4
Sin
2,88
0,516
0,001129
R²
3,28
1,1
0,001124
R²
6,0728
0,8111
0,001141
5
Cos
2,36
-1,5
0,001177
0,26384
2,96
-2,98
0,001495
0,27408
5,3267
-2,8712
0,001154
6
Sin
2,97
2,97
0,0015
P
2,77
-1,68
0,003
p
3,1081
3,8871
0,001504
7
sin
2,71
-2,44
0,003004
2,3727E-40
2,7
2,8
0,002111
1,7577E-42
2,7912
-3,2104
0,002986
Fig. 11. The results of measurements approximated by model consisting of seven sinusoids with the use Excel Solver tool - it's hard to count them. The increments pattern have changes, one just 0,027 mm from the edge of the otolith nucleus.
Fig. 12. The base periodicity: unite of daily increments with nine daily - number of them along nucleus radius.
180
190
200
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230
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250
260
0
0,01
0,02
0,03
0,04
nucleus edge
gęst. optyczna fragmentu promienia jądra, N = 713 pomiarów
solver Σ1-7(An*sin(2πx/Tn+φn))+const.
mm
solver: A1=4,52, φ1=-2,64, T1=0,00789; A2=3,42, φ2=-0,08, T2=0,00091; A3=3,31, φ3=3,11, T3=0,00446; A4=6,07, φ4=0,81, T4=0,00114; A5=5,33, φ5=-2,87, T5=0,00115; A6=3,11, φ6=3,89, T6=0,0015; A7=2,79, φ7=-3,21, T7=0,00299; stała =235,1
Σ(yₑ-yₓ)²=72257,5
optical density of nucleus radius, N=713 measurements solver tool Σ1-7(An*sin(2πx/Tn+φn))+const.
225
235
245
0
0,01
0,02
0,03
0,04
nucleus edge
Σ1,2(Asin(2πx/T +φ))+cons
mm
A1=4,52, φ1=-2,64 T1=0,00789; A2=3,42, φ2=-0,08, T2=0,00091; stała =235,1

8. Fig. 13. Sinusoidal components: the daily cycle and circadian, their increments number in the section of nucleus radius of otolith - they overlap (they can be passing at the same layer on the joint surface goes in negative and positive curvatures).
Fig. 14. Sinusoidal components: multiple echo effects (reflections from the parallel walls): several daily cycles - their increments number in the section of nucleus radius of otolith. Mixing the direct signal with the reflected - delayed signal. In otolith sound speed is 3.5 times higher than in the endolymph.
Adding to model the four-hour periodicity, y6=0,37sin( x[mm] +4,82)+235,1 , Fig. 8, increases the fit of the model (increases the amplitude of circadian growth). But no eliminates changes in the formula growth – f.e. abnormal one at 0.027 mm distance from the nucleus edge, Fig. 11. These optical density profiles of otolith nuclei of Ps. georgianus larvae complement previously presented otolith profiles of optical density and widths of daily increments for juvenes and adults this species (1): 0.00091 mm for larvae; 0.00284 mm for juvenes; 0.0034 mm for adults.
Champsocephalus gunnarii.
Moving the 503 measurements of increments periodicity in relation to adjacent otolith increments determined the average daily otolith growth for larvae of Ch. gunnarii amounting to 0.0024 mm. Average daily increments of this width in the measured radius otolith equals to 0.43 mm can be 180. Harmonic analysis Lomb-Scargl’s, did not give the results of such a strong periodicity for these daily increments. They were drowned out by large number of strong signals with periods of 0.03 mm.
227
229
231
233
235
237
239
241
0
0,01
0,02
0,03
0,04
nucleus edge
y2=3,39sin(2πx/0,001+-0,08)+235,1
y4=6,84sin(2πx/0,001+0,85)+235,1
y5=6,1cos(2πx/0,001+-2,98)+235,1
mm
A2=3,42, φ2=-0,08, T2=0,00091; A4=6,07, φ4=0,81, T4=0,00114; A5=5,33, φ5=-2,87, T5=0,00115; const. =235,1
231
233
235
237
239
0
0,01
0,02
0,03
0,04
nucleus edge
y3=3,36cos(2πx/0,004+3,02)+235,1
y6=3,06sin(2πx/0,002+4,01)+235,1
y7=2,05sin(2πx/0,003+-1,41)+235,1
A3=3,31, φ3=3,11, T3=0,00446; A6=3,11, φ6=3,89, T6=0,0015; A7=2,79, φ7=-3,21, T7=0,00299; const. =235,1
mm

9. Fig. 15. Decomposition of spatial periodicity of optical density of otoliths for sinus and cosine components. Meaning of symbols as in Fig. 7.
Diurnal periodicity in the optical density profile of otoliths Ch. gunnarii really stand Redfit model for red noise. Because the measurement range is large in relation to the growth cycles, it was partitioned into 3 equal segments for averaging the resulting spectrum, which reduces the noise (4). In the model, the number of points on the frequency axis is concentrated twice.
Despite negative a runs test for fit to the red-noise model, it was used to investigate the statistical significance of the component peaks. Simulations 99% of Monte Carlo model showed that there are only two significant spectral peaks: daily and 54 daily increments. These sinusoids over goes 99.88% False-alarm level, which includes probability of random to be generated by noise, Fig. 16.
Fig. 16. The spectrum of sinusoids in red noise model REDFIT. A. Meaning of symbols as in Fig. 8.
0
10
20
30
40
50
60
70
80
90
100
110
120
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,2
p(random) = 1,396E-47
Moc ~A²
Theor.AR(1)
poziom 0,01 = 12,29
poziom 0,05 = 10,65
Frequency
Peak1 f =0.0030998 cycle per 0.0004 mm, or per 4 h., 0.129 mm/cycle, or 54 days/cycle
Peak2 f =0.00077495 cycle per 0.0004 mm, or per 4 h., 0.52 mm/cycle, or 215 days/cycle
Peak3 f =0.0023249 cycle per 0.0004 mm, or per 4 h.
0.172 mm/cycle, lub 72 days/cycle
Peak4 f =0.0015499 cycle per 0.0004 mm, or per 4 h.
0.258 mm/cycle, or 108 days/cycle
Peak5 f =0.0080595 cycle per 0.0004 mm, or per 4 h.
0.05 mm/cycle, or 21 days/cycle
Peak6 f =0.012477 cycle per 0.0004 mm, or per 4 h.
0.032 mm/cycle, or 13 days/cycle
Peak7 f =0.014492 cycle per 0,0004 mm, or per 4 h.
0.028 mm/cycle, or 12 days/cycle
Peak8 f =0.023869 cycle per 0.0004 mm, or per 4 h.
0.017 mm/cycle, or 7 days/cycle
Peak12 f =0.16669 cycle per 0.0004 mm, or per 4 h. 0.002 mm/cycle,
or 1 day/cycle
Peak11 f =0.11268 cycle per 0.0004 mm, or per 4 h.; 0.004 mm/cycle, or 1 day/cycle
Peak10 f =0.08943 cycle per 0.0004 mm, or per 4 h.
0.0045 mm/cycle, or 2 days/cykl
Peak9 f =0.054324 cycle per 0.0004 mm, or per 4 h.
0.007 mm/cycle, or 3 days/cycle
power ~A²
Theor. AR(1)
level 0.01=12.29
level 0.05=10.65
9
10
6
5
3
4
8
2
7
1
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,2
0,21
0,22
Moc ~A²
95%χ²
99%χ²
kryt.χ²
95%M.Carlo
99%M.Carlo
Teor.AR(1)
Frequency
Peak₃ f =0.16667 cycle per 0.0004 mm, or per 4 h.; 0.0024 mm/cycle, or 1 d./cycle;
Peak₄ f =0.17162 cycle per 0.0004 mm, or per 4 h.; 0,002 mm/cycle, or 1 d./cycle;
Peak₅ f =0.11276 cycle per 0.0004 mm, or per 4 h.; 0.003547 mm/cycle, or 1 day/cycle
Peak₆ f =0.089219 cycle per 0.0004 mm, or per 4 h.; 0.004483 mm/cycle, or 2 days/cycle;
Peak₇ f =0.21747 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 1 day/cycle;
Peak₈ f =0.18401 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 1 day/cycle;
Peak₉ f =0.0080545 cycle per 0.0004 mm, or per 4 h.; 0.05 mm/cycle, or 21 days/cycle;
Peak₁₀ f =0.012392 cycle per 0.0004 mm, or per 4 h.; 0.0323 mm/cycle, lub 13 days/cycle;
Peak₁ f =0.0030979 cycle per 0.0004 mm, or per 4 h.; 0,129 mm/cycle, or 54 d./cycle;
Peak₂ f =0.20694 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 0.8 day/cycle,
τ: 7,0239, Bnwith: 0,001499, False-alarm: 99,88; oversample: 2; segments: 3
p(random) = 5,96E-48; Runs test for fit to the red noise model: k=348 is outside 5% acceptance interval: 379-431.
strong periodic components.
power ~A²
95%χ²
99%χ²
Crit.level χ²
95%Monte Carlo
99%Monte Carlo
Theoret. AR(1)

10. Fig. 17. The spectrum of sinusoids in red noise model REDFIT. B. Meaning of symbols as in Fig. 8.
Obtained above the frequency of daily increments, being the subject of studies, provide data about the period of this process, which is a cyclic record of periodic signals that cause cyclical changes in component concentrations and pressure of endolymph.
The peak frequency of daily increments from research reached 0.16667 cycles per unit - per 0.0004 mm, intervals of which measurement was performed. Hence one cycle - the daily growth period of otolith Ch. gunnarii is 0.0024 mm. Thus obtained periods of other peaks were used to determine the component equations of periodicity in the otolith increments.
Fig. 18. The empirical record of optical density from otoliths of Ch. gunnarii and harmonic characteristics of the two components from their cyclical increments: daily, 0.0024 mm wide and weekly, 0.02 mm of the displaced series of 503 measurements.
6
5
3
4
8
2
7
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0,085
0,095
0,105
0,115
0,125
0,135
0,145
0,155
0,165
0,175
0,185
0,195
0,205
0,215
Moc ~A²
95%99%kryt.χ²
95%M.Carlo
99%M.Carlo
Teor.AR(1)
Frequency
Peak₃ f =0.16667 cycle per 0.0004 mm, or per 4 h.; 0.0024 mm/cycle, or 1 day/cycle;
Peak₄ f =0.17162 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 1 day/cycle;
Peak₅ f =0.11276 cycle per 0.0004 mm, or per 4 h.; 0.003547 mm/cycle, or 1 day/cycle
Peak₆ f =0.089219 cycle per 0.0004 mm, or per 4 h.; 0.004483 mm/cycle, or 2 days/cycle;
Peak₇ f =0.21747 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 1 day/cycle;
Peak₈ f =0.18401 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 1 day/cycle;
Peak₂ f =0.20694 cycle per 0.0004 mm, or per 4 h.; 0.002 mm/cycle, or 0.8 day/cycle
τ: 7,0239, Bnwith: 0,001499, False-al.: 99,88;
oversample: 2; segments: 3
p(random) = 5,96E-48;
Runs test for fit to the red noise model:
k=348 is outside 5% acceptance interval: 379-431.
strong periodic components
power ~A²
95%χ²
99%χ²
Crit.level χ²
95%Monte Carlo
99%Monte Carlo
Theoret. AR(1)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
0,1
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,2
Moc ~A²
y₁=29,27sin(2πx/0,026+0,37)+85
Σᵢ₌₁₋₂(Aᵢ*sin(2πx/Tᵢ+φᵢ))+const.
A₁=29,27, φ₁=0,37, T₁=0,0257; A₂=24,6, φ₂=-0,119, T₂=0,0024; ;
mm
Power ~A²

11. Fig. 19. The empirical record of optical density from otoliths of Ch. gunnarii and harmonic characteristics of the nine components from their cyclical increments of the displaced series of 503 measurements.
Fig. 20. Fitting harmonic equations to empirical records of optical density in the otoliths Ch. gunnarii, PAST determine function of circadian increments - green line (R2 = 0.77 is for above function that have a fixed sinusoid of daily increments - No. 8. Without of this daily sinusoid harmonic obtains higher R2). With Solver tool using minimizing the sum of squared deviations periodicity of daily increments is determined from empirical data as the seventh component of the model (double line).
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
0,1
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,2
Moc ~A²
Σᵢ₌₁₋₉(Aᵢ*sin(2πx/Tᵢ+φᵢ))+const.
mm
solver: A₁=29,27, φ₁=0,37, T₁=0,02574; A₂=2,98, φ₂=1,72, T₂=0,00193; A₃=24,6, φ₃=-0,12, T₃=0,0024; A₄=28,8, φ₄=2,22, T₄=0,00233; A₅=14,8, φ₅=-0,15, T₅=0,00355; A₆=4,1, φ₆=0,06, T₆=0,00448; A₇=3,24, φ₇=0,2, T₇=0,00184; A₈=10,2, φ₈=-2,74, T₈=0,00217; A₉=5,96, φ₉=-2,38,
T₉=-0,0458, const = 85
Σ(yₑ-yᵢ)²=871443
Power ~A²
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
0,1
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,2
Moc ~A²
Σᵢ₌₁₋₉(Aᵢ*sin(2πx/Tᵢ+φᵢ))+const.
dopasowane, nieznane A, T, φ:
y₇=3,33sin(2πx/0,0021-6,78)+85
mm
unknown A, T, φ: A₁=24,2, φ₁=2,19, T₁=0,0248; A₂=25,1, φ₂=-0,242, T₂=0,0331; A₃=17,1, φ₃=-0,881, T₃=0,0662; A₄=16,4, φ₄=-1, T₄=0,0209; A₅=11,7, φ₅=-2,7, T₅=0,0124; A₆=7,44, φ₆=0,437, T₆=0,0058; A₇=7,13, φ₇=-0,876, T₇=0,0083; A₈=1,97, φ₈=-2,22, T₈=0,0024;
A₉=18,26, φ₉=-9,64, T₉=-0,07012; const. =85; Σ(yₑ-yᵢ)²=330042
A₉=5,96, φ₉=-2,38, T₉=0,0064;
const. =89,05
solver: A₁=31,08, φ₁=1, T₁=0,0263; A₂=3,88, φ₂=-2,5, T₂=0,00417; A₃=9,18, φ₃=-3,5, T₃=0,01566; A₄=9,09, φ₄=3,57, T₄=-0,00812; A₅=0,07, φ₅=3,19, T₅=-0,00076; A₆=4,12,
φ₆=-3,27, T₆=0,005; A₇=3,33, φ₇=-6,78, T₇=0,0021; A₈=9,31, φ₈=-7,28, T₈=0,00577
y fitted, unknown A, T, φ
Power ~A²

12. Fig. 21. Harmonic model fitted to the profile of optical density along of otolith radius of Ch. gunnarii appointed the same formula of a harmonic function as for the moving fragment. It is composed of a two sinusoids: daily increment peak No 3 and 2 weekly increments – peak No 1. First starting from the otolith center, daily increments are slightly narrower and should be measured separately. However, the nucleus of Ch. gunnarii is small, so the problem with the lack of separate measurements is not large: it seems that from 0 to 0.05 mm instead of 2 periods there are 3.
In spite of the various changes in optical density, eg due to environmental changes, the basic pattern of daily increments remains the same along the whole profile of optical density on otolith radius. There are 179 of them (similar to numbers obtained in the profile displacement). Daily peaks describes y3=3,82sin( x[mm] -24,83)+90.
Tab. 2. Sinusoids parameters fit to 503 measurements (Ch. gunnarii), daily increments in bold.
From spectral analysis of PAST.
From fitting sinusoids of PAST.
From minimizing the sum of squared deviations (solver)
No
f
A
φ
T
Const.
f
A
φ
T
Const.
A
φ
T
Const.
1
sin
13.7
2.37
0.129119
81.85
cos
24.2
2.19
0.02481
89.05
29.28568448
0.366804749
0.025738099
85
2
sin
2.98
1.72
0.001932
χ²
sin
25.1
-0.242
0.033090
χ²
3.133956642
1.062888152
0.001932927
Σ(yₑ-yᵢ)²
3
cos
24.6
-0.119
0.002399
222570
cos
17.1
-0.881
0.06622
160410
1.444064802
-21.4882443
0.002399952
465807
4
sin
28.8
2.22
0.002330
R²
sin
16.4
-1
0.020890
R²
-0.239585307
-16.98166601
0.002330731
5
cos
14.8
-0.153
0.003547
0.684
cos
11.7
-2.7
0.012400
0.77196
3.66552516
0.744563692
0.003547357
6
sin
4.1
0.0589
0.0045
P
sin
7.44
0.437
0.0058
P
-3.185641848
-11.31575925
0.00448335
7
sin
3.24
0.203
0.001839
4.8E-114
sin
7.13
-0.876
0.008268
1.54E-148
-0.988937091
-3.94738222
0.001839334
8
sin
10.2
-2.74
0.002173
Akaike IC
sin
1.97
-2.22
0.002399
Akaike IC
1.019973039
-17.74347848
0.002173795
9
222538
sin
5.96
-2.38
0.006401
160410
4.853205895
6.200844788
0.006401000
Tab. 3. Sinusoids parameters fit to all 1614 measurements (Ch. gunnarii), daily increments in bold.
From minimizing the sum of squared deviations (solver)
No
f
A
φ
T
Const.
1
sin
11.34678543
-1.825153025
0.024976809
89.61
2
sin
2.250685872
2.289589021
0.001937818
Σ(yₑ-yᵢ)²
3
cos
3.817085043
-24.83181682
0.00238518
3577583
4
sin
-0.222998561
-24.96633606
0.00229975
5
cos
3.601751213
-1.707692003
0.003529968
6
sin
-2.177121558
-1.413467857
0.004584774
7
sin
-1.203566121
-5.202701474
0.001832624
8
sin
2.602419876
-15.24427788
0.00218312
9
sin
2.512937926
4.448054353
0.006367814
0
20
40
60
80
100
120
140
160
180
200
220
240
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Moc ~A²
Σᵢ₌₁₋₉(Aᵢsin(2πx/Tᵢ+φᵢ))+const.
mm
solver: A₁=11,35, φ₁=-1,83, T₁=0,02498; A₂=2,25, φ₂=2,29, T₂=0,00194; A₃=3,82, φ₃=-24,83, T₃=0,00239; A₄=-0,22, φ₄=-24,97, T₄=0,0023; A₅=3,6, φ₅=-1,71, T₅=0,00353; A₆=-2,18, φ₆=-1,41, T₆=0,00458; A₇=-1,2, φ₇=-5,2, T₇=0,00183;
A₈=2,6, φ₈=-15,24, T₈=0,00218; A₉=2,51, φ₉=4,45, T₉=0,00637; const. =90; Σ(yₑ-yᵢ)²=3577583
Power ~A²

13. Discussion
Diurnal increments in otolith satisfy the conditions to describe them as signals periodically variable. Signals – signs of daily increments are multiple repeated at intervals of time: as their name in daily cycle and several repeated at interval of space - here, depending on the species and physiological period: forward every 0.0024 mm for juvenes of Ch. gunnari, or every 0.0009 mm for larvae, 0.0028 mm for juvenes; 0.0034 mm for adults of Ps. georgianus (1).
The above harmonic parameters of the otolith formation - daily creation of matter describe the cyclical changes in environment, tidal, in magnetic field which, when run in the air are not saved. In endolymph and in liquid two crystal, that form the otolith surface, above cyclical changes are captured in the pattern of daily increments. Endolymph transfers vibrations and waves that act on otolith and on the particles forming them. One theory for perception of sound is based on creating in endolymph a standing wave forming the local density and thinning, and thus charge changes determining the arrangement of otolith substrates. Arrangement laid down by one, short impulse (at higher concentration parallel arrangament) later does not change, because it is captured, recorded in a protein network and as a disorder causes a further growth of the crystal. Design orientations of microparticles formed by micro-changes further determine the macro form.
Otoliths from larvae are almost spherical, Fig. 1 (1). Otoliths grow with surface layers of mixed crystals of collagen network with aragonite. Ingrowing layers form curved surfaces proper to liquid crystals. Thus, they must be formed, depending on the ambient pressure. The larvae inside the eggs at the early stages of setting up bodies, as eggs float freely, or lie. In a motionless otoliths as detectors of motion, acceleration and environment vibration grow in spherical surfaces because their dipolar substrates are distributed almost everywhere in the endolymph with the same probability, at orientation in accordance with distribution of positive and negative charges, radially around the sphere. For propagated in the sea vibrations, environment noises that to their perception are transferred into the marine animals microscopic otoliths with their high density could be their receiver. Otoliths transfer waves with speed 3.5 times faster than sea water (~5000m/s).
Endolymph has a concentration and tributary of components regulated that keeps its function and metabolism at the appropriate level. Regulating concentration of the otolith components in the endolymph to equalize the speed of extracting them out from its to the process of crystallization with appropriate its speed, keeps the growth crystal surface at constant diameter with constant dimensions. This determine stability increments and the range of variation determined by changes in endolimfie. When the crystallization rate is higher than the rate of removing substrates from endolymph, the crystal diameter is increasing and vice- versa (10) [str. 456]. Process of otolith growth depends on laying the collagen fibers in the matrix in which their gaps aragonite crystallizes in sizes limited by these gaps.
Analyzed vibration, background noise in endolymph, carry cycling to densities of its components, to distribution of charges and to other parameters of the endolymph property, that changes the orientation of the dipoles of otoliths precursors. There are subject to cyclic concentration, accelerating crystallization alternately by recrystallization, facilitating the alignment of both component and the interpenetration into crystallized matrix of otolith.

14. For small larvae of Antarctic fish, the vital information about the environment provide to them vibration and noise from the sea: about the presence of ice, about the presence of krill forming various concentrations depending on the environment (11), and thus the background noise. This information allows them to locate food, threats, hiding. Because of that from the start of growth of larval runs specialization of organ vibration perception - otoliths to these sounds and vibrations, which determines their free life in the sea (12). Organisms without otolith, or statoliths is sedentary, associated with the bottom (13). For fish larvae may be less important sounds generated by large marine organisms, but with the development, and through development – growth of otolith, this organ of perception will recognize them. Width of daily increments of Ps. georgianus larvae developing in the eggs are narrower than those for juvenile and adult fish (1), living in different zones of the sea. But for juveniles of Ch. gunnari and Ps. georgianus widths of their increments are similar, as they share similar environment.
The analyzed species perform diurnal vertical migrations, thus periodically changing the acoustic field. With increasing depth sound amplitude decreases (for pressures of 15 dB from 0.00086 to 0.00026 cm, for pressures of 70 dB from 0.48251 to 0.14641 cm) and the speed of sound decreases.
With the depth noise from the waves disappear and there are conditions that increase the possibility of the perception of weak signals of biological activity. Generally fish are responsive to audio signals, when they are to noise in the ratio of not less than 1/5. Sounds specific to feeding 5000 Hz pressure of 50 N/m2 are received under no noise, from a distance of 70 m (with noise waving 1 ° B from the 3.5 m). Spawning sounds 100 Hz with pressure of 1000 N/m2 are received from a distance of 300 m (at wave of 1 ° B from 30 m, 2 ° B from 10 m, 4 ° B from the 9 m). Environmental noise can help to hear, could result in changes that increase hearing sensitivity.
Acoustic waves from different sources carry information about the factors changing them in the form specific to those factors emitted frequency and sound pressure. Various changes of acoustic waves from a fixed known source may provide information on factor specific changing the radiated waves (differently amended by the air bubbles, differently from schools of fish or krill and even differently by the bottom, also depending on whether it is muddy, sandy or rocky). Collection this information for the receiving marine organisms provides picture of the environment changes corresponding to the image from the light waves in the atmosphere for organisms equipped with organs of sight.
Krill in large densities protects larvae of fish, giving them the environment to develop, changing the physico - chemical properties of Antarctic waters on favorable to the development of larvae and juvenes present in its clusters, Fig. 22, (14). High densities of krill form a "moving, living ocean", causing bio-turbulent waters of size effects almost equal to the effect of mixing water in the tide, or with wind action (15; 16) – frequencies for wind are above 100 Hz and sound pressure 48 ÷ 20 ÷ -5 dB. Noise from turbulent flow of water masses also mean the presence of krill - which form there its higher concentrations (11). Ice cover also modifies the sea water, provides shelter and allows fish to extend their propagation for food. Noise caused by moving ice are frequencies 9 ÷ 5000 Hz, wherein the waters of completely ice-covered frequencies are from 1 to 10000 Hz with sound pressure of 45 dB to -15 dB, in which from 40 to 1000 Hz sound pressure run above the average of the minimum the sound pressure of own

15. noise of sea, i.e. in the range from 20 to 5 dB, in the remaining range is less than that average.
In the marine environment above noise carry information on changes in the environment, which is mostly in cyclical nature – noise of ice edge is cyclical daily from tides that stand out at the coast of Antarctica, with possibly reflection in the oscillation of the southern edge of the ice cover.
Fig. 22. Oscillation of the southern edge of the ice cover and occurrence of larvae of fish in the krill fishery (marked by arrows) carried out in the ice marginal zone between South Orkney and Elephant Island during an expedition on rv. "Professor Siedlecki" in season 1988/89 (17). Arrows – krill catches containing fish larvae of the following species: Ch. aceratus (40,66); Ch. gunnarii (41); Chionodraco rastrospinosus (41,66,78); Chaenodraco wilsoni (69,73,74); Dissostichus eleginoides (73); Pleurogramma antarcticum (40,74); Gryodraco antarcticus (40,73,74,82); Neopagetopsis sp. (55); Neopagetopsis ionach (56,71,73); Trematomus eulepidotus (65); Notolepis coatsi (67,71); Pagetopsis sp. (69,78,82); Notothenia sp. (69); Notothenia larseni (73;74); Pagetopsis macropterus (73); Electrona carlsbergi (78).
To environmental cycles, with the development of organisms synchronize physiological cycles, which at the beginning of synchronization may be reflected in circadian cycles - due to the complex process of adapting many of the currently formed biological, internal functions in the larval development.
It appears that for fishes does not exist a practical lower limit of sensed sounds. For the range of infrasound (sound waves of low frequency) function of receptors take over the organs of touch.
Even more simple statolith enable bivalve Donax to choose from complex sounds generated by the waves, the right kind of wave to use it to achieve success in reproduction.
Conclusions
Results are probable: larvae physiological cycles are usually circadian - at above model for fish larvae the highest amplitudes are characterized by circadian cycles, A4, A5. They may result from the transformation of the larvae to adapt to the daily cycle.
Time sinusoid parameters employed to describe time increments of otolith does not necessarily correspond with the parameters of the acoustic waves. Three theories of perception of sounds shall indicate more work in this direction.
Sine wave of daily increments separated component from sinusoids model, enables fast measurement of their number and width.
Obtained similar values of mean daily increment in three independent methods:
1. search for the minimum difference in optical density of otoliths when relative moving their profiles to each other.
kg/h
3 kg/h
14 kg/h
78
82
74
73
71
40
41
69
67
66
65
56
55

16. 2. harmonic analysis of REDFIT (4).
3. fit of the harmonic model to empirical data by minimizing the sum of squared deviations. Use solver tool - the most accurate method.
Records of optical density otolith increments can be a valuable material for the study of biological, physical and evolutionary (13) in the marine environment.
However, at noise emissions are weak, treated as irrelevant and discarded from the analysis, in the appropriate scale for the micro-level organizations can be important synchronisers of cyclicality - particularly for a large number of larvae - thus determining the mass gain, the success of generations and even existence of a species being closely related to the free way of life in vast, poor at food and raw space of Antarctic Ocean.
Lack of synchronization conditioning for species their lifestyle adapted to the environment may be partly responsible for the disappearance of that timing-dependent species. No cover surface of the ocean by ice reaching in the 70's up to Georgia Pd, coincides with the long kept decline of fish Pseudochaenichthys georgianus by 75% - despite a ban on their fishing. Meanwhile, the open ocean and bottom species does not associated with the neighborhood of ice edge are present in an amount does not less, but even more (18).
Cited works
1. Traczyk, R.J. Zastosowanie analizy mikroprzyrostów i morfologii otolitów georgianki (Pseudochaenichthys georgianus Norman, 1939) z rejonu Georgii Południowej (Antarktyka) do określenia wieku, wzrostu oraz ważniejszych okresów rozwoju ryb. Kraków : Ryszard Traczyk In Altum, ISBN: 978-83-62841-10-3, 2013.
2. Kota, M. Basics of Image Processing and Analysis. EMBL Heidelberg : Centre for Molecular & Cellular Imaging, 2013. Copyright 2006 - 2013, Kota Miura (http://cmci.embl.de).
3. Parkes, G., I. Everson, J. Anderson, Z. Cielniaszek, J. Szlakowski, R. Traczyk. Report of the UK/Polish fish stock assessment survey around South Georgia in January 1990. London : Imp.Coll. of Sci. & Techn., 1990, str. 20.
4. Hammer, Ø. Time series analysis with Past. Oslo : Natural History Museum, University of Oslo., 2010.
5. Schulz, M., M. Mudelsee. REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series. GE : Computers & Geosciences 28, 421-426., 2002.
6. Kaczkowska, A. Transformacje Fouriera, DFT, FFT, Periodogram Lomba-Scargle’a. Gdańsk : Politechnika Gdańska, 2009.
7. Kozłowski, E. Analiza spektralna. Wrocław : GUM, Zakł. Fizykocemii, http://bip.gum.gov.pl/, 2013.
8. NASA. Algorithm Documentation. NJ : NASA Exoplanet Archive, 2012. http://exoplanetarchive.ipac.caltech.edu/applications/Periodogram/docs/Algorithms.html.
9. UF. Lomb-Scargle Periodogram. FL : University of Florida, EEL, 6537, 2009.
10. Anonymous, 1983. Encyklopedia Fizyki Współczesnej. W-wa : PWN, 1983. str. 1072.
11. Witek, Z., J. Kalinowski, A. Grelowski. Formation of Antarctic Krill Concentrations in Relation to Hydrodynamic Processes and Social Behaviour. [aut. książki] D. Sahrhage. Antarctic Ocean and Resources Variability. Berlin : Springer-Verlag, 1988.
12. Mitrinowicz - Modrzejewska, A. Fizjologia i patologia głosu, słuchu i mowy. W-wa. : Państw. Zakł. Wyd. Lek., 1963. str. 307.
13. Traczyk, R. Migrations of Antarctic fish Pseudochaenichthys georgianus Norman, 1939 in the Scotia Sea. Hobart : CCAMLR. WG-FSA-12/68 Rev. 1, 2012.
14. Ślósarczyk, W., Z. Cielniaszek. Postlarval and juvenile fish (Pisces, Perciformes and

17. Myctophiformes) in the Antarctic Peninsula region the Antarctic Peninsula region, 1983/1984. 1-2. W-wa : Pol. Polar Res., 1985. strony 159-165. Tom 6.
15. Harmon, K. Shrimpy Sea Life May Mix Oceans as Much as Tides and Winds Do. bio-ocean- mixing. [Online] Scientific American, 29 VII 2009. http://www.scientificamerican.com/article.cfm?id=bio-ocean-mixing.
16. Pyper, W. Krill mix up the ocean. Tasmania : Australian Antarctic Magazine, 2008.
17. Traczyk, R. The occurrence of krill Euphausia superba in the floating ice edge zone and some its biological data. http://georgianka.strefa.pl/krill_oc.html. [Online] INTERIA.PL S.A., 1993.
18. Traczyk, R. Economic competition for high profits from Antarctic living resources in protection area and Mercury contaminants of fish from outside. Hobart : CCAMLR WG-EMM-13/04, 2.1.1, 2013.