SC/D06/J5

ESTIMATION OF G(0) BASED ON THE SIGHTING SURVEY DATA AND
COVARIATES INFORMATION OF JARPA
Takashi MATSUISHI1, Hisatoshi IKEDA1, Shigetoshi NISHIWAKI2
1

Faculty of Fisheries Sciences, Hokkaido University

2

ICR

ABSTRACTThe estimation model of g(0) was developed using the data of whales
sighting survey and surfacing information of whale without the independent
observer experiment. Moving coordinate system was defined as the position ofobserver was fixed at origin. The surfacing probability in a unit time Δs andsurfacing detection probability function Q were also defined. The function of Q is
considered many covariates such as sighting survey condition, group size of whaleand so on. g(0) was estimated by maximizing the likelihood of Q on the data ofprimary observation information of whales group. Results of simulation
examination suggest that unbiased estimators were obtained from this method.
g(0) of JARPA sighting survey on the area IV was estimated by the proposed
method. The g(0) were estimated as 0.5 - 0.7 and were significantly different toone.

Key words: g(0), non-IO, surfacing probability, Covariates

Line transect methods are widely used to estimate density and population of cetaceans. The
traditional line transect methods have strong assumption that the probability of objects detection on
the track line, g(0), is 100%. However, this assumption does not match in whale survey (Doi et al.,
1983; Hiby, 1986; Øien, 1990; Schweder et al., 1991b; Cooke, 1994). If the whales on the track line
are not detected, the number of whale is underestimated with traditional line transect method.
Therefore to estimate g(0) is necessary to obtain unbiased number of whale. The independent
observer (IO) experiment and the estimation model of g(0) are widely used. On the other hand,
many sighting surveys have not performed IO because of the limitation of facilities or observers on
the survey vessel.
A few studies have been reported on the g(0) estimation without IO experiments. Doi et al. (1983)
estimated g(0) = 0.699 for Antarctic minke whale by using Monte-Carlo simulation. Kasamatsu andJoyce (1995) estimated g(0) of odontocete by using simulation method. Okamura (2003) examined
the accuracy and bias of g(0) estimated from the part of IO model which can apply to the survey
data of traditional line transect from generated pseudo-observation data by using simulation. He
reported that true g(0) can be estimated from the model added average of surfacing interval andterm of group size. These studies assumed that the conditions during the surveys were constant.
However, some studies suggested that survey conditions affect the observation efficiency of thesurvey. Doi et al. (1983) indicated that effort of observation differs by the angle between track line1and the target. Barlow (2001) presented that perpendicular distance was affected by group size,
Beaufort scale, cue, vessel, year and survey area. Murase et al. (2004) reported that detection
distance was shorter when Beaufort scale was larger and cue was body than brow. The estimation ofg(0) added these factors is not studied enough.
The population estimation of the Antarcic minke whale on the area IV by using traditional linetransect method from JARPA data had a twice discrepancy between the estimated population of
97/98 and 99/00. One of the reason of this discrepancy has been thought to be the difference of g(0)or survey conditions.
In the current study, a g(0) estimation model without independent observer program was proposed
to estimate population density of cetacean from the data of cetacean sighting survey and covariatesof survey conditions such as Beaufort, visibility and so on. The impact of such covariates was also
examined by using pseudo-observation data generated from Monte-Carlo simulation. Furthermore,
the model was applied for sighting data of JARPA on area IV and estimated g(0) and the populationof Antarctic minke whale.

METHODSg(0) estimation model using non-IO data
Survey vessel was fixed at the origin of moving coordinate system and was headed to the Y-axis. This X-Y
coordinate indicates the relative position of whale groups (including whale group which the size of it equals toone) to the vessel. Assuming the vessel speed v is enough faster than whales one, a position of whale group isdescribed as ( x, y − vτ ) , where ( x, y ) is whale position on time t, and τ is elapsed time from time t (Fig. 1).
Whale behavior was categorized into two; surfacing and diving. The probability of surfacing or diving wereassumed to be independent to the vessel position and time. A surfacing whale has a possibility to be observed by
the observers on the vessel, as far as y < ylimit , where ylimit is a large vertical distance enough not to be able todetect.
Expectation surfacing probability Δs was defined as the average probability that a whale surface in the intervalof Δτ from the time that the whale pass a certain point ( x, y ) . Assuming the behavior of whale is independent
on time or relative position from the vessel, Δs is a constant. The derivation of Δs was described in appendix.
We defined the surfacing detection probability subject to the whale surfaces at ( x, y ) , Q ( x, y, Η | a) , whereΗ = {η1 ,L,η m } is a vector of covariates and assumed to be constant for a whale group which enter the areay < ylimit , a = {a1 ,L, al } is a vector of free parameters. Q is a function describing probability, thus
0 ≤ Q ≤ 1.

For making the discrete model on Y axis, the distance ylimit is divided in c bands evenly. The width of the bandis calculated as ylimit c and the speed of vessel was set at v. Then, the time to pass a band Δτ is calculated asfollows;Δτ =ylimitc⋅v(1)

The interval of band i was defined as(i − 1)Δτv ≤ y ≤ iΔτv, i = 1,L, c .

(2)where i is interval number, i=1 is the nearest interval to vessel on the origin (Fig. 2).
Here, we redefined expected surfacing detection probability that a whale on the line with the perpendiculardistance x, was detected in the band of i if it surfaced on that band; Q( x, (i − 0.5) Δτv, Η | a) , and abbreviated toQi ( x , Η | a ) .

2

Furthermore probability that observer primary detected whale on the line with perpendicular distance x or the
primary detection probability q i ( x, Η | Δs, a) was defined as follows. A whale group on ( x, ylimit ) achieves to( x, y limit − Δτv) after time interval Δτ . It passed band i=c and expected surfacing probability is Δs from thedefinition of it. Therefore, detection probability on the band c is given by
qc ( x, Η | Δs, a ) = Δs ⋅ Qc ( x, Η | a) .

(3)

qc −1 is calculated as follows;q c −1 ( x, Η | Δs, a ) = (1 − q c ) ⋅ Δs ⋅ Qc −1 ( x, Η | a )Similarly, qi is obtained as recurrence equation,

(4)

c
⎛
⎞
qi ( x, Η | Δs, a ) = ⎜1 − ∑ qk ⎟ ⋅ Δs ⋅ Qi ( x, Η | a) .
⎝ k =i+1 ⎠(5)

The g(x) is the detection probability from y limit to y=0 and equals∑ q (x ) . That is
c

i =1

i

c

g ( x, Η | Δs, a) = ∑ q i ( x, Η | Δs, a ) .

(6)

i =1

Note qi >c = 0 , as equation (5) is derived from Q and Δs , g(x) can be calculated from them.
In order to obtain g(x) from Equation (5), the parameters a were estimated by maximum likelihood method using
detection probability density distribution, q pdf , obtained from q under particular Η as follows;

qpdf ,i j (x j , Η j | Δs j , a ) =cylimit⋅1⋅ qi (x j , Η j | Δs j , a ) ,w(Η j | Δs j , a) j

(7)

where ij is detected interval of whale group j, xj, Η j and Δs j is detected perpendicular distance, covariatesand expected surfacing probability of whale group j respectively and w is effective strip width under Η j andΔs j that is calculated from∞w(Η | Δs, a) = ∫ g ( x, Η | Δs, a)dx

(8)

−∞

In equation (7),cylimit⋅1
was the value to normalize q. Log-likelihood function for detection wasw(Η j | Δs, a)calculated by{}

LL(a | Δs, Η, x, i ) = ∑ ln qpdf ,i (x j , Η j | Δs j , a )
n

j =1

j

{

}

⎡
c ⎤
= ∑ ⎢ln qi j (x j , Η j | Δs j , a ) − ln{w(Η j | Δs j , a)}+ ln⎥
ylimit ⎦j =1 ⎣
n(9)g(0) was estimated from maximum likelihood estimators a * that was calculated from the equation (9), as
follows;c(g ( x = 0, Η | Δs, a ) = ∑ qi x = 0, Η | Δs, a*
*

)

(10)

i =1

*

*

Furthermore group density D , individual density Dind around the track line and number of whale was derived
by
n

1
,
*
j =1 w ( Η j | Δs j , a ) ⋅ L

D* = ∑

(11)

*
j

3

n

sj

j =1

w (Η j | Δs j , a * ) ⋅ L

*
Dind
=∑

(12)

*
j

*
N * = A ⋅ Dind

(13)
respectively, where sj is group size of j, and A is survey area. Confidence interval of them was estimated by
Bootstrap method.

Simulation
The simulation tests were conducted in order to examine that this procedure estimate properly, and to examine theeffect of covariates using pseudo-observation data.
The data was generated as follows. A virtual sea was created with 15 nmile width (the range of X-axis is –7.5 to
7.5 nmile), and with 200 nmile length (the range of Y-axis is 0 to 200 nmile) on computer. The sea length was
divided in c=1000 bands. The vessel fixed at the origin and was headed to the Y-axis. The virtual sea moved
backward along Y-axis with speed v = 11.5kt. The vessel searched on the sea surface (the range of X-axis was -7.5
to 7.5 nmile and Y-axis is 0 to ylimit =7.5 nmile) and detected a surfacing whale with surfacing detection
probability as follows;

{

}

a1
⎡ ⎛ a dist ⋅ 1 + a sin 2 (| ang |) ⋅
⎞ ⎤
0
2
⎟ ⎥
⎢ ⎜
Q(dist , ang | Η, a) = exp ⎢− ⎜
⎟ ⎥
2
⎛
⎞
⋅ (1 + a4 η1 ) ⋅ (1 + a5 η 2 )⎟ ⎥ (14)⎜⎢ ⎜a3 ( s −1) ⎟⎝1 + e⎠⎠ ⎥⎦⎢⎣ ⎝a 0 = 3.5, a1 = 0.8, a 2 = 0.6, a3 = 0.1, a 4 = 0.6, a 5 = 0.7
where dist and ang are the distance and angle from the vessel to the whale respectively and two covariates
η1 = {0,1,2,3,4} and η 2 = {0,1} were defined in the vector Η . The whale groups of size s=1,2,…,5 were

distributed randomly with density D(s) (0.05, 0.027, 0.013, 0.007 and 0.003 ind./nmile2 respectively) and repeatedsurfacing and diving at the same point on the virtual sea according to surfacing interval p.d.f. as fallows;f (τ , s ) =⎡ τ ⎤1
exp ⎢−⎥λ ( s)⎣ λ ( s) ⎦(15)

λ ( s) = 166.92 ⋅ e −0.25 s
In this model, the parameter function λ ( s ) were adjusted as the average surfacing interval of a whale groupwith s=1 is 130 sec. If whale surfaced, the next surfacing timing was selected randomly according to f (τ , s ) .
Each whale group j carried covariates factors Η j , hence one group has two types of covariates η1 = {0,1,2,3,4}
and η 2 = {0,1} and the size = {1,2,3,4,5} . Therefore 50 types of different group were appeared on the virtualsea. If whale passed across the X-axis, the whale was removed once and a new whale was generated at (x, 200)
where x was selected randomly. When observer detected a whale group, the detection distance dist, angle ang,group size s and covariates Η j was recorded. The observation was terminated when the number of detection
reached to n=400, and survey length L was recorded. Same survey was repeated 30 times.
From the virtual sighting data obtained above, g(0) and density was estimated using g(0) estimation model under
the true Q described equation (14) and compared with true g(0) and density. The true g(0) was calculated fromtrue Q using equation (10). True individual density was calculated from Dind =Dmax∑ sD(s) . Individual density
s =1

*
ind

was calculated using estimated g*(0), effective strip width w* and length of track line L. Furthermore, theaverage, coefficient of variance and quartile (25%ile, 50%ile, 75%ile) was calculated by estimated values from 30
sighting data respectively.
For examining the misspecification of the model of Q, another Q model was also used which leave out covariate
terms from equation (14), as follows;

4

a1
⎡ ⎛
2
⎛
⎞⎞ ⎤
2
⎟ ⎥
Q(dist , ang | Η, a) = exp ⎢− ⎜⎜ a 0 dist ⋅ 1 + a 2 sin (| ang |) ⋅ ⎜a ( s −1) ⎟⎟⎝1+ e 3
⎠⎠ ⎥⎦
⎢⎣ ⎝

{

}

(16)

*

The result was compared using the quartile of Dind and g*(0) with true value.
Application on the JARPA data
The SSV sighting data, survey condition data and survey effort data of JARPA from 95/96 to 01/02 only in area IV
were used for the application.
For the survey condition data which is the additional data of the sighting data, detection distance, detection angle,
size of whale group, cue of the detection , Beaufort scale, wind speed, and pack ice coverage were used for thefull model as the covariates and significant covariates were selected by using GLM by year. Length of track line,
area and ship velocity was also used following Hakamada et al. (2001). A total of 60 individual and 657 surfacing
data was obtained from the chasing experiment on 90/91 to 96/97 and was used for the application.
The confidence intervals of the estimators were obtained by using bootstrap method. Because of the small numberof replication (30), the errors of the estimators were assumed to be log-normal distribution and the 95%
confidence interval were estimated from the distribution function fitted to the 30 estimators obtained from thebootstrap samples.

RESULT
From the result of simulation examinations, unbiased g(0)s were estimated on all combination ofsize and other covariates under true Q model was given (Fig. 4). The CVs are 0.30 - 9.18%. True
g(0)s were included between 25%ile and 75%ile of the estimated g(0) under all combination.
g(0)s from limited Q model were estimated to nearly similar value in the same size in spite of thevariation of covariates (Fig. 5). 48 of 50 true g(0)s were out of the range of quartiles of estimatedg(0).
From the analysis of the chasing experiment of JARPA on 90/91 to 96/97, the average surfacing
interval was obtained as Table 1. Survey length of each year and stratum used in this study was
shown in Fig. 6. Average group size in each year and stratum was shown in Fig. 7. Fig. 8 shows theaverage Beaufort index.
From the GLM analysis, angle, size and cue were selected for all four years and Beaufort, windvelocity or pack ice coverage were selected for some years as the significant covariates.
The estimated g(0) value was shown in Fig. 9. In 95/96, g(0) was estimated about 0.5 and was lower
than the other years. In 97/98 to 01/02, g(0) were estimated quite same value as 0.7. The differences
between estimated g(0) and 1 were significant for all years.
Fig. 10 and 11 show the estimated group density and individual density. The estimated number ofgroup and number of individuals are shown in Fig. 12 and 13.
The total population estimated from the current study was about 40% higher than the traditional
estimators (Hakamada et al. 2001)(Fig. 14). Especially in 1995, the population was estimated twice
comparing the traditional one.

DISCUSSION
As a result of the simulation study, it is suggested that the unbiased and precise estimators of g(0)and total density of whale group were obtained from the g(0) estimation model of the current study.
The estimators of g(0) without covariates were widely scattered from the true values, against theestimators from covariates were precise. Doi et al. (1983) indicated that effort of observation differs

5

by the angle between track line and the target. Barlow (2001) presented that perpendicular distance
was affected by group size, Beaufort scale, cue, vessel, year and survey area. Murase et al. (2004)
reported that detection distance was shorter when Beaufort scale was larger and cue was body thanbrow. Although our simulation expressed somewhat extreme situations, these reports and our result
suggested that the covariates should be considered as far as no-IO estimation model.
A discrete model was used for the estimation. One of the advantages of discrete model was theflexibility of the model. Even if the problem was complicated, the discrete model could describe it
flexibly. Although it takes more time to calculate using the discrete model than analytical models, it
is not serious problem by using recent powerful PC.
For further study, the validation or sensitivity of the assumption of Q(0,0) and Δs should be
examined. Moreover, the estimation of parameters concerning with the covariates is thought to be
more robust if it is estimated from the all year pooled data in stead of year by year. Furthermore, theother covariates such as visibility etc should be incorporated for further study.

REFERENCE
Borchers, D.L., Buckland, S.T., Goedhart, P.W., Clarke, E.D. and Hedley, S.L. (1998). Horvitz-Thompson
Estimators for Double-Platform Line Transect Surveys. Biometrics 54: 1221-1237.
Cooke, J. G. (1994). A Re-analysis of Experiential Data to Estimate G(0) for Shipboard Surveys of Minke Whalesin the North Atlantic. SC/46/NA8 presented to the IWC Scientific Committee (unpublished): 1-20.
Doi, T., Kasamatsu, F. and Nakano, T. (1983). Further Simulation Studies on Sighting by Introducing Both
Concentration of Sighting Effort by Angle and Aggregations of Minke Whales in the Antarctic. Rep. Int.
Whal. Commn. 33: 403-412.
Hakamada, T., Matsuoka, K., Nishiwaki, S., Murase, H. and Tanaka, S. (2001). Abundance Trend of Southern
Hemisphere Minke Whales in Areas IV and V Obtained from JARPA Data. SC/53/IA12 presented to the
IWC Scientific Committee (unpublished): 1-35.
Hiby, A.R. (1986). Results of a Hazard Rate Model Relevent to Experiments on the 1984/85 IDCR Minke Whale
Assessment Cruise. Rep. Int. Whal. Commn. 36: 497-498.
Joyce, G.G. (1982). Blow Patterns as Sighting Cues for Censusing Minke Whales in Antarctic Waters. Rep. Int.
Whal. Commn. 32: 787-790.
Joyce, G.G. and Øien, N. (1989). Surfacing Rates of Minke Whales in Norwegian Waters. Rep. Int. Whal. Commn.
39: 431-434.
Øien, N. (1990). Estimates of g(0) for Minke Whales Based of an Independent Observer Experment During the
Norwegian Sightings Surveys in July 1988. Rep. Int, Whal. Commn. 42: 575-581.
Øien, N., Folkow, L. and Lydersen, C. (1990). Dive Time Experiment on Minke Whales in Norwegian Watars
During the 1988 Season. Rep. Int. Whal. Commn. 40: 337-341.
Saito, T., Ikeda, H., Matsuishi, T. and Nishiwaki, S. (2004). Surfacing Interval of Minke Whale Balaenoptera
bonaerensis in Antarctic. Bull. Fish. Sci. Hokkaido Univ. 55: 7-10.
Schweder, T., Øien, N. and Høst, G. (1991b). Estimates of Detection Probability for Shipboard Surveys ofNortheastern Atlantic Minke Whales, based on a Parallel Ship Experiment. Rep. Int, Whal. Commn. 41:
417-432.
Stern, S.J. (1992). Surfacing Rates and Surfacing Patterns on Minke Whales (Balaenoptera acutorostrata) off
Central California, and Probability of a Whale Surfacing within Visual Range. Rep. Int. Whal. Commn. 42:
379-385.
Stockin, K.A., Fairbairns, R.S., Parsons, E.C.M. and Sims, D.W. (2001). Effects of Diel and Seasonal Cycles on
the Dive Duration of the Minke Whale (Balaenoptera acutorostrata). J. Mar. Biol. Ass. U.K. 81: 189-190.
Ward, A.J. and Hiby, A.R. (1987). Analysis of Cue-Counting and Blow Estimation Experiments Carried Out
during the 1985/86 IWC/IDCR Minle Whale Assessment Cruise. Rep. Int. Whal. Commn. 37: 259-262.

6

Table 1. Effective data number obtained from experiment ofchasing behavior research in JARPA. The data was used to
calculate the surfacing interval p.d.f, f s (τ , s ) . 2.5% and 97.5%
represent the lower or upper of 95% confidence interval of
average.

Size
1
2
3
5
Total

N
223
177
123
134
657

Average
131.0
114.0
55.3
46.4

(x, ylimit )

2.5%
114.3
96.7
41.9
39.4

97.5%
147.8
131.3
68.6
53.4

Y
(x,i=c)whale (x, y)vv
Δτ sec .

surfacingposition(x,i=2)(x,i=1)x

X
ship

Fig. 1.

Assumption of model. X and Y axis are relative coordinate that represents a poision of whale
group from a research ship. A whale at (x, y) moves to the inverse direction of Y axis with shipvelocity v. An open circle represents diving whale position and close circle surfacing. ylimit means
the limit distance of detection. In this study, a segment line coordinate, (x, i) i={1,2,…,c}, wasdefined, where c is number of segments between y=0 to y=ylimit. The whale on border of a segmentline can pass the segment for sec. calculated by v and unit of segment line length.

7

Y(x, ylimit )q c = Δs ⋅ Qc
q c −1 = Δs ⋅ Qc −1 ⋅ (1 − q c )q c − 2 = Δs ⋅ Qc − 2 ⋅ (1 − (q c + q c −1 ))c −1

q1 = Δs ⋅ Q1 ⋅ (1 − ∑ q c −i +1 )i =1xFig. 2.

X

Calculation process of detection probability qi. A whale group farter than (x, ylimit) can’t detect andthe surfacing probability between a segment line is constant Δs . Therefore the detection
probability qc for first segment i=c is calculated by product of Δs and Qc that is detection
probability when whale is surfacing. Furthermore, the probability qc-1 for the next segment, i=c-1,
is calculated by product of Δs , Qc and undetected whale rate, 1-qc.

15 nmilev

Yv
200 nmile

ylimit
7.5 nmile

dist

X
ship
Fig. 3.

The define field of research area in the data generator of simulation. Whales distribute randomly
in strip area and move to the inverse direction of Y axis with ship velocity v. Ship is fixed at Origin,
O. A Research ship can detect the surfacing whales into the gray zone with detection probability
Q(dist | Η )

8

1

(a)

Estimated g(0)

0.9
0.8
0.7

Size 10.6

Size 2Size 30.5

Size 4Size 5

0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

True g(0)Fig. 4.

Result of estimated g(0)s from true Q model under each 50 covariates combination. Error barmeans 25-75%ile.

1(a)Estimated g(0)

0.9
0.8
0.7

Size 1Size 2Size 30.6
0.5

Size 4Size 5

0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

True g(0)Fig.5.

Result of estimated g(0)s from limited Q model under 50 covariates combination. Error bar means
25-75%ile.

9

NW
SW
PB

NE
SE
7

1200

NW
SW
PB

6
Ave. of Observed group size(ind1000

L (nmile)

800
600
400
200
0

5
4
3
2
1
0

1995/96 1997/98 1999/00 2001/02
year

1995/96

5.0

NW
SW
PB

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

3.5

3.0

2.5
1999/00

2001/02

NE
SE

4.0

1997/98

1999/00

Fig. 7. Average of detected group size.

4.5

1995/96

1997/98
year

Fig. 6. Survey line length of each year andstratum.

Ave. of beaufort index

NESE

2001/02

g(0)
1995

1997

year

1999

2001

Year

Fig. 9.

Fig. 8. Average of Beaufort index.

Estimated g(0) from the data of JARPA
in area IV by using the proposed
method in the current study.
.

10

0.40
NW
SW
PB

0.35

0.7

↑PB: 0.665

NESE

D ind(ind./Area)

0.30
D (groups/Area)

NW
SW
PB

0.6

0.25
0.20
0.15

0.5
0.4
0.3
0.2

0.10

0.1

0.05

0.0

0.00
1995/96

1997/98

1999/00

1995/96

2001/02

1997/98

25,000

NW
SW
PB

2001/02

Fig. 11. Estimated density of individuals.

Fig. 10. Estimated density of groups.

30,0001999/00
year

year

NE
SE
50,00020,00040,00015,00030,000

N ind(ind.)

N (group)↑PB: 1.31

NESE

10,000
5,000

NW
SW
PB

NE
SE

1995/96

1997/98

20,00010,000--1995/96

1997/98

1999/00

2001/02

1999/00

2001/02

year

year

Fig. 12. Number of group.

Fig. 13. Number of individual.

250,000

N(ind.)

200,000

150,000
100,000

50,000
0

1995

1997

1999

2001

Year

Fig. 14. Estimated population of Antarctic minke whale in area IV from JARPA data. Black line indicate
the estimated population of the current study, and the gray line is that of the traditional linetransect method (Hakamada et al. 2001). Error bars shows 95% confidence interval.

11

APPENDIX

Δs was calculated only from surfacing interval time of whale. It was assumed that an observer can detect one
group by a cue (blow or surfaced body) that was made by a member of the group.
The surfacing interval time was defined as the time interval of cues made by a group. Probability density functionof surfacing interval time is defined as f s (τ , s | Α) , where τ is the surfacing interval time, s is the group sizeand Α is the parameter vector. The cumulative probability density function is denote as Fs (τ , s | Α) . The ratioof groups that are diving for time τ is given as 1 − Fs (τ , s ) . The probability that the group surface once in theinterval of Δτ subject to the group is diving at time τ was given as follows;

Fs (τ + Δτ , s) − Fs (τ , s )1 − Fs (τ , s)

(17)

The ratio that a group has stayed diving for time t at any time is proportional to 1 − Fs (t , s ) . Then probability
density of diving time t is described as follows;∞

1 − Fs (t , s )(18)∫0 {1 − Fs (τ , s)}dτ
From these equations and assumptions, Δs is calculated by formula of expectation below;

Δs ( s ) = ∫∞0Fs (τ + Δτ , s ) − Fs (τ , s )
dτ
−
F
τ
s
1
(
,
)
s
∫ {1 − Fs (τ , s)}dτ∞

1 − Fs (τ , s )

⋅

0

∞

∫ {F (τ + Δτ , s) − F (τ , s)}dτ∫ {1 − F (τ , s)}dτ

= 0

s

s

(19)

∞

s

0

Δτ

∫ {1 − F (τ , s)}dτ=∫ {1 − F (τ , s)}dτ
s

0
∞

s

0

Furthermore expectation of τ ≥ 0 equals

E[τ | s ] = ∫ {1 − Fs (τ , s)}dτ
∞

(20)

0

It follows that Δs is describedΔτ∫ {1 − F (τ , s)}dτΔs ( s ) =
0

s

(21)

E[τ | s ]12