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Pham Van Huan (2003): Extremum sea levels in Vietnam coast. VNUH, Journal of Science, T. XIX, No 1, pp. 22-38

Module by: Phạm Văn Huấn. E-mail the author

Pham Van Huan: EXTREMUM SEA LEVELS IN VIETNAM COAST. VNUH Journal of Science, T. XIX, No 1, pp. 22-38, 2003

Abstract: A review of the investigations on the sea level changes in South-china sea is presented and the methods of approximate calculation of theoretical tidal extremes were explained in detail.

The sea level changes near Vietnam coast due to global warming and other effects is evaluated to be from 1 to 3 mm per year.

For seven stations with full set of harmonic constants determined the theoretical extreme heights of tidal level by predicting hourly tide heights in a 20-year period. For other nineteen stations with 11 harmonic constants of main tidal constituents the theoretical astronomical extreme levels were calculated by the iteration method. The comparison showed a good agreement between two methods.

The empirical extreme analysis was carried out for 25 tide gauges along Vietnam coast to evaluate the design values of sea level of different rare frequencies.

The analysis also showed that the tidal extremes and design level values of 20-year return period are of the same range. The level values of longer return period are affected mainly by floods and surges.

Introduction

The extreme sea levels are study subject of many purposes. The maximal and minimal values of sea levels and their occurrence probabilities are taken into account in designing hydrotechnical structures.

The theory of extreme analysis of statistical mathematics is applied to the hydrometeorology with different distributions of the observed series of climatic and hydrological parameters [5,7]. The main concepts of these methods will be presented in section 2.1.

In the case that observed series of sea level are not long enough to apply the procedures of extreme analysis theory, that usually happen in the design investigations in the coastal zone and estuaries, one may use theoretical extreme values of purely tidal levels.

In many practical problems the minimal theoretical level is assumed to be the zero depth in tidal seas. This level can be calculated by subtracting maximal low height of tide due to astronomical conditions from mean sea level. In some countries this value is determined by analyzing a predicted series of tidal heights 19-year long, one choose the lowest height among all low waters in the series. In Russia the minimal theoretical level is determined by known method of Vladimirsky.

Vladimirsky method gives an analytical solution of the problem with harmonic constants of 8 main tidal constituents. The rest tidal constituents are taken into account approximately. Recently the calculations can be performed rapidly in computers, evaluating extreme heights of tide can be carried out by more detailed schemes and the accuracy is improved by withdrawing a non-restricted number of tide constituents into consideration [6]. Section 2.2 will explain in details a scheme to implement this method in practice and in section 3 will presented the application results to obtain maximal characteristics of sea level in some region of Vietnam coast.

The observation of sea level along Vietnam coast is mainly carried out by a system of tidal gauges of the Vietnam Hydrometeorological Service. Generally speaking, up to now the number of tidal gauges that belongs to Vietnam waters is not many and the number of observation years is not long enough. So there is no much deal with the behavior of sea level in general and the empirical calculations of level extremes in special.

In some rare works there reported the results of analyzing changeableness of sea level and the estimating the trend of sea level rise in the base of analysis of observed series of sea level some years long. The spectrum analysis [2] showed that besides the semiannual and annual periods, in the almost of tidal gauges oscillations of period of 6 to 10 years and longer exist (figure 1).

Table 1 lists the results of estimation of the sea level rise by trend analysis with monthly mean level [2-4]. It is followed that the summary effect by the global warming and oscillations of sea bed in region of Vietnam coast causes a rate of level rise about 1 to 3 mm per year.

A full cumbersome calculation of level extremes was performed in [1]. In this report firstly listed series of monthly average, maximal and minimal levels for all gauges along Vietnam coast up to middle of ninetieth. The extreme analysis was carried out by an asymptotic Gumbel function of probability distribution of the extremes.

Figure 1: Spectrum of sea level at tidal gauges Hon Dau and Quy Nhon
Figure 1 (graphics1.png)
Table 1: Table 1. Rate of sea level rise at some points along Vietnam coast
Gauge Co-ordinates Observation years Trend (mm/year)
Hon Dau 20 ° 40 ' N 106 ° 50 ' E 20 ° 40 ' N 106 ° 50 ' E size 12{"20" rSup { size 8{ circ } } "40"'N - "106" rSup { size 8{ circ } } "50"'E} {} 1957-1994 2,1
Cua Cam 20 ° 45 ' N 106 ° 50 ' E 20 ° 45 ' N 106 ° 50 ' E size 12{"20" rSup { size 8{ circ } } "45"'N - "106" rSup { size 8{ circ } } "50"'E} {} 1961-1992 2,7
Da Nang 16 ° 06 ' N 108 ° 13 ' E 16 ° 06 ' N 108 ° 13 ' E size 12{"16" rSup { size 8{ circ } } "06"'N - "108" rSup { size 8{ circ } } "13"'E} {} 1978-1994 1,2
Quy Nhon 13 ° 45 ' N 109 ° 13 ' E 13 ° 45 ' N 109 ° 13 ' E size 12{"13" rSup { size 8{ circ } } "45"'N - "109" rSup { size 8{ circ } } "13"'E} {} 1976-1994 0,9
Vung Tau 10 ° 20 ' N 107 ° 04 ' E 10 ° 20 ' N 107 ° 04 ' E size 12{"10" rSup { size 8{ circ } } "20"'N - "107" rSup { size 8{ circ } } "04"'E} {} 1979-1994 3,2

The method of study

Extremes analysis with empirical data

Assume ViVi size 12{V rSub { size 8{i} } } {} the values of incidental variable VV size 12{V} {} at time ii size 12{i} {} and

X(m)=maxV1,V2,...,VmX(m)=maxV1,V2,...,Vm size 12{X rSup { size 8{ \( m \) } } ="max" left lbrace V rSub { size 8{1} } , V rSub { size 8{2} } , "." "." "." , V rSub { size 8{m} } right rbrace } {}; X(m)=minV1,V2,...,VmX(m)=minV1,V2,...,Vm size 12{X rSub { size 8{ \( m \) } } ="min" left lbrace V rSub { size 8{1} } , V rSub { size 8{2} } , "." "." "." , V rSub { size 8{m} } right rbrace } {}.

One is often interest in estimation the probability with which maximal or minimal value exceeds a threshold, P{X(m)>x}P{X(m)>x} size 12{P lbrace X rSup { size 8{ \( m \) } } >x rbrace } {} or P{X(m)<x}P{X(m)<x} size 12{P lbrace X rSub { size 8{ \( m \) } } <x rbrace } {}. If the observations on the hydrometeorological parameters are independent and distribute differently due to distribution function F(x)=P{Vix}F(x)=P{Vix} size 12{F \( x \) =P lbrace V rSub { size 8{i} } <= x rbrace } {}, the precise distribution of maximum and minimum can be expressed:

P{X(m)x}=[F(x)]mP{X(m)x}=[F(x)]m size 12{P lbrace X rSup { size 8{ \( m \) } } <= x rbrace = \[ F \( x \) \] rSup { size 8{m} } } {} and P{X(m)x}=1[1F(x)]mP{X(m)x}=1[1F(x)]m size 12{P lbrace X rSub { size 8{ \( m \) } } <= x rbrace =1 - \[ 1 - F \( x \) \] rSup { size 8{m} } } {} (1)

The extremes analysis theory says that with the enough length of sample mm size 12{m} {}, the probability distribution of the normalized maximum Y(m)=(X(m)um)/bmY(m)=(X(m)um)/bm size 12{Y rSup { size 8{ \( m \) } } = \( X rSup { size 8{ {} rSub { size 6{ \( m \) } } } } - u rSub {m} size 12{ \) /b rSub {m} }} {}, bm>0bm>0 size 12{b rSub { size 8{m} } >0} {} can be approximated by one of the three following forms of asymptotic function

Table 2
G 1 ( y ) = exp ( e y ) G 1 ( y ) = exp ( e y ) size 12{G rSub { size 8{1} } \( y \) ="exp" \( - e rSup { size 8{ - y} } \) } {} . . . (Gumbel function) . . .  
G 2 ( y ) = exp ( y 1 / k ) , y > 0, k < 0 G 2 ( y ) = exp ( y 1 / k ) , y > 0, k < 0 size 12{G rSub { size 8{2} } \( y \) ="exp" \( - y rSup { size 8{1/k} } \) ," "y>0," "k<0} {} . . . (Frechet function) . . . (2)
G 3 ( y ) = exp [ ( y ) 1 / k ] , y < 0, k > 0 G 3 ( y ) = exp [ ( y ) 1 / k ] , y < 0, k > 0 size 12{G rSub { size 8{3} } \( y \) ="exp" \[ - \( - y \) rSup { size 8{1/k} } \] ," "y<0," "k>0} {} . . . (Weibull function) . . .  

and similar for the minimal value

Table 3
H 1 ( y ) = 1 exp ( e y ) H 1 ( y ) = 1 exp ( e y ) size 12{H rSub { size 8{1} } \( y \) =1 - "exp" \( - e rSup { size 8{ - y} } \) } {} . . . . . . . . . .  
H 2 ( y ) = 1 exp [ ( y ) 1 / k ] , y < 0, k < 0 H 2 ( y ) = 1 exp [ ( y ) 1 / k ] , y < 0, k < 0 size 12{H rSub { size 8{2} } \( y \) =1 - "exp" \[ - \( - y \) rSup { size 8{1/k} } \] ," "y<0," "k<0} {} . . . . . . . . . . (3)
H 3 ( y ) = 1 exp ( y 1 / k ) , y > 0, k > 0 H 3 ( y ) = 1 exp ( y 1 / k ) , y > 0, k > 0 size 12{H rSub { size 8{3} } \( y \) =1 - "exp" \( - y rSup { size 8{1/k} } \) ," "y>0," "k>0} {} . . . . . . . . . .  

These different forms of asymptotic functions are dependent to the shape of the trail of probability distribution F(x)F(x) size 12{F \( x \) } {} (the right side for the maxima and the left side for the minima). In practice the sample conditions (the homogeneity, the independence and the dimension) influence on the precision of the approximation by the above asymptotic functions.

Asymptotic extreme distributions include three parameters: kk size 12{k - {}} {}shape parameter, umum size 12{u rSub { size 8{m} } - {}} {} local parameter and bmbm size 12{b rSub { size 8{m} } - {}} {} scale parameter.

Often, instead of estimating the distribution of maxima (or minima), one executes a diverse problem: determine a design value, i. e. a value xp(m)xp(m) size 12{x rSub { size 8{p} } rSup { size 8{ \( m \) } } } {} such as

PX(m)xp(m)=pPX(m)xp(m)=p size 12{P left lbrace X rSup { size 8{ \( m \) } } <= x rSub { size 8{p} } rSup { size 8{ \( m \) } } right rbrace =p} {}. (4)

Otherwise xp(m)xp(m) size 12{x rSub { size 8{p} } rSup { size 8{ \( m \) } } } {} is the quantile pp size 12{p} {} of extreme distribution. Besides, one converts the probability of the design value xpxp size 12{x rSub { size 8{p} } } {} to return period T=1/(1p)T=1/(1p) size 12{T=1/ \( 1 - p \) } {}, where TT size 12{T - {}} {} the time to be expected that threshold xpxp size 12{x rSub { size 8{p} } } {} is exceeded for the first time, or the average time between two above threshold events.

Using the asymptotic extreme distribution the design values can be easily expressed. For example, with Gumbel distribution, one has:

yp=G11(p)=log(logp)yp=G11(p)=log(logp) size 12{y rSub { size 8{p} } =G rSub { size 8{1} } rSup { size 8{ - 1} } \( p \) = - "log" \( - "log"p \) } {}. (5)

Consequently, design value estimate with return period T=(1p)1T=(1p)1 size 12{T= \( 1 - p \) rSup { size 8{ - 1} } } {} years of the extreme variable XX size 12{X} {} may be calculated knowing parameters uu size 12{u} {} and bb size 12{b} {}:

xp=byp+uxp=byp+u size 12{x rSub { size 8{p} } = ital "by" rSub { size 8{p} } +u} {}, (6)

where ypyp size 12{y rSub { size 8{p} } } {} is also called “normalized design value”.

A question of principle in the application of extremes analysis theory is the precision of the approximation (2) or (3), i. e. the question on the rate of convergence of precise distribution of extremes F(m)F(m) size 12{F rSup { size 8{ \( m \) } } } {} to the asymptotic one, in practical aspect, the precision of design value xpxp size 12{x rSub { size 8{p} } } {} estimated by asymptotic distribution in comparison with it's real value (but often unknown) xp(m)xp(m) size 12{x rSub { size 8{p} } rSup { size 8{ \( m \) } } } {}.

The methods of estimation of extreme distribution aim at settlement the question on the initial series, the relatively short length of initial series. Tibor Farago and Richard W. Kats [5] explain different methods to estimate the extreme parameters and determine design values and their estimate accuracy. Section 3.3 presents the results obtained by applying these methods to series of annually maximal and minimal levels of some tidal gauges along Vietnam coast.

Method of computing extreme values of tide

The tidal height above the mean level may be expressed by the following formula

zt=ifiHicosϕizt=ifiHicosϕi size 12{z rSub { size 8{t} } = Sum cSub { size 8{i} } {f rSub { size 8{i} } H rSub { size 8{i} } "cos"ϕ rSub { size 8{i} } } } {}, (7)

where fifi size 12{f rSub { size 8{i} } - {}} {} the reduce coefficients depended on longitude of the rising knot of lunar orbit; HiHi size 12{H rSub { size 8{i} } - {}} {} the average amplitudes and ϕiϕi size 12{ϕ rSub { size 8{i} } - {}} {} the phase of tidal constituents.

Depending on the tidal feature, the height of tide may achieve the extremes when longitude of the rising knot of lunar orbit N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {} (for diurnal tide) or N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {} (for semidiurnal tide). In these conditions ( N=0°,180°N=0°,180° size 12{N=0 rSup { size 8{ circ } } , "180" rSup { size 8{ circ } } } {}) the phases of tidal constituents are expressed through astronomical parameters in table 2.

Table 4: Table 2. Expressions of phases and reduce coefficients f of tidal constituents [6]
Tidal constituent Phase, ϕϕ size 12{ϕ} {} ff size 12{ size 9{f}} {} for N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {} ff size 12{f} {}for N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {}
M 2 M 2 size 12{M rSub { size 8{2} } } {} 2t + 2h 2s g M 2 2t + 2h 2s g M 2 size 12{2t+2h - 2s - g rSub { size 8{M rSub { size 6{2} } } } } {} 0,963 1,038
S 2 S 2 size 12{S rSub { size 8{2} } } {} 2t g S 2 2t g S 2 size 12{2t - g rSub { size 8{S rSub { size 6{2} } } } } {} 1,000 1,000
N 2 N 2 size 12{N rSub { size 8{2} } } {} 2t + 2h 3s + p g N 2 2t + 2h 3s + p g N 2 size 12{2t+2h - 3s+p - g rSub { size 8{N rSub { size 6{2} } } } } {} 0,963 1,037
K 2 K 2 size 12{K rSub { size 8{2} } } {} 2t + 2h g K 2 2t + 2h g K 2 size 12{2t+2h - g rSub { size 8{K rSub { size 6{2} } } } } {} 1,317 0,748
K 1 K 1 size 12{K rSub { size 8{1} } } {} t + h + 90 ° g K 1 t + h + 90 ° g K 1 size 12{t+h+"90" rSup { size 8{ circ } } - g rSub { size 8{K rSub { size 6{1} } } } } {} 1,113 0,882
O 1 O 1 size 12{O rSub { size 8{1} } } {} t + h 2s 90 ° g O 1 t + h 2s 90 ° g O 1 size 12{t+h - 2s - "90" rSup { size 8{ circ } } - g rSub { size 8{O rSub { size 6{1} } } } } {} 1,183 0,806
P 1 P 1 size 12{P rSub { size 8{1} } } {} t h 90 ° g P 1 t h 90 ° g P 1 size 12{t - h - "90" rSup { size 8{ circ } } - g rSub { size 8{P rSub { size 6{1} } } } } {} 1,000 1,000
Q 1 Q 1 size 12{Q rSub { size 8{1} } } {} t + h 3s + p 90 ° g Q 1 t + h 3s + p 90 ° g Q 1 size 12{t+h - 3s+p - "90" rSup { size 8{ circ } } - g rSub { size 8{Q rSub { size 6{1} } } } } {} 1,183 0,806
M 4 M 4 size 12{M rSub { size 8{4} } } {} 4t + 4h 4s g M 4 4t + 4h 4s g M 4 size 12{4t+4h - 4s - g rSub { size 8{M rSub { size 6{4} } } } } {} 0,928 1,077
MS 4 MS 4 size 12{ ital "MS" rSub { size 8{4} } } {} 4t + 2h 2s g MS 4 4t + 2h 2s g MS 4 size 12{4t+2h - 2s - g rSub { size 8{ ital "MS" rSub { size 6{4} } } } } {} 0,963 1,038
M 6 M 6 size 12{M rSub { size 8{6} } } {} 6t + 6h 6s g M 6 6t + 6h 6s g M 6 size 12{6t+6h - 6s - g rSub { size 8{M rSub { size 6{6} } } } } {} 0,894 1,118
Sa Sa size 12{ ital "Sa"} {} h g Sa h g Sa size 12{h - g rSub { size 8{ ital "Sa"} } } {} 1,000 1,000
SSa SSa size 12{ ital "SSa"} {} 2h g SSa 2h g SSa size 12{2h - g rSub { size 8{ ital "SSa"} } } {} 1,000 1,000

In table 2, tt size 12{t - {}} {} average zone time from midnight; hh size 12{h - {}} {} average longitude of the Sun; ss size 12{s - {}} {} average longitude of the Moon; pp size 12{p - {}} {} average longitude of lunar orbit perigee; gigi size 12{g rSub { size 8{i} } - {}} {} special initial phase related to the Greenwich longitude.

The extreme heights of tide may be computed from (7) if the values of astronomical parameters t,h,st,h,s size 12{t, h, s} {} and pp size 12{p} {}, which form a combination corresponding to an extreme condition, are known. Investigating on extremes the function z(t,h,s,p)z(t,h,s,p) size 12{z \( t, h, s, p \) } {} from (7), we obtain a system of four equations with four unknowns t,h,st,h,s size 12{t, h, s} {} and pp size 12{p} {} whose values determine the extreme condition of the tidal height:

2M 2 sin ϕ M 2 + 2S 2 sin ϕ S 2 + 2N 2 sin ϕ N 2 + 2K 2 sin ϕ K 2 + K 1 sin ϕ K 1 + O 1 sin ϕ O 1 + P 1 sin ϕ P 1 + Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 4 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 = 0 2M 2 sin ϕ M 2 + 2S 2 sin ϕ S 2 + 2N 2 sin ϕ N 2 + 2K 2 sin ϕ K 2 + K 1 sin ϕ K 1 + O 1 sin ϕ O 1 + P 1 sin ϕ P 1 + Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 4 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 = 0 alignc { stack { size 12{2M rSub { size 8{2} } "sin"ϕ rSub { size 8{M rSub { size 6{2} } } } +2S rSub {2} size 12{"sin"ϕ rSub {S rSub { size 6{2} } } } size 12{+2N rSub {2} } size 12{"sin"ϕ rSub {N rSub { size 6{2} } } } size 12{+2K rSub {2} } size 12{"sin"ϕ rSub {K rSub { size 6{2} } } } size 12{+{}}} {} # size 12{K rSub { size 8{1} } "sin"ϕ rSub { size 8{K rSub { size 6{1} } } } +O rSub {1} size 12{"sin"ϕ rSub {O rSub { size 6{1} } } } size 12{+P rSub {1} } size 12{"sin"ϕ rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"sin"ϕ rSub {Q rSub { size 6{1} } } } size 12{+{}}} {} # size 12{4M rSub { size 8{4} } "sin"ϕ rSub { size 8{M rSub { size 6{4} } } } +4 ital "MS" rSub {4} size 12{"sin"ϕ rSub { ital "MS" rSub { size 6{4} } } } size 12{+6M rSub {6} } size 12{"sin"ϕ rSub {M rSub { size 6{6} } } } size 12{ {}=0}} {} } } {}

2M 2 sin ϕ M 2 + 2N 2 sin ϕ N 2 + 2K 2 sin ϕ K 2 + K 1 sin ϕ K 1 + O 1 sin ϕ O 1 + P 1 sin ϕ P 1 + Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 4 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 + Sa sin ϕ Sa + 2 SSa sin ϕ SSa = 0 2M 2 sin ϕ M 2 + 2N 2 sin ϕ N 2 + 2K 2 sin ϕ K 2 + K 1 sin ϕ K 1 + O 1 sin ϕ O 1 + P 1 sin ϕ P 1 + Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 4 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 + Sa sin ϕ Sa + 2 SSa sin ϕ SSa = 0 alignc { stack { size 12{2M rSub { size 8{2} } "sin"ϕ rSub { size 8{M rSub { size 6{2} } } } +2N rSub {2} size 12{"sin"ϕ rSub {N rSub { size 6{2} } } } size 12{+2K rSub {2} } size 12{"sin"ϕ rSub {K rSub { size 6{2} } } } size 12{+K rSub {1} } size 12{"sin"ϕ rSub {K rSub { size 6{1} } } } size 12{+{}}} {} # size 12{O rSub { size 8{1} } "sin"ϕ rSub { size 8{O rSub { size 6{1} } } } +P rSub {1} size 12{"sin"ϕ rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"sin"ϕ rSub {Q rSub { size 6{1} } } } size 12{+4M rSub {4} } size 12{"sin"ϕ rSub {M rSub { size 6{4} } } } size 12{+{}}} {} # size 12{4 ital "MS" rSub { size 8{4} } "sin"ϕ rSub { size 8{ ital "MS" rSub { size 6{4} } } } +6M rSub {6} size 12{"sin"ϕ rSub {M rSub { size 6{6} } } } size 12{+ ital "Sa""sin"ϕ rSub { ital "Sa"} } size 12{+2 ital "SSa""sin"ϕ rSub { ital "SSa"} } size 12{ {}=0 }} {} } } {}

2M 2 sin ϕ M 2 + 3N 2 sin ϕ N 2 + 2O 1 sin ϕ O 1 + 3Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 2 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 = 0 2M 2 sin ϕ M 2 + 3N 2 sin ϕ N 2 + 2O 1 sin ϕ O 1 + 3Q 1 sin ϕ Q 1 + 4M 4 sin ϕ M 4 + 2 MS 4 sin ϕ MS 4 + 6M 6 sin ϕ M 6 = 0 alignc { stack { size 12{2M rSub { size 8{2} } "sin"ϕ rSub { size 8{M rSub { size 6{2} } } } +3N rSub {2} size 12{"sin"ϕ rSub {N rSub { size 6{2} } } } size 12{+2O rSub {1} } size 12{"sin"ϕ rSub {O rSub { size 6{1} } } } size 12{+3Q rSub {1} } size 12{"sin"ϕ rSub {Q rSub { size 6{1} } } } size 12{+{}}} {} # size 12{4M rSub { size 8{4} } "sin"ϕ rSub { size 8{M rSub { size 6{4} } } } +2 ital "MS" rSub {4} size 12{"sin"ϕ rSub { ital "MS" rSub { size 6{4} } } } size 12{+6M rSub {6} } size 12{"sin"ϕ rSub {M rSub { size 6{6} } } } size 12{ {}=0}} {} } } {}

N2sinϕN2+Q1sinϕQ1=0N2sinϕN2+Q1sinϕQ1=0 size 12{N rSub { size 8{2} } "sin"ϕ rSub { size 8{N rSub { size 6{2} } } } +Q rSub {1} size 12{"sin"ϕ rSub {Q rSub { size 6{1} } } } size 12{ {}=0}} {} (8)

where M2=fM2HM2,S2=fS2HS2,...,SSa=fSSaHSSa.M2=fM2HM2,S2=fS2HS2,...,SSa=fSSaHSSa. size 12{M rSub { size 8{2} } =f rSub { size 8{M rSub { size 6{2} } } } H rSub {M rSub { size 6{2} } } size 12{," "S rSub {2} } size 12{ {}=f rSub {S rSub { size 6{2} } } } size 12{H rSub {S rSub { size 6{2} } } } size 12{, "." "." "." ," " ital "SSa"=f rSub { ital "SSa"} } size 12{H rSub { ital "SSa"} } size 12{ "." }} {}

If the approximate values of astronomical parameters corresponding to extreme condition (t',h',s',p')(t',h',s',p') size 12{ \( { {t}} sup { ' }, { {h}} sup { ' }, { {s}} sup { ' }, { {p}} sup { ' } \) } {} are known, we may lead equations (8) to a linear form by Taylor expansion. When approximate values of the unknown are sufficiently close to the exact values (to,ho,so,po)(to,ho,so,po) size 12{ \( t rSub { size 8{o} } ,h rSub { size 8{o} } ,s rSub { size 8{o} } ,p rSub { size 8{o} } \) } {} the expansion can be restricted in first order items.

With designations of corrections to the approximate values of astronomical parameters as following

Δt = t o t ' ; Δs = s o s ' ; Δt = t o t ' ; Δs = s o s ' ; size 12{Δt=t rSub { size 8{o} } - { {t}} sup { ' };" "Δs=s rSub { size 8{o} } - { {s}} sup { ' };} {}

Δh = h o h ' ; Δp = p o p ' , Δh = h o h ' ; Δp = p o p ' , size 12{Δh=h rSub { size 8{o} } - { {h}} sup { ' };" "Δp=p rSub { size 8{o} } - { {p}} sup { ' },} {}

the result of the expansion is a system of four linear equations with diagonally symmetric coefficient matrix in order to find Δt,Δh,Δs,ΔpΔt,Δh,Δs,Δp size 12{Δt,``Δh,``Δs,``Δp} {}:

a 1 Δt + b 1 Δh + c 1 Δs + d 1 Δp + l 1 = 0 a 1 Δt + b 1 Δh + c 1 Δs + d 1 Δp + l 1 = 0 size 12{a rSub { size 8{1} } Δt+b rSub { size 8{1} } Δh+c rSub { size 8{1} } Δs+d rSub { size 8{1} } Δp+l rSub { size 8{1} } =0} {}

b 1 Δt + b 2 Δh + c 2 Δs + d 2 Δp + l 2 = 0 b 1 Δt + b 2 Δh + c 2 Δs + d 2 Δp + l 2 = 0 size 12{b rSub { size 8{1} } Δt+b rSub { size 8{2} } Δh+c rSub { size 8{2} } Δs+d rSub { size 8{2} } Δp+l rSub { size 8{2} } =0} {}

c 1 Δt + c 2 Δh + c 3 Δs + d 3 Δp + l 3 = 0 c 1 Δt + c 2 Δh + c 3 Δs + d 3 Δp + l 3 = 0 size 12{c rSub { size 8{1} } Δt+c rSub { size 8{2} } Δh+c rSub { size 8{3} } Δs+d rSub { size 8{3} } Δp+l rSub { size 8{3} } =0} {}

d1Δt+d2Δh+d3Δs+d4Δp+l4=0d1Δt+d2Δh+d3Δs+d4Δp+l4=0 size 12{d rSub { size 8{1} } Δt+d rSub { size 8{2} } Δh+d rSub { size 8{3} } Δs+d rSub { size 8{4} } Δp+l rSub { size 8{4} } =0} {} (9)

a 1 = 4M 2 cos { ϕ M 2 ' + 4S 2 cos { ϕ S 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' + P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 16 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; a 1 = 4M 2 cos { ϕ M 2 ' + 4S 2 cos { ϕ S 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' + P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 16 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; alignl { stack { size 12{a rSub { size 8{1} } =4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +4S rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {S rSub { size 6{2} } } } size 12{+4N rSub {2} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+4K rSub {2} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{2} } } } size 12{+{}}} {} # size 12{K rSub { size 8{1} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{K rSub { size 6{1} } } } +O rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{+P rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+{}}} {} # size 12{"16"M rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } +"16" ital "MS" rSub {4} size 12{"cos {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{+"36"M rSub {6} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

b 1 = 4M 2 cos { ϕ M 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 8 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; b 1 = 4M 2 cos { ϕ M 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 8 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; alignc { stack { size 12{b rSub { size 8{1} } =4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +4N rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+4K rSub {2} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{2} } } } size 12{+K rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{1} } } } size 12{+{}}} {} # size 12{O rSub { size 8{1} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{O rSub { size 6{1} } } } - P rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+"16"M rSub {4} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{4} } } }} {} # size 12{+8 ital "MS" rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{ ital "MS" rSub { size 6{4} } } } +"36"M rSub {6} size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

c 1 = 4M 2 cos { ϕ M 2 ' 6N 2 cos { ϕ N 2 ' 2O 1 cos { ϕ O 1 ' 3Q 1 cos { ϕ Q 1 ' 16 M 4 cos { ϕ M 4 ' 8 MS 4 cos { ϕ MS 4 ' 36 M 6 cos { ϕ M 6 ' ; c 1 = 4M 2 cos { ϕ M 2 ' 6N 2 cos { ϕ N 2 ' 2O 1 cos { ϕ O 1 ' 3Q 1 cos { ϕ Q 1 ' 16 M 4 cos { ϕ M 4 ' 8 MS 4 cos { ϕ MS 4 ' 36 M 6 cos { ϕ M 6 ' ; alignc { stack { size 12{c rSub { size 8{1} } = - 4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } - 6N rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{ - 2O rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{ - 3Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{ - {}}} {} # size 12{"16"M rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } - 8 ital "MS" rSub {4} size 12{"cos {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{ - "36"M rSub {6} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

d 1 = 2N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; d 1 = 2N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; size 12{d rSub { size 8{1} } =2N rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{N rSub { size 6{2} } } } +Q rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{;}} {}

l 1 = 2M 2 sin { ϕ M 2 ' + 2S 2 sin { ϕ S 2 ' + 2N 2 sin { ϕ N 2 ' + 2K 2 sin { ϕ K 2 ' + K 1 sin { ϕ K 1 ' + O 1 sin { ϕ O 1 ' + P 1 sin { ϕ P 1 ' + Q 1 sin { ϕ Q 1 ' + 4M 4 sin { ϕ M 4 ' + 4 MS 4 sin { ϕ MS 4 ' + 6M 6 sin { ϕ M 6 ' ; l 1 = 2M 2 sin { ϕ M 2 ' + 2S 2 sin { ϕ S 2 ' + 2N 2 sin { ϕ N 2 ' + 2K 2 sin { ϕ K 2 ' + K 1 sin { ϕ K 1 ' + O 1 sin { ϕ O 1 ' + P 1 sin { ϕ P 1 ' + Q 1 sin { ϕ Q 1 ' + 4M 4 sin { ϕ M 4 ' + 4 MS 4 sin { ϕ MS 4 ' + 6M 6 sin { ϕ M 6 ' ; alignc { stack { size 12{l rSub { size 8{1} } =2M rSub { size 8{2} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +2S rSub {2} size 12{"sin {" ital {ϕ}} sup { ' } rSub {S rSub { size 6{2} } } } size 12{+2N rSub {2} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+2K rSub {2} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{2} } } } size 12{+{}}} {} # size 12{K rSub { size 8{1} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{K rSub { size 6{1} } } } +O rSub {1} size 12{"sin {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{+P rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+{}}} {} # size 12{4M rSub { size 8{4} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } +4 ital "MS" rSub {4} size 12{"sin {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{+6M rSub {6} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

b 2 = 4M 2 cos { ϕ M 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' + P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 4 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' + Sa cos { ϕ Sa ' + 4 SSa cos { ϕ SSa ' ; b 2 = 4M 2 cos { ϕ M 2 ' + 4N 2 cos { ϕ N 2 ' + 4K 2 cos { ϕ K 2 ' + K 1 cos { ϕ K 1 ' + O 1 cos { ϕ O 1 ' + P 1 cos { ϕ P 1 ' + Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 4 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' + Sa cos { ϕ Sa ' + 4 SSa cos { ϕ SSa ' ; alignc { stack { size 12{b rSub { size 8{2} } =4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +4N rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+4K rSub {2} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{2} } } } size 12{+K rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{1} } } } size 12{+{}}} {} # size 12{O rSub { size 8{1} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{O rSub { size 6{1} } } } +P rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+"16"M rSub {4} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{4} } } } size 12{+{}}} {} # size 12{4 ital "MS" rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{ ital "MS" rSub { size 6{4} } } } +"36"M rSub {6} size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{+ ital "Sa""cos {" ital {ϕ}} sup { ' } rSub { ital "Sa"} } size 12{+4 ital "SSa""cos {" ital {ϕ}} sup { ' } rSub { ital "SSa"} } size 12{;}} {} } } {}

c 2 = 4M 2 cos { ϕ M 2 ' 6N 2 cos { ϕ N 2 ' 2O 1 cos { ϕ O 1 ' 3Q 1 cos { ϕ Q 1 ' 16 M 4 cos { ϕ M 4 ' 4 MS 4 cos { ϕ MS 4 ' 36 M 6 cos { ϕ M 6 ' ; c 2 = 4M 2 cos { ϕ M 2 ' 6N 2 cos { ϕ N 2 ' 2O 1 cos { ϕ O 1 ' 3Q 1 cos { ϕ Q 1 ' 16 M 4 cos { ϕ M 4 ' 4 MS 4 cos { ϕ MS 4 ' 36 M 6 cos { ϕ M 6 ' ; alignc { stack { size 12{c rSub { size 8{2} } = - 4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } - 6N rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{ - 2O rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{ - 3Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{ - {}}} {} # size 12{"16"M rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } - 4 ital "MS" rSub {4} size 12{"cos {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{ - "36"M rSub {6} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

d 2 = 2N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; d 2 = 2N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; size 12{d rSub { size 8{2} } =2N rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{N rSub { size 6{2} } } } +Q rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{;}} {}

l 2 = 2M 2 sin { ϕ M 2 ' + 2N 2 sin { ϕ N 2 ' + 2K 2 sin { ϕ K 2 ' + K 1 sin { ϕ K 1 ' + O 1 sin { ϕ O 1 ' P 1 sin { ϕ P 1 ' + Q 1 sin { ϕ Q 1 ' + 4M 4 sin { ϕ M 4 ' + 2 MS 4 sin { ϕ MS 4 ' + 6M 6 sin { ϕ M 6 ' + Sa sin { ϕ Sa ' + 2 SSa sin { ϕ SSa ' ; l 2 = 2M 2 sin { ϕ M 2 ' + 2N 2 sin { ϕ N 2 ' + 2K 2 sin { ϕ K 2 ' + K 1 sin { ϕ K 1 ' + O 1 sin { ϕ O 1 ' P 1 sin { ϕ P 1 ' + Q 1 sin { ϕ Q 1 ' + 4M 4 sin { ϕ M 4 ' + 2 MS 4 sin { ϕ MS 4 ' + 6M 6 sin { ϕ M 6 ' + Sa sin { ϕ Sa ' + 2 SSa sin { ϕ SSa ' ; alignc { stack { size 12{l rSub { size 8{2} } =2M rSub { size 8{2} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +2N rSub {2} size 12{"sin {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+2K rSub {2} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{2} } } } size 12{+K rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {K rSub { size 6{1} } } } size 12{+{}}} {} # size 12{O rSub { size 8{1} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{O rSub { size 6{1} } } } - P rSub {1} size 12{"sin {" ital {ϕ}} sup { ' } rSub {P rSub { size 6{1} } } } size 12{+Q rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+4M rSub {4} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{4} } } } size 12{+{}}} {} # size 12{2 ital "MS" rSub { size 8{4} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{ ital "MS" rSub { size 6{4} } } } +6M rSub {6} size 12{"sin {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{+ ital "Sa""sin {" ital {ϕ}} sup { ' } rSub { ital "Sa"} } size 12{+2 ital "SSa""sin {" ital {ϕ}} sup { ' } rSub { ital "SSa"} } size 12{;}} {} } } {}

c 3 = 4M 2 cos { ϕ M 2 ' + 9N 2 cos { ϕ N 2 ' + 4O 1 cos { ϕ O 1 ' + 9Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 4 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; c 3 = 4M 2 cos { ϕ M 2 ' + 9N 2 cos { ϕ N 2 ' + 4O 1 cos { ϕ O 1 ' + 9Q 1 cos { ϕ Q 1 ' + 16 M 4 cos { ϕ M 4 ' + 4 MS 4 cos { ϕ MS 4 ' + 36 M 6 cos { ϕ M 6 ' ; alignc { stack { size 12{c rSub { size 8{3} } =4M rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } +9N rSub {2} size 12{"cos {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{+4O rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{+9Q rSub {1} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{+{}}} {} # size 12{"16"M rSub { size 8{4} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } +4 ital "MS" rSub {4} size 12{"cos {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{+"36"M rSub {6} } size 12{"cos {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

d 3 = 3N 2 cos { ϕ N 2 ' 3Q 1 cos { ϕ Q 1 ' ; d 3 = 3N 2 cos { ϕ N 2 ' 3Q 1 cos { ϕ Q 1 ' ; size 12{d rSub { size 8{3} } = - 3N rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{N rSub { size 6{2} } } } - 3Q rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{;}} {}

l 3 = 2M 2 sin { ϕ M 2 ' 3N 2 sin { ϕ N 2 ' 2O 1 sin { ϕ O 1 ' 3Q 1 sin { ϕ Q 1 ' 4M 4 sin { ϕ M 4 ' 2 MS 4 sin { ϕ MS 4 ' 6M 6 sin { ϕ M 6 ' ; l 3 = 2M 2 sin { ϕ M 2 ' 3N 2 sin { ϕ N 2 ' 2O 1 sin { ϕ O 1 ' 3Q 1 sin { ϕ Q 1 ' 4M 4 sin { ϕ M 4 ' 2 MS 4 sin { ϕ MS 4 ' 6M 6 sin { ϕ M 6 ' ; alignc { stack { size 12{l rSub { size 8{3} } = - 2M rSub { size 8{2} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{2} } } } - 3N rSub {2} size 12{"sin {" ital {ϕ}} sup { ' } rSub {N rSub { size 6{2} } } } size 12{ - 2O rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {O rSub { size 6{1} } } } size 12{ - 3Q rSub {1} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{ - {}}} {} # size 12{4M rSub { size 8{4} } "sin {" ital {ϕ}} sup { ' } rSub { size 8{M rSub { size 6{4} } } } - 2 ital "MS" rSub {4} size 12{"sin {" ital {ϕ}} sup { ' } rSub { ital "MS" rSub { size 6{4} } } } size 12{ - 6M rSub {6} } size 12{"sin {" ital {ϕ}} sup { ' } rSub {M rSub { size 6{6} } } } size 12{;}} {} } } {}

d 4 = N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; d 4 = N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; size 12{d rSub { size 8{4} } =N rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{N rSub { size 6{2} } } } +Q rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{;}} {}

l 4 = N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; l 4 = N 2 cos { ϕ N 2 ' + Q 1 cos { ϕ Q 1 ' ; size 12{l rSub { size 8{4} } =N rSub { size 8{2} } "cos {" ital {ϕ}} sup { ' } rSub { size 8{N rSub { size 6{2} } } } +Q rSub {1} size 12{"cos {" ital {ϕ}} sup { ' } rSub {Q rSub { size 6{1} } } } size 12{;}} {}

ϕi'ϕi' size 12{ { {ϕ}} sup { ' } rSub { size 8{i} } - {}} {} phase of the tidal constituents computed through approximate values of the astronomical parameters t',h',s't',h',s' size 12{ { {t}} sup { ' }, { {h}} sup { ' }, { {s}} sup { ' }} {} and p'.p'. size 12{ { {p}} sup { ' } "." } {}

In order to compute the values of astronomical corresponding to extreme condition (to,ho,so,po)(to,ho,so,po) size 12{ \( t rSub { size 8{o} } , h rSub { size 8{o} } , s rSub { size 8{o} } , p rSub { size 8{o} } \) } {} with a given accuracy the iteration method may be used. If any correction among (Δt,Δh,Δs,Δp)(Δt,Δh,Δs,Δp) size 12{ \( Δt, Δh, Δs, Δp \) } {} obtained from solving system (9) exceeds in magnitude a given value δδ size 12{ lline δ rline } {} the solution will repeated and then in order to compute the coefficients of equations (9) we will use the phases ϕi''ϕi'' size 12{ { {ϕ}} sup { '' } rSub { size 8{i} } } {} computed through the values corrected of astronomical parameters:

t ' ' = t ' + Δ t ' ; { s ' ' = s ' + Δ s ' ; { h ' ' = h ' + Δ h ' ; { p ' ' = p ' + Δ p ' . t ' ' = t ' + Δ t ' ; { s ' ' = s ' + Δ s ' ; { h ' ' = h ' + Δ h ' ; { p ' ' = p ' + Δ p ' . size 12{ { {t}} sup { '' }= { {t}} sup { ' }+Δ { {t}} sup { ' };" {" ital {s}} sup { '' }= { {s}} sup { ' }+Δ { {s}} sup { ' };" {" ital {h}} sup { '' }= { {h}} sup { ' }+Δ { {h}} sup { ' };" {" ital {p}} sup { '' }= { {p}} sup { ' }+Δ { {p}} sup { ' } "." } {}

The loop is repeated until all corrections (Δt,Δh,Δs,Δp)(Δt,Δh,Δs,Δp) size 12{ \( Δt, Δh, Δs, Δp \) } {} obtained in solution step kk size 12{k} {} of system (9) become less than δδ size 12{ lline δ rline } {}:

Δt(k),Δh(k),Δs(k),Δp(k)<δΔt(k),Δh(k),Δs(k),Δp(k)<δ size 12{ lline Δt rSup { size 8{ \( k \) } } rline , lline Δh rSup { size 8{ \( k \) } } rline , lline Δs rSup { size 8{ \( k \) } } rline , lline Δp rSup { size 8{ \( k \) } } rline < lline δ rline } {}.

Table 5: Table 3. Values of astronomical parameters approximately corresponding extreme condition [6] for semidiurnal tide
Astronomical parameters Minimal level condition Maximal level condition
t ' t ' size 12{ { {t}} sup { ' }} {} t 1 ' = 90 ° + 0,5 g S 2 t 1 ' = 90 ° + 0,5 g S 2 size 12{ { {t}} sup { ' } rSub { size 8{1} } ="90" rSup { size 8{ circ } } +0,5g rSub { size 8{S rSub { size 6{2} } } } } {} t 1 ' = 180 ° + 0,5 g S 2 t 1 ' = 180 ° + 0,5 g S 2 size 12{ { {t}} sup { ' } rSub { size 8{1} } ="180" rSup { size 8{ circ } } +0,5g rSub { size 8{S rSub { size 6{2} } } } } {}
  t 2 ' = 270 ° + 0,5 g S 2 t 2 ' = 270 ° + 0,5 g S 2 size 12{ { {t}} sup { ' } rSub { size 8{2} } ="270" rSup { size 8{ circ } } +0,5g rSub { size 8{S rSub { size 6{2} } } } } {} t 2 ' = 0,5 g S 2 t 2 ' = 0,5 g S 2 size 12{ { {t}} sup { ' } rSub { size 8{2} } =0,5g rSub { size 8{S rSub { size 6{2} } } } } {}
h ' h ' size 12{ { {h}} sup { ' }} {} 0,5 ( g K 2 g S 2 ) 0,5 ( g K 2 g S 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - g rSub {S rSub { size 6{2} } } size 12{ \) }} {} 0,5 ( g K 2 g S 2 ) 0,5 ( g K 2 g S 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - g rSub {S rSub { size 6{2} } } size 12{ \) }} {}
s ' s ' size 12{ { {s}} sup { ' }} {} 0,5 ( g K 2 g M 2 ) 0,5 ( g K 2 g M 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - g rSub {M rSub { size 6{2} } } size 12{ \) }} {} 0,5 ( g K 2 g M 2 ) 0,5 ( g K 2 g M 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - g rSub {M rSub { size 6{2} } } size 12{ \) }} {}
p ' p ' size 12{ { {p}} sup { ' }} {} 0,5 ( g K 2 3g M 2 + 2g N 2 ) 0,5 ( g K 2 3g M 2 + 2g N 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - 3g rSub {M rSub { size 6{2} } } size 12{+2g rSub {N rSub { size 6{2} } } } size 12{ \) }} {} 0,5 ( g K 2 3g M 2 + 2g N 2 ) 0,5 ( g K 2 3g M 2 + 2g N 2 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{2} } } } - 3g rSub {M rSub { size 6{2} } } size 12{+2g rSub {N rSub { size 6{2} } } } size 12{ \) }} {}

If initial approximate values (t',h',s',p')(t',h',s',p') size 12{ \( { {t}} sup { ' }, { {h}} sup { ' }, { {s}} sup { ' }, { {p}} sup { ' } \) } {} close to real values (to,ho,so,po)(to,ho,so,po) size 12{ \( t rSub { size 8{o} } , h rSub { size 8{o} } , s rSub { size 8{o} } , p rSub { size 8{o} } \) } {} the iteration rapidly converges. These values of astronomical parameters corresponding to extreme condition may be calculate through four diurnal or semidiurnal tidal constituents depending on tide feature. The extreme condition for four semidiurnal tidal constituents and diurnal constituents is determined by the following expressions:

- For semidiurnal tide: ϕM2=ϕS2=ϕN2=ϕK2=ϕϕM2=ϕS2=ϕN2=ϕK2=ϕ size 12{ϕ rSub { size 8{M rSub { size 6{2} } } } =ϕ rSub {S rSub { size 6{2} } } size 12{ {}=ϕ rSub {N rSub { size 6{2} } } } size 12{ {}=ϕ rSub {K rSub { size 6{2} } } } size 12{ {}=ϕ}} {}.

- For diurnal tide: ϕK1=ϕO1=ϕP1=ϕQ1=ϕϕK1=ϕO1=ϕP1=ϕQ1=ϕ size 12{ϕ rSub { size 8{K rSub { size 6{1} } } } =ϕ rSub {O rSub { size 6{1} } } size 12{ {}=ϕ rSub {P rSub { size 6{1} } } } size 12{ {}=ϕ rSub {Q rSub { size 6{1} } } } size 12{ {}=ϕ}} {},

where ϕ=180°+nϕ=180°+n size 12{ϕ="180" rSup { size 8{ circ } } +2π`n - {}} {} for the lowest level and ϕ=360°+nϕ=360°+n size 12{ϕ="360" rSup { size 8{ circ } } +2π`n - {}} {} for the highest level.

From these expressions follow the formulae for computing the approximate values of astronomical parameters (t',h',s',p')(t',h',s',p') size 12{ \( { {t}} sup { ' }, { {h}} sup { ' }, { {s}} sup { ' }, { {p}} sup { ' } \) } {} corresponding the extreme conditions (tables 3 to 6).

In order to compute approximate value of average zone time t't' size 12{ { {t}} sup { ' }} {} there are two expressions for the lowest condition and highest condition separately, since for semidiurnal tide one day has two high waters and two low waters. The choice of formula used in concrete case must be reference to the sign of supplementary coefficients BB size 12{B} {} and CC size 12{C} {} (table 4). Coefficients BB size 12{B} {} and CC size 12{C} {} are computed by following formulae:

B=O1cosα1+P1cosα2+Q1cosα3+K1cosα4B=O1cosα1+P1cosα2+Q1cosα3+K1cosα4 size 12{B=O rSub { size 8{1} } "cos"α rSub { size 8{1} } +P rSub { size 8{1} } "cos"α rSub { size 8{2} } +Q rSub { size 8{1} } "cos"α rSub { size 8{3} } +K rSub { size 8{1} } "cos"α rSub { size 8{4} } } {};

C=O1sinα1+P1sinα2+Q1sinα3+K1sinα4C=O1sinα1+P1sinα2+Q1sinα3+K1sinα4 size 12{C=O rSub { size 8{1} } "sin"α rSub { size 8{1} } +P rSub { size 8{1} } "sin"α rSub { size 8{2} } +Q rSub { size 8{1} } "sin"α rSub { size 8{3} } +K rSub { size 8{1} } "sin"α rSub { size 8{4} } } {}; (10)

α1=gM20,5gK2gO190°;α3=gN20,5gK2gQ190°α1=gM20,5gK2gO190°;α3=gN20,5gK2gQ190° size 12{α rSub { size 8{1} } =g rSub { size 8{M rSub { size 6{2} } } } - 0,5g rSub {K rSub { size 6{2} } } size 12{ - g rSub {O rSub { size 6{1} } } } size 12{ - "90" rSup { circ } } size 12{;" "α rSub {3} } size 12{ {}=g rSub {N rSub { size 6{2} } } } size 12{ - 0,5g rSub {K rSub { size 6{2} } } } size 12{ - g rSub {Q rSub { size 6{1} } } } size 12{ - "90" rSup { circ } }} {};

α2=gS20,5gK2gP190°;α4=0,5gK2gK1+90°α2=gS20,5gK2gP190°;α4=0,5gK2gK1+90° size 12{α rSub { size 8{2} } =g rSub { size 8{S rSub { size 6{2} } } } - 0,5g rSub {K rSub { size 6{2} } } size 12{ - g rSub {P rSub { size 6{1} } } } size 12{ - "90" rSup { circ } } size 12{;" "α rSub {4} } size 12{ {}=0,5g rSub {K rSub { size 6{2} } } } size 12{ - g rSub {K rSub { size 6{1} } } } size 12{+"90" rSup { circ } }} {} (11)

Table 6: Table 4. Conditions of the lowest and highest level [6]
Conditions of the lowest level Conditions of the highest level
t1't1' size 12{t rSub { size 8{1} } rSup { size 8{'} } } {} when C>0C>0 size 12{C>0} {}t2't2' size 12{t rSub { size 8{2} } rSup { size 8{'} } } {} when C<0C<0 size 12{C<0} {} t1't1' size 12{t rSub { size 8{1} } rSup { size 8{'} } } {} when B<0B<0 size 12{B<0} {}t2't2' size 12{t rSub { size 8{2} } rSup { size 8{'} } } {} when B>0B>0 size 12{B>0} {}
Table 7: Table 5. Values of astronomical parameters approximately corresponding to extreme condition [6] for diurnal tide
Astronomical parameters Conditions of the lowest level Conditions of the highest level
t ' t ' size 12{ { {t}} sup { ' }} {} 0,5 ( g K 1 g P 1 ) 0,5 ( g K 1 g P 1 ) size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - g rSub {P rSub { size 6{1} } } size 12{ \) }} {} 0,5 ( g K 1 + g P 1 ) + 180 ° 0,5 ( g K 1 + g P 1 ) + 180 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } +g rSub {P rSub { size 6{1} } } size 12{ \) +"180" rSup { circ } }} {}
h ' h ' size 12{ { {h}} sup { ' }} {} 0,5 ( g K 1 g P 1 ) + 90 ° 0,5 ( g K 1 g P 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - g rSub {P rSub { size 6{1} } } size 12{ \) +"90" rSup { circ } }} {} 0,5 ( g K 1 g P 1 ) + 90 ° 0,5 ( g K 1 g P 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - g rSub {P rSub { size 6{1} } } size 12{ \) +"90" rSup { circ } }} {}
s ' s ' size 12{ { {s}} sup { ' }} {} 0,5 ( g K 1 g O 1 ) + 90 ° 0,5 ( g K 1 g O 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - g rSub {O rSub { size 6{1} } } size 12{ \) +"90" rSup { circ } }} {} 0,5 ( g K 1 g O 1 ) + 90 ° 0,5 ( g K 1 g O 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - g rSub {O rSub { size 6{1} } } size 12{ \) +"90" rSup { circ } }} {}
p ' p ' size 12{ { {p}} sup { ' }} {} 0,5 ( g K 1 3g O 1 + 2g Q 1 ) + 90 ° 0,5 ( g K 1 3g O 1 + 2g Q 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - 3g rSub {O rSub { size 6{1} } } size 12{+2g rSub {Q rSub { size 6{1} } } } size 12{ \) +"90" rSup { circ } }} {} 0,5 ( g K 1 3g O 1 + 2g Q 1 ) + 90 ° 0,5 ( g K 1 3g O 1 + 2g Q 1 ) + 90 ° size 12{0,5 \( g rSub { size 8{K rSub { size 6{1} } } } - 3g rSub {O rSub { size 6{1} } } size 12{+2g rSub {Q rSub { size 6{1} } } } size 12{ \) +"90" rSup { circ } }} {}

The choice of reduce coefficients to compute values fHfH size 12{ ital "fH"} {} is depended on the tide feature:

1) For semidiurnal tide, if HK1+HO1HM20,5HK1+HO1HM20,5 size 12{ { {H rSub { size 8{K rSub { size 6{1} } } } +H rSub {O rSub { size 6{1} } } } over { size 12{H rSub {M rSub { size 6{2} } } } } } size 12{<0,5}} {} then ff size 12{f} {} is chosen for N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {};

2) For diurnal tide, if HK1+HO1HM21,5HK1+HO1HM21,5 size 12{ { {H rSub { size 8{K rSub { size 6{1} } } } +H rSub {O rSub { size 6{1} } } } over { size 12{H rSub {M rSub { size 6{2} } } } } } size 12{>1,5}} {} then ff size 12{f} {} is chosen for N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {};

3) For mixed tide, if 0,5<HK1+HO1HM21,50,5<HK1+HO1HM21,5 size 12{0,5< { {H rSub { size 8{K rSub { size 6{1} } } } +H rSub {O rSub { size 6{1} } } } over { size 12{H rSub {M rSub { size 6{2} } } } } } size 12{<1,5}} {} then we must use the values of astronomical parameters for both semidiurnal tide (table 3) and diurnal tide (table 5). When compute with astronomical parameters of diurnal tide choose ff size 12{f} {} for N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {}, when compute with astronomical parameters of semidiurnal tide choose ff size 12{f} {} for N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {} and N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {}. The highest level and the lowest level obtained by three variants will be accepted to be the extremes.

We also compute the approximate values of astronomical parameters corresponding the extreme conditions by Vladimirsky method; this method applied for 8 tide constituents. In Vladimirsky method the extreme heights of tide is determined by consequently choosing values ϕK1ϕK1 size 12{ϕ rSub { size 8{K rSub { size 6{1} } } } } {} in the interval from 0°0° size 12{0 rSup { size 8{ circ } } } {} to 360°360° size 12{"360" rSup { size 8{ circ } } } {}:

H = K 1 cos ϕ K 1 + K 2 cos ( K 1 + a 4 ) + R 1 + R 2 + R 3 H = K 1 cos ϕ K 1 + K 2 cos ( K 1 + a 4 ) + R 1 + R 2 + R 3 size 12{H=K rSub { size 8{1} } "cos"ϕ rSub { size 8{K rSub { size 6{1} } } } +K rSub {2} size 12{"cos" \( 2ϕ rSub {K rSub { size 6{1} } } } size 12{+a rSub {4} } size 12{ \) + lline R rSub {1} size 12{+R rSub {2} } size 12{+R rSub {3} } rline }} {}

L=K1cosϕK1+K2cos(K1+a4)R1+R2+R3L=K1cosϕK1+K2cos(K1+a4)R1+R2+R3 size 12{L=K rSub { size 8{1} } "cos"ϕ rSub { size 8{K rSub { size 6{1} } } } +K rSub {2} size 12{"cos" \( 2ϕ rSub {K rSub { size 6{1} } } } size 12{+a rSub {4} } size 12{ \) - lline R rSub {1} size 12{+R rSub {2} } size 12{+R rSub {3} } rline }} {} (12)

where

R 1 = M 2 2 + O 1 2 + 2M 2 O 1 cos τ 1 ; R 2 = S 2 2 + P 1 2 + 2S 2 P 1 cos τ 2 ; R 1 = M 2 2 + O 1 2 + 2M 2 O 1 cos τ 1 ; R 2 = S 2 2 + P 1 2 + 2S 2 P 1 cos τ 2 ; alignl { stack { size 12{R rSub { size 8{1} } = sqrt {M rSub { size 8{2} } rSup { size 8{2} } +O rSub { size 8{1} } rSup { size 8{2} } +2M rSub { size 8{2} } O rSub { size 8{1} } "cos"τ rSub { size 8{1} } } " ;"} {} # R rSub { size 8{2} } = sqrt {S rSub { size 8{2} } rSup { size 8{2} } +P rSub { size 8{1} } rSup { size 8{2} } +2S rSub { size 8{2} } P rSub { size 8{1} } "cos"τ rSub { size 8{2} } } " ;" {} } } {}

τ 1 = ϕ K 1 + a 1 ; τ 2 = ϕ K 1 + a 2 ; τ 3 = ϕ K 1 + a 3 ; a 1 = g K 1 + g O 1 g M 2 ; a 2 = g K 1 + g P 1 g S 2 ; a 3 = g K 1 + g Q 1 g N 2 ; a 4 = 2g K 1 + g K 2 180 ° . τ 1 = ϕ K 1 + a 1 ; τ 2 = ϕ K 1 + a 2 ; τ 3 = ϕ K 1 + a 3 ; a 1 = g K 1 + g O 1 g M 2 ; a 2 = g K 1 + g P 1 g S 2 ; a 3 = g K 1 + g Q 1 g N 2 ; a 4 = 2g K 1 + g K 2 180 ° . alignl { stack { size 12{τ rSub { size 8{1} } =ϕ rSub { size 8{K rSub { size 6{1} } } } +a rSub {1} size 12{"; "τ rSub {2} } size 12{ {}=ϕ rSub {K rSub { size 6{1} } } } size 12{+a rSub {2} } size 12{"; "τ rSub {3} } size 12{ {}=ϕ rSub {K rSub { size 6{1} } } } size 12{+a rSub {3} } size 12{;}} {} # size 12{a rSub { size 8{1} } =g rSub { size 8{K rSub { size 6{1} } } } +g rSub {O rSub { size 6{1} } } size 12{ - g rSub {M rSub { size 6{2} } } } size 12{"; "a rSub {2} } size 12{ {}=g rSub {K rSub { size 6{1} } } } size 12{+g rSub {P rSub { size 6{1} } } } size 12{ - g rSub {S rSub { size 6{2} } } } size 12{"; "}} {} # size 12{a rSub { size 8{3} } =g rSub { size 8{K rSub { size 6{1} } } } +g rSub {Q rSub { size 6{1} } } size 12{ - g rSub {N rSub { size 6{2} } } } size 12{"; "a rSub {4} } size 12{ {}=2g rSub {K rSub { size 6{1} } } } size 12{+g rSub {K rSub { size 6{2} } } } size 12{ - "180" rSup { circ } } size 12{ "." }} {} } } {}

The choice of reduce coefficients to compute values fHfH size 12{ ital "fH"} {} is also made as the above recommendations, i. e. with the semidiurnal tide ff size 12{f} {} is chosen for N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {}, with diurnal tide ff size 12{f} {} is chosen for N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {}. With mixed tide the computation is performed with ff size 12{f} {} for N=180°N=180° size 12{N="180" rSup { size 8{ circ } } } {} and N=0°N=0° size 12{N=0 rSup { size 8{ circ } } } {} and than the lowest and highest values in two variants will be the extreme levels.

If compute extreme levels with 8 tide constituents then the last results are obtained directly from the expressions (12). In the case other constituents are taken into the computations, we must reference to values (ϕK1)min(ϕK1)min size 12{ \( ϕ rSub { size 8{K rSub { size 6{1} } } } \) rSub {"min"} } {} and (ϕK1)max(ϕK1)max size 12{ \( ϕ rSub { size 8{K rSub { size 6{1} } } } \) rSub {"max"} } {} from analyzing (12) to compute the astronomical parameters corresponding extreme conditions t,h,s,pt,h,s,p size 12{t, h, s, p} {} and use them as the approximations to compute the coefficients of equations (9).

The conditions of the lowest level:

t = 0,5 ( ε 2 ) min + g S 2 + 90 ° ; t = 0,5 ( ε 2 ) min + g S 2 + 90 ° ; size 12{t=0,5 left [ \( ε rSub { size 8{2} } \) rSub { size 8{"min"} } +g rSub { size 8{S rSub { size 6{2} } } } right ]+"90" rSup { circ } size 12{;}} {}

h = ( ϕ K 1 ) min + g K 1 0,5 ( ε 2 ) min + g S 2 180 ° ; h = ( ϕ K 1 ) min + g K 1 0,5 ( ε 2 ) min + g S 2 180 ° ; size 12{h= \( ϕ rSub { size 8{K rSub { size 6{1} } } } \) rSub {"min"} size 12{+g rSub {K rSub { size 6{1} } } } size 12{ - 0,5 left [ \( ε rSub {2} size 12{ \) rSub {"min"} } size 12{+g rSub {S rSub { size 6{2} } } } right ]} size 12{ - "180" rSup { circ } } size 12{;}} {}

s = ( ϕ K 1 ) min + g K 1 0,5 ( ε 1 ) min + g M 2 180 ° ; s = ( ϕ K 1 ) min + g K 1 0,5 ( ε 1 ) min + g M 2 180 ° ; size 12{s= \( ϕ rSub { size 8{K rSub { size 6{1} } } } \) rSub {"min"} size 12{+g rSub {K rSub { size 6{1} } } } size 12{ - 0,5 left [ \( ε rSub {1} size 12{ \) rSub {"min"} } size 12{+g rSub {M rSub { size 6{2} } } } right ]} size 12{ - "180" rSup { circ } } size 12{;}} {}

p = ( ϕ K 1 ) min + g K 1 1,5 ( ε 1 ) min + g M 2 + ( ε 3 ) min g N 2 180 ° ; p = ( ϕ K 1 ) min + g K 1 1,5 ( ε 1 ) min + g M 2 + ( ε 3 ) min g N 2 180 ° ; size 12{p= \( ϕ rSub { size 8{K rSub { size 6{1} } } } \) rSub {"min"} size 12{+g rSub {K rSub { size 6{1} } } } size 12{ - 1,5 left [ \( ε rSub {1} size 12{ \) rSub {"min"} } size 12{+g rSub {M rSub { size 6{2} } } } right ]} size 12{+ left [ \( ε rSub {3} size 12{ \) rSub {"min"} } size 12{ - g rSub {N rSub { size 6{2} } } } right ]} size 12{ - "180" rSup { circ } } size 12{;}} {}

and the conditions of the highest level:

t = 0,5 ( ε 2 ) max + g S 2 ; t = 0,5 ( ε 2 ) max + g S 2 ; size 12{t=0,5 left [ \( ε rSub { size 8{2} } \) rSub { size 8{"max"} } +g rSub { size 8{S rSub { size 6{2} } } } right ];} {}

h = ( ϕ K 1 ) max + g K 1 0,5 ( ε 2 ) max + g S 2 90 ° ; h = ( ϕ K 1 ) max + g K 1