extremum sea levels in vietnam coast

2.1. Extremes analysis with empirical data

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

X(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,...,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} size 12{P lbrace X rSup { size 8{ {} rSub { size 6{ ( m ) } } } } >x rbrace } {} or 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{Vi≤x} 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)]m size 12{P lbrace X rSup { size 8{ ( m ) } } <= x rbrace = [ F ( x ) ] rSup { size 8{m} } } {} and P{X(m)≤x}=1−[1−F(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 m size 12{m} {}, the probability distribution of the normalized maximum Y(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>0 size 12{b rSub { size 8{m} } >0} {} can be approximated by one of the three following forms of asymptotic function

and similarly for the minimal value

These different forms of asymptotic functions are dependent to the shape of the trail of probability distribution 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: k− size 12{k - {}} {}shape parameter, um− size 12{u rSub { size 8{m} } - {}} {} local parameter and bm− 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) size 12{x rSub { size 8{p} } rSup { size 8{ ( m ) } } } {} such as

PX(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) size 12{x rSub { size 8{p} } rSup { size 8{ ( m ) } } } {} is the quantile p size 12{p} {} of extreme distribution. Besides, one converts the probability of the design value xp size 12{x rSub { size 8{p} } } {} to return period T=1/(1−p) size 12{T=1/ ( 1 - p ) } {}, where T− size 12{T - {}} {} the time to be expected that threshold xp 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=G1−1(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=(1−p)−1 size 12{T= ( 1 - p ) rSup { size 8{ - 1} } } {} years of the extreme variable X size 12{X} {} may be calculated knowing parameters u size 12{u} {} and b size 12{b} {}:

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

where yp 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) size 12{F rSup { size 8{ ( m ) } } } {} to the asymptotic one, in practical aspect, the precision of design value xp size 12{x rSub { size 8{p} } } {} estimated by asymptotic distribution in comparison with it's real value (but often unknown) 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.

2.2. Method of computing extreme values of tide

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

zt=∑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 fi− size 12{f rSub { size 8{i} } - {}} {} the reduce coefficients depended on longitude of the rising knot of lunar orbit; Hi− size 12{H rSub { size 8{i} } - {}} {} the average amplitudes and ϕi− 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° size 12{N=0 rSup { size 8{ circ } } } {} (for diurnal tide) or N=180° size 12{N="180" rSup { size 8{ circ } } } {} (for semidiurnal tide). In these conditions ( 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.

In table 2 t− size 12{t - {}} {} average zone time from midnight; h− size 12{h - {}} {} average longitude of the Sun; s− size 12{s - {}} {} average longitude of the Moon; p− size 12{p - {}} {} average longitude of lunar orbit perigee; gi− 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,s size 12{t, h, s} {} and p size 12{p} {}, which form a combination corresponding to an extreme condition, are known. Investigating on extremes the function 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,s size 12{t, h, s} {} and p size 12{p} {} whose values determine the extreme condition of the tidal height:

2M2sinϕM2+2S2sinϕS2+2N2sinϕN2+2K2sinϕK2+K1sinϕK1+O1sinϕO1+P1sinϕP1+Q1sinϕQ1+4M4sinϕM4+4MS4sinϕMS4+6M6sinϕM6=02M2sinϕM2+2N2sinϕN2+2K2sinϕK2+K1sinϕK1+O1sinϕO1+P1sinϕP1+Q1sinϕQ1+4M4sinϕM4+4MS4sinϕMS4+6M6sinϕM6+SasinϕSa+2SSasinϕSSa=02M2sinϕM2+3N2sinϕN2+2O1sinϕO1+3Q1sinϕQ1+4M4sinϕM4+2MS4sinϕMS4+6M6sinϕM6=0N2sinϕN2+Q1sinϕQ1=0}}}}}}}} size 12{alignl { stack { left none 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{+{}} {} # right rbrace left none K rSub {1} size 12{"sin"ϕ rSub {K rSub { size 6{1} } } } size 12{+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{+{}} {} # right rbrace left none 4M rSub {4} size 12{"sin"ϕ rSub {M rSub { size 6{4} } } } size 12{+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} {} # right rbrace left none 2M rSub {2} size 12{"sin"ϕ rSub {M 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{+K rSub {1} } size 12{"sin"ϕ rSub {K rSub { size 6{1} } } } size 12{+{}} {} # right rbrace left none 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{+4M rSub {4} } size 12{"sin"ϕ rSub {M rSub { size 6{4} } } } size 12{+{}} {} # right rbrace left none 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{+ ital "Sa""sin"ϕ rSub { ital "Sa"} } size 12{+2 ital "SSa""sin"ϕ rSub { ital "SSa"} } size 12{ {}=0" "} {} # right rbrace left none 2M rSub {2} size 12{"sin"ϕ rSub {M rSub { size 6{2} } } } size 12{+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{+{}} {} # right rbrace left none 4M rSub {4} size 12{"sin"ϕ rSub {M rSub { size 6{4} } } } size 12{+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} {} # right rbrace left none N rSub {2} size 12{"sin"ϕ rSub {N rSub { size 6{2} } } } size 12{+Q rSub {1} } size 12{"sin"ϕ rSub {Q rSub { size 6{1} } } } size 12{ {}=0} {} # right rbra } } rbrace } {} (8)

where 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') 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) 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

the result of the expansion is a system of four linear equations with diagonally symmetric coefficient matrix:

AX+λ=0 size 12{ ital "AX"+λ=0} {}, (9)

where

a1b1c1d1b2c2d2c3d3d4rdli∥∥∥∥∥∥A= size 12{A=alignr { stack { ldline matrix { a rSub { size 8{1} } {} # b rSub { size 8{1} } {} # c rSub { size 8{1} } {} # d rSub { size 8{1} } {} } {} # rdline ldline matrix { {} # b rSub { size 8{2} } {} # c rSub { size 8{2} } {} # d rSub { size 8{2} } {} } {} # rdline ldline matrix { {} # {} # c rSub { size 8{3} } {} # d rSub { size 8{3} } {} } {} # rdline ldline matrix { {} # {} # {} # d rSub { size 8{4} } {} } {} # rdli } } "" ;" "X= ldline matrix { Δt {} ## Δh {} ## Δs {} ## Δp } rdline ;" "λ= ldline matrix { l rSub { size 8{1} } {} ## l rSub { size 8{2} } {} ## l rSub { size 8{3} } {} ## l rSub { size 8{4} } } rdline } {};