Hölder Scales of Sea Level

The statistics of sea level is essential in the field of geosciences, ranging from ocean dynamics to climates. The fractal properties of sea level, such as long-range dependence LRD or longmemory, 1/f noise behavior, and self-similarity SS , are known. However, the description of its multiscale behavior as well as local roughness with the Hölder exponent h t from a view of multifractional Brownian motion mBm is rarely reported, to the best of our knowledge. In this research, we will exhibit that there is the multiscale property of sea level based on h t s of sea level data recorded by the National Data Buoy Center NDBC at six stations in the Florida and Eastern Gulf of Mexico. The contributions of this paper are twofold as follows. i Hölder exponent of sea level may not change with time considerably at small time scale, for example, daily time scale, but it varies significantly at large time scale, such as at monthly time scale. ii The dispersion of the Hölder exponents of sea level may be different at different stations. This implies that the Hölder roughness of sea level may be spatial dependent.


Introduction
The study of sea level fluctuations plays a role in geosciences 1-3 .There are two categories of time scales of sea level.One is for yearly data with time scales in one yr, or 10 yr, or more; see, for example, 4-16 .The other is about data with time scales hourly, daily, weekly, or monthly; see, for example, 17-39 .The former generally relates to the study of trend of relative mean sea level with respect to global and Earth or planetary changes, for example, in the filed of climates, while the latter is usually associated with the research of local dynamics of sea level in the aspects of navigations, coastal engineering, tide power production, ship design, and so forth.Our research uses the hourly sea level data recorded by NDBC 40 .
In addition to LRD, there is another essential property of processes in geosciences, called self-similarity SS ; see, for example, 67-84 .By SS, we mean that a random function x t satisfies the property given by x t a H x at , ∀a > 0, t > 0, 1.2 where is the equality in distribution, H ∈ 0, 1 is the Hurst parameter that measures SS, and a is a scale 61, 83-86 .Note that the term SS implies the roughness or irregularity of a random function 86 .If x t satisfies 1.2 , it is globally self-similar.That is, its irregularity characterized by H keeps the same for all t > 0 87 , corresponding the case of monofractal 88, 89 .
Since the global SS implies the same value of H for all t, it may be too restrictive to describe real data in engineering and sciences to use a monofractal model.Therefore, multifractal models are desired in various fields of sciences and engineering; see, for example, 85-92 and references therein, including those in geosciences; see, for example, 93-110 , just citing a few.From a view of multifractal, a random function that is not self-similar may be of local self-similarity LSS .
There are several ways of describing multifractality of a random function based on various definitions of dimensions, such as the Minkowski dimension, the Rényi dimension, the Hausdorff dimension, the packing dimension, the box-counting dimension, and the correlation dimension 86, 89, 90, 111-114 .In this paper, we adopt the H ölder exponent 0 < h t < 1 in multifractional Brownian motion mBm introduced by Peltier and Levy-Vehel 115 .Taking into account h t in mBm, therefore, one may use the following: x t a h t x at , ∀a > 0, t > 0, 1.3 to characterize the LSS property of a locally self-similar random function x t on a point-bypoint basis.We call the LSS or local roughness characterized by h t the H ölder roughness in this paper.The applications of h t attract increasing interests of researchers in sciences and technologies, ranging from teletraffic to geophysics; see, for example, 116-134 , simply mentioning a few.This paper aims at investigating the H ölder multiscales H ölder scales for short of sea level.By H ölder scales, we mean the time scales described by the H ölder exponents in mBm.The contributions of this paper are in two aspects.On the one hand, we will reveal that variations of h t of sea level may be indistinctively at small time scale, for example, daily time scale, but h t of sea level varies significantly at large time scale, such as at monthly time scale.On the other hand, we will exhibit that the dispersion of the H ölder exponents of sea level may usually be spatial dependent.The remaining paper is organized as follows.Data used in this research are briefed in Section 2. The method for describing the H ölder exponent in mBm is explained in Section 3. Results of data processing and discussions are given in Section 4, which is followed by conclusions.

Data
NDBC is a part of the US National Weather Service NWS 135 .It provides scientists with data for their scientific research, including significant wave height and water level 136 .We use the data measured at stations named LKWF1, LONF1, SAUF1, SMKUF1, SPGF1, and VENF1, respectively.In terms of the names of measurement stations, LKWF1 implies the station at Lake Worth, FL 137 ; the station LONF1 is the one at Long Key, FL 138 ; the station SAUF1 is at St. Augustine, FL 139 ; SMKUF1 is the station at Sombrero Key, FL 140 ; SPGF1 is at Settlement Point, GBI 141 ; and VENF1 is at Venice, FL 142 .They are located in the Florida and Eastern Gulf of Mexico.The data are under the directory of Water Level, which are publicly accessible 143 , referring Gilhousen 144 as an instance of research using the data by NDBC.
All data were hourly recorded with ten separate devices indexed by TGn n 01, 02, . . ., 10 .Without losing generality, this research utilizes the data from the device TG01.Denote the data series by x s yyyy t , where s is the name of the measurement station and yyyy stands for the index of year.Denote by h s yyyy t its corresponding h t at the station s in the year of yyyy.For example, x lkwf1l 2002 t and h lkwf1l 2002 t , respectively, represent the measured sea level time series and its h t at the station LKWF1 in 2002.
If the recorded data are labeled by 99, they are taken as outliers, which are not involved in the computations.In this case, they are replaced with the mean of that series.NDBC suggests that 10 ft should be subtracted from every level series x s yyyy t 145 .By taking into account this suggestion in the computation of h t , we modify x s yyyy t by subtracting 10 ft and denote y s yyyy t modified data of sea level.That is, y s yyyy t x s yyyy t − 10.

Methodology
Let B t be the standard Brownian motion.Then, B t satisfies the following properties.
i The increments B τ t − B t are Gaussian.
ii E B τ t − B t 0 and iii If the first item on the right hand of 3.3 is taken as the zero-input response of the system that generates B H t for t > 0, we may regard the fBm as the convolution of the impulse function t H−1/2 /Γ H 1/2 and dB t /dt 169 .Therefore, 3.3 may be rewritten by where * is the operator of convolution and 3.5 It may be interesting to note that t H−1/2 /Γ H 1/2 is a special case of the operators of fractional order discussed by Mikusinski 170, Equation 59.1 .
The function B H t has the following properties.
i B H 0 0.
ii The increments B H t t 0 − B H t 0 are Gaussian.
iii Its structure function is given by where In addition, it satisfies the self-similarity expressed by 1.2 , which implies that B H t is globally self-similar.Consequently, there is a limitation that its self-similarity or roughness keeps the same for all t > 0. To release such a limitation, one may adopt the tool of the mBm equipped with the H ölder exponent h t ; see, for example, 115, 119, 133 .In fact, the mBm is a generalization of fBm by replacing the Hurst parameter H in 3.3 with a continuous function h t that satisfies H : 0, ∞ → 0, 1 ; see 87, 115-134, 171-182 .Denote the mBm by X t .Then, Considering the local growth of the increment process of X t , one may write a sequence given by where m is the largest integer not exceeding N/k.Then, h t at point t j/ N − 1 is given by The above is the expression of applying mBm to investigate h t of sea level time series, which measures the H ölder roughness of sea level on a point-by-point basis.

Observations and Discussions
We demonstrate h t s of sea level series x smkf1 2008 t at the time scales of day, week, and month, respectively.

H ölder Roughness at Daily Time Scale
Figure 1 indicates 4 daily series of sea level at the station SMKUF1 from Jan. 1 to Jan. 4 in 2008.Figure 2 demonstrates their corresponding H ölder exponents.From Figure 2, we see that 4h t s of daily series of sea level vary with time insignificantly.Therefore, we obtain the remark below.
Remark 4.1.The H ölder exponents of sea level at the daily time scale, that is, 24 hours, may not vary significantly.This may imply that h t ≈ h t τ if τ ≤ 24 hours.Remark 4.2.The H ölder exponents of sea level at the weekly time scale, that is, 168 hours, may not vary considerably enough.

H ölder Roughness at Monthly Time Scale
Figure 5 illustrates 4 monthly series of sea level at the station SMKUF1 in 2008.Their corresponding H ölder exponents are indicated in Figure 6.From Figure 6, we see the following.
Remark 4.3.The H ölder exponents of sea level at the monthly time scale vary with time significantly.

Variation of H ölder Roughness at Large Time Scale
We now investigate the H ölder exponents of sea level at large time scale.By large time scale, we mean that the scale is around month or larger.Figure 7  We summarize the variances of the H ölder exponents of test data in Tables 7, 8, 9, 10, 11 and 12.

Discussions
Generally, the H ölder exponents of sea level series are time varying.They are considerably at large time scales but insignificantly at small time scales.In addition, their variations are in general spatial dependent as the Tables 7-12 exhibit.For instance, in 2002, Var h t varies, in the form of magnitude of order, from 10 −3 to 10 −4 at different stations.This motivates us to take the spatial-time modeling of H ölder roughness of sea level as our possible future work.Finally, we note that the meaning of the term of local roughness of a random function is the same as that of local self-similarity 60, 65, 86 .Thus, according to 1.2 , Remarks 4.1-4.3exhibit the self-similarity of sea level at small and large time scales, respectively.

Conclusions
We have presented our results in the H ölder exponents of sea level in the Florida and Eastern Gulf of Mexico.The present results reveal an interesting phenomenon of time scales of sea level.To be precise, the H ölder exponents of sea level may not vary considerably at small time scales, such as daily time scale, but vary with time significantly at large time scale, such as monthly time scale.Moreover, our research exhibits that variations of the H ölder exponents of sea levels may be spatial dependent.Though the research is with the data in Florida and Eastern Gulf of Mexico, the results may be useful for further exploring general properties of the H ölder scales and roughness of sea level. 0

Four
weekly series of sea level at the station SMKUF1 in Jan. 2008 are shown in Figure 3. Their corresponding H ölder exponents are plotted in Figure 4.They appear monotonically increase, see Figures 4 b and 4 d , or decrease, see Figures 4 a and 4 c .In general, they imply the following remark.

Figure 3 :
Figure 3: Weekly sea level at the station SMKUF1 on January 2008.a .x smkf1 2008 t in the 1st week in Jan. 2008.b .x smkf1 2008 t in the 2nd week in Jan. 2008.c .x smkf1 2008 t in the 3rd week on January 2008.d .x smkf1 2008 t in the 4th week in Jan. 2008.
Kolmogorov introduced a class of random functions the covariance function of which is now recognized as the one of fractional Brownian motion fBm 146, Theorem 6 .Note that, for a random function x t , the function f τ expressed by Matérn 147, page 51 .It is usually called variogram in geosciences 148-157 .In the field of fluid mechanics, it is named structure function 158-161 .Yaglom derived fBm based on the theory of structure functions 162 .In this paper, we use the fBm introduced by Bandelbrot and van Ness based on fractional calculus 163 .It is well known that B t is nondifferentiable in the domain of ordinary functions 164-166 .In the domain of generalized functions, however, it is differentiable 167, 168 .Denote the fBm by B H t .Based on the Weyl's fractional derivative or integral 163 , it is expressed by In nonoverlapping intervals t 1 , t 2 and t 3 , t 4 , the increments B t 4 -B t 3 and B t 2 -B t 1 are independent.iv B 0 0 and B t is continuous at t 0.
d Figure 2 a indicates the sea level series The variances of the H ölder exponents of sea level at different observation stations may be considerably different.Monthly sea level at the station SMKUF1 in 2008.a .x smkf1 2008 t on January 2008.b .x smkf1 2008 t in March 2008.c .x smkf1 2008 t on July 2008.d .x smkf1 2008 t on October 2008.
d Figure 4: H ölder exponents of weekly sea level at the station SMKUF1 in January 2008.a .h smkf1 2008 t in the 1st week on January 2008.b .h smkf1 2008 t in the 2nd week in Jan. 2008.c .h smkf1 2008 t in the 3rd week in Jan. 2008.d .h smkf1 2008 t in the 4th week in Jan. 2008.

Table 7 :
Variances of the H ölder exponents at LKWF1.

Table 8 :
Variances of the H ölder exponents at LONF1.

Table 9 :
Variances of the H ölder exponents at SAUF1.

Table 10 :
Variances of the H ölder exponents at SMKF1.

Table 11 :
Variances of the H ölder exponents at SPGF1.

Table 12 :
Variances of the H ölder exponents at VENF1.