2.1. Extremes analysis with empirical data

Assume $V_i$ the values of incidental variable $V$ at time $i$ and

$X^{(m)}=\text{max}\left\{V_1,V_2,\text{.}\text{.}\text{.},V_m\right\}$; $X_{(m)}=\text{min}\left\{V_1,V_2,\text{.}\text{.}\text{.},V_m\right\}$.

One is often interest in estimation the probability with which maximal or minimal value exceeds a threshold, $P\{X^{{}_{(m)}}>x\}$ or . If the observations on the hydrometeorological parameters are independent and distribute differently due to distribution function , the precise distribution of maximum and minimum can be expressed:

and (1)

The extremes analysis theory says that with the enough length of sample $m$, the probability distribution of the normalized maximum $Y^{(m)}=(X^{{}_{(m)}}-u_m)/b_m$, $b_m>0$ can be approximated by one of the three following forms of asymptotic function

$G_1(y)=\text{exp}(-e^{-y})$	(Gumbel function)
	(Frechet function)	(2)
	(Weibull function)

and similarly for the minimal value

$H_1(y)=1-\text{exp}(-e^{-y})$
		(3)
$H_3(y)=1-\text{exp}(-y^{1/k}),y>\mathrm{0,}k>0$

These different forms of asymptotic functions are dependent to the shape of the trail of probability distribution $F(x)$ (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: $k-$shape parameter, $u_m-$ local parameter and $b_m-$ 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 $x_p^{(m)}$ such as

. (4)

Otherwise $x_p^{(m)}$ is the quantile $p$ of extreme distribution. Besides, one converts the probability of the design value $x_p$ to return period $T=1/(1-p)$, where $T-$ the time to be expected that threshold $x_p$ 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:

$y_p=G_1^{-1}(p)=-\text{log}(-\text{log}p)$. (5)

Consequently, design value estimate with return period $T=(1-p{)}^{-1}$ years of the extreme variable $X$ may be calculated knowing parameters $u$ and $b$:

$x_p={\text{by}}_p+u$, (6)

where $y_p$ 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 $F^{(m)}$ to the asymptotic one, in practical aspect, the precision of design value $x_p$ estimated by asymptotic distribution in comparison with it's real value (but often unknown) $x_p^{(m)}$.

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

$z_t=\sum _i{f}_i{H}_i\text{cos}{\varphi }_i$, (7)

where $f_i-$ the reduce coefficients depended on longitude of the rising knot of lunar orbit; $H_i-$ the average amplitudes and ${\varphi }_i-$ 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 $N=0^{°}$ (for diurnal tide) or $N={\text{180}}^{°}$ (for semidiurnal tide). In these conditions ( $N=0^{°},{\text{180}}^{°}$) the phases of tidal constituents are expressed through astronomical parameters in table 2.

In table 2 $t-$ average zone time from midnight; $h-$ average longitude of the Sun; $s-$ average longitude of the Moon; $p-$ average longitude of lunar orbit perigee; $g_i-$ special initial phase related to the Greenwich longitude.

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

Expressions of phases and reduce coefficients of tidal constituents [6]

Tidal constituent	Phase, $\varphi$
Reduce coefficient, $f$
$N=0^{°}$	$N={\text{180}}^{°}$
$M_2$	$\mathrm{2t}+\mathrm{2h}-\mathrm{2s}-g_{M_2}$	0,963	1,038
$S_2$	$\mathrm{2t}-g_{S_2}$	1,000	1,000
$N_2$	$\mathrm{2t}+\mathrm{2h}-\mathrm{3s}+p-g_{N_2}$	0,963	1,037
$K_2$	$\mathrm{2t}+\mathrm{2h}-g_{K_2}$	1,317	0,748
$K_1$	$t+h+{\text{90}}^{°}-g_{K_1}$	1,113	0,882
$O_1$	$t+h-\mathrm{2s}-{\text{90}}^{°}-g_{O_1}$	1,183	0,806
$P_1$	$t-h-{\text{90}}^{°}-g_{P_1}$	1,000	1,000
$Q_1$	$t+h-\mathrm{3s}+p-{\text{90}}^{°}-g_{Q_1}$	1,183	0,806
$M_4$	$\mathrm{4t}+\mathrm{4h}-\mathrm{4s}-g_{M_4}$	0,928	1,077
${\text{MS}}_4$	$\mathrm{4t}+\mathrm{2h}-\mathrm{2s}-g_{{\text{MS}}_4}$	0,963	1,038
$M_6$	$\mathrm{6t}+\mathrm{6h}-\mathrm{6s}-g_{M_6}$	0,894	1,118
$\text{Sa}$	$h-g_{\text{Sa}}$	1,000	1,000
$\text{SSa}$	$\mathrm{2h}-g_{\text{SSa}}$	1,000	1,000

$\begin{array}{}{\mathrm{2M}}_2\text{sin}{\varphi }_{M_2}+{\mathrm{2S}}_2\text{sin}{\varphi }_{S_2}+{\mathrm{2N}}_2\text{sin}{\varphi }_{N_2}+{\mathrm{2K}}_2\text{sin}{\varphi }_{K_2}+\\ K_1\text{sin}{\varphi }_{K_1}+O_1\text{sin}{\varphi }_{O_1}+P_1\text{sin}{\varphi }_{P_1}+Q_1\text{sin}{\varphi }_{Q_1}+\\ {\mathrm{4M}}_4\text{sin}{\varphi }_{M_4}+4{\text{MS}}_4\text{sin}{\varphi }_{{\text{MS}}_4}+{\mathrm{6M}}_6\text{sin}{\varphi }_{M_6}=0\\ {\mathrm{2M}}_2\text{sin}{\varphi }_{M_2}+{\mathrm{2N}}_2\text{sin}{\varphi }_{N_2}+{\mathrm{2K}}_2\text{sin}{\varphi }_{K_2}+K_1\text{sin}{\varphi }_{K_1}+\\ O_1\text{sin}{\varphi }_{O_1}+P_1\text{sin}{\varphi }_{P_1}+Q_1\text{sin}{\varphi }_{Q_1}+{\mathrm{4M}}_4\text{sin}{\varphi }_{M_4}+\\ 4{\text{MS}}_4\text{sin}{\varphi }_{{\text{MS}}_4}+{\mathrm{6M}}_6\text{sin}{\varphi }_{M_6}+\text{Sa}\text{sin}{\varphi }_{\text{Sa}}+2\text{SSa}\text{sin}{\varphi }_{\text{SSa}}=0\\ {\mathrm{2M}}_2\text{sin}{\varphi }_{M_2}+{\mathrm{3N}}_2\text{sin}{\varphi }_{N_2}+{\mathrm{2O}}_1\text{sin}{\varphi }_{O_1}+{\mathrm{3Q}}_1\text{sin}{\varphi }_{Q_1}+\\ {\mathrm{4M}}_4\text{sin}{\varphi }_{M_4}+2{\text{MS}}_4\text{sin}{\varphi }_{{\text{MS}}_4}+{\mathrm{6M}}_6\text{sin}{\varphi }_{M_6}=0\\ N_2\text{sin}{\varphi }_{N_2}+Q_1\text{sin}{\varphi }_{Q_1}=0\\ \begin{array}{}\}\}\}\}\}\}\}\}\end{array}\\ \end{array}$ (8)

where $M_2=f_{M_2}{H}_{M_2},S_2=f_{S_2}{H}_{S_2},\text{.}\text{.}\text{.},\text{SSa}=f_{\text{SSa}}{H}_{\text{SSa}}\text{.}$

If the approximate values of astronomical parameters corresponding to extreme condition $(t^{\text{'}},h^{\text{'}},s^{\text{'}},p^{\text{'}})$ 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 $(t_o,h_o,s_o,p_o)$ 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:

$\text{AX}+\lambda =0$, (9)

where

$\begin{array}{}\begin{array}{cccc}{a}_1& b_1& c_1& d_1\end{array}\\ \begin{array}{cccc}& b_2& c_2& d_2\end{array}\\ \begin{array}{cccc}& & c_3& d_3\end{array}\\ \begin{array}{cccc}& & & d_4\end{array}\\ \\ \mathrm{rdli}\\ \\ \\ \begin{array}{}\parallel \parallel \parallel \parallel \parallel \parallel \end{array}\\ A=\\ \end{array}$;

$\begin{array}{}{b_1={\mathrm{4M}}_2\text{cos {}\varphi }_{M_2}^{\text{'}}+{\mathrm{4N}}_2{\text{cos {}\varphi }_{N_2}^{\text{'}}\end{array}+{\mathrm{4K}}_2{\text{cos {}\varphi }_{K_2}^{\text{'}}+K_1{\text{cos {}\varphi }_{K_1}^{\text{'}}+{+O_1\text{cos {}\varphi }_{O_1}^{\text{'}}-P_1{\text{cos {}\varphi }_{P_1}^{\text{'}}+Q_1{\text{cos {}\varphi }_{Q_1}^{\text{'}}+\text{16}{M}_4{\text{cos {}\varphi }_{M_4}^{\text{'}}+{+8{\text{MS}}_4\text{cos {}\varphi }_{{\text{MS}}_4}^{\text{'}}+\text{36}{M}_6{\text{cos {}\varphi }_{M_6}^{\text{'}};$

${\varphi }_i^{\text{'}}-$ phase of the tidal constituents computed through approximate values of the astronomical parameters $t^{\text{'}},h^{\text{'}},s^{\text{'}}$ and $p^{\text{'}}\text{.}$

In order to compute the values of astronomical corresponding to extreme condition $\left(t_o,h_o,s_o,p_o\right)$ with a given accuracy the iteration method may be used. If any correction among $\left(\mathrm{\Delta t},\mathrm{\Delta h},\mathrm{\Delta s},\mathrm{\Delta p}\right)$ obtained from solving system (9) exceeds in magnitude a given value $\mid \delta \mid$ the solution will repeated and then in order to compute the coefficients of equations (9) we will use the phases ${\varphi }_i^{\text{'}\text{'}}$ computed through the values corrected of astronomical parameters:

The loop is repeated until all corrections $\left(\mathrm{\Delta t},\mathrm{\Delta h},\mathrm{\Delta s},\mathrm{\Delta p}\right)$ obtained in solution step $k$ of system (9) become less than $\mid \delta \mid$:

.

Values of astronomical parameters approximately corresponding extreme condition [6]

Semidiurnal tide
Astronomical parameters	Minimal level condition	Maximal level condition
$t^{\text{'}}$
$t_1^{\text{'}}={\text{90}}^{°}+\mathrm{0,5}{g}_{S_2}$	$t_1^{\text{'}}={\text{180}}^{°}+\mathrm{0,5}{g}_{S_2}$
$t_2^{\text{'}}={\text{270}}^{°}+\mathrm{0,5}{g}_{S_2}$	$t_2^{\text{'}}=\mathrm{0,5}{g}_{S_2}$
$h^{\text{'}}$	$\mathrm{0,5}(g_{K_2}-g_{S_2})$
$s^{\text{'}}$	$\mathrm{0,5}(g_{K_2}-g_{M_2})$
$p^{\text{'}}$	$\mathrm{0,5}(g_{K_2}-{\mathrm{3g}}_{M_2}+{\mathrm{2g}}_{N_2})$

If initial approximate values $\left(t^{\text{'}},h^{\text{'}},s^{\text{'}},p^{\text{'}}\right)$ close to real values $\left(t_o,h_o,s_o,p_o\right)$ 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: ${\varphi }_{M_2}={\varphi }_{S_2}={\varphi }_{N_2}={\varphi }_{K_2}=\varphi$.

－ For diurnal tide: ${\varphi }_{K_1}={\varphi }_{O_1}={\varphi }_{P_1}={\varphi }_{Q_1}=\varphi$,

where $\varphi ={\text{180}}^{°}+\mathrm{2\pi n}-$ for the lowest level and $\varphi ={\text{360}}^{°}+\mathrm{2\pi n}-$ for the highest level.

From these expressions follow the formulae for computing the approximate values of astronomical parameters $\left(t^{\text{'}},h^{\text{'}},s^{\text{'}},p^{\text{'}}\right)$ corresponding the extreme conditions (tables 3－6).

In order to compute approximate value of average zone time $t^{\text{'}}$ 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 $B$ and $C$ (table 4). Coefficients $B$ and $C$ are computed by following formulae:

$\begin{array}{}B=O_1\text{cos}{\alpha }_1+P_1\text{cos}{\alpha }_2+Q_1\text{cos}{\alpha }_3+K_1\text{cos}{\alpha }_4\\ C=O_1\text{sin}{\alpha }_1+P_1\text{sin}{\alpha }_2+Q_1\text{sin}{\alpha }_3+K_1\text{sin}{\alpha }_4\\ \begin{array}{}\}\end{array}\\ \end{array}$ (10)

$\begin{array}{}{\alpha }_1=g_{M_2}-\mathrm{0,5}{g}_{K_2}-g_{O_1}-{\text{90}}^{°};{\alpha }_3=g_{N_2}-\mathrm{0,5}{g}_{K_2}-g_{Q_1}-{\text{90}}^{°};\\ {\alpha }_2=g_{S_2}-\mathrm{0,5}{g}_{K_2}-g_{P_1}-{\text{90}}^{°};{\alpha }_4=\mathrm{0,5}{g}_{K_2}-g_{K_1}+{\text{90}}^{°}\text{.}\\ \begin{array}{}\}\end{array}\\ \end{array}$ (11)

Conditions of the lowest and highest level [6]

Conditions of the lowest level	Conditions of the highest level
$t_1^{\text{'}}$ when $C>0$$t_2^{\text{'}}$ when	$t_1^{\text{'}}$ when $t_2^{\text{'}}$ when $B>0$

Values of astronomical parameters approximately corresponding to extreme condition [6]

Diurnal tide
Astronomical parameters	Conditions of the lowest level	Conditions of the highest level
$t^{\text{'}}$	$\mathrm{0,5}(g_{K_1}-g_{P_1})$	$\mathrm{0,5}(g_{K_1}+g_{P_1})+{\text{180}}^{°}$
$h^{\text{'}}$	$\mathrm{0,5}(g_{K_1}-g_{P_1})+{\text{90}}^{°}$
$s^{\text{'}}$	$\mathrm{0,5}(g_{K_1}-g_{O_1})+{\text{90}}^{°}$
$p^{\text{'}}$	$\mathrm{0,5}(g_{K_1}-{\mathrm{3g}}_{O_1}+{\mathrm{2g}}_{Q_1})+{\text{90}}^{°}$

The choice of reduce coefficients to compute values $\text{fH}$ is depended on the tide feature:

1) For semidiurnal tide, if then $f$ is chosen for $N={\text{180}}^{°}$;

2) For diurnal tide, if $\left(H_{K_1}+H_{O_1}\right)H_{M_2}\mathrm{1,5}$ then $f$ is chosen for $N=0^{°}$;

3) For mixed tide, if 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 $f$ for $N=0^{°}$, when compute with astronomical parameters of semidiurnal tide choose $f$ for $N=0^{°}$ and $N={\text{180}}^{°}$. 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 ${\varphi }_{K_1}$ in the interval from 0° to 360°:

$\begin{array}{}H=K_1\text{cos}{\varphi }_{K_1}+K_2\text{cos}({\mathrm{2\varphi }}_{K_1}+a_4)+\mid R_1+R_2+R_3\mid \\ L=K_1\text{cos}{\varphi }_{K_1}+K_2\text{cos}({\mathrm{2\varphi }}_{K_1}+a_4)-\mid R_1+R_2+R_3\mid \\ \begin{array}{}\}\end{array}\end{array}$ (12)

where

The choice of reduce coefficients to compute values $\text{fH}$ is also made as the above recommendations, i. e. with the semidiurnal tide $f$ is chosen for $N={\text{180}}^{°}$, with diurnal tide $f$ is chosen for $N=0^{°}$. With mixed tide the computation is performed with $f$ for $N={\text{180}}^{°}$ and $N=0^{°}$ 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 $({\varphi }_{K_1}{)}_{\text{min}}$ and $({\varphi }_{K_1}{)}_{\text{max}}$ from analyzing (12) to compute the astronomical parameters corresponding extreme conditions $t,h,s,p$ and use them as the approximations to compute the coefficients of equations (9).

The conditions of the lowest level:

and the conditions of the highest level:

where

The last extreme values of tide with arbitrary number of constituents are determined from equation (7) using the values of the astronomical parameters $t_o,h_o,s_o,p_o$ corrected by the iteration method.

However, it is worth to make a note that computation of approximate values of astronomical parameters $t^{\text{'}},h^{\text{'}},s^{\text{'}},p^{\text{'}}$ by formulae in tables 3 and 5 is much simpler than Vladimirsky method when the computation involve more than 8 tide constituents

Thus the procedure of computing extreme levels may be performed due to two following schemes:

1) Regardless what is the number of tidal constituents, due to formulae in tables 3 and 6 determine the approximate values of the astronomical parameters correspondig to extreme conditions, than correct these values by the iteration method. Compute the extreme levels by equation (7).

2) Compute the extreme levels with 8 tidal constituents by Vladimirsky method. If the number of constituents is bigger 8 then compute the approximate values of astronomical parameters of extreme condition for 8 constituents by Vladimirsky method and correct them by the iteration method. Compute the extreme levels by equation (7).

In some cases when shallow water constituents have such a big magnitude that causes the approximate values of astronomical parameters computed by formulae in tables 3 and 5 or by Vladimirsky method insufficiently closed to real values $\left(t_o,h_o,s_o,p_o\right)$ to meet the convergence of the iteration process in solution system (7). If a given number of iteration steps (for example, 8 or 10 steps) do not provide the convergence of the results, one may use the method of consequent approximations.

Each shallow water constituent of big magnitude is decomposed into a number of constituents of the smaller magnitude:

$f_i{H}_i\text{cos}{\varphi }_i=n\frac{1}{n}{f}_i{H}_i\text{cos}{\varphi }_i\text{.}$ (13)

The number $n$ is stated depending on the magnitude of tidal constituents. The solution is performed in some steps, each step include all the calculations to correct the values of astronomical parameters, i. e. forming and solving system (9) and iterative calculations as when to correct the astronomical extreme conditions at a given step. For solving in first step astronomical parameters are computed by the formulae in tables 3 and 5 or by Vladimirsky method; further, their values obtained in each steps are used as initial values for the next step. The magnitude of shallow water constituents is increased from step to step to the full magnitude (for example, in first approximation step the magnitude is chosen as $\frac{1}{n}{f}_i{H}_i$, in second step － $\frac{2}{n}{f}_i{H}_i$, in step $n$ － $f_i{H}_i$.