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 long memory, 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.
1. 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].
Since the pioneering work of Hurst on time series with long-range dependence (LRD) is observed in the Nile Basin [41], the LRD property of time series in geosciences has been widely observed; see, for example, [42–59]. By LRD, one means that the covariance function C(τ) of time series x(t) decays so slowly such that
(1.1)∫0∞C(τ)dτ=∞,
where τ is time lag and C(τ)=E[x(t+τ)x(t)]. Therefore, LRD is a global property of time series [60–66].
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
(1.2)x(t)≜aHx(at),∀a>0,t>0,
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 Hölder 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:
(1.3)x(t)≜ah(t)x(at),∀a>0,t>0,
to characterize the LSS property of a locally self-similar random function x(t) on a point-by-point basis. We call the LSS or local roughness characterized by h(t) the Hölder 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 Hölder multiscales (Hölder scales for short) of sea level. By Hölder scales, we mean the time scales described by the Hölder 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 Hölder 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 Hölder exponent in mBm is explained in Section 3. Results of data processing and discussions are given in Section 4, which is followed by conclusions.
2. 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,
(2.1)y_s_yyyy(t)=x_s_yyyy(t)-10.
Tables 1, 2, 3, 4, 5 and 6 list those data.
Measured data at LKWF1.
Series name
Record date and time
L (record length)
x_lkwf1_1996(t)
0:00, 1 Jan.–23:00, 31 Dec. 1996
8208
x_lkwf1_1997(t)
0:00, 1 Jan.–23:00, 31 Dec. 1997
7776
x_lkwf1_1998(t)
0:00, 1 Jan.–23:00, 31 Dec. 1998
8736
x_lkwf1_1999(t)
0:00, 1 Jan.–23:00, 31 Dec. 1999
8760
x_lkwf1_2000(t)
0:00, 1 Jan.–17:00, 26 Feb. 2000
1362
x_lkwf1_2001(t)
17:00, 8 Aug.–23:00, 31 Dec. 2001
2972
x_lkwf1_2002(t)
0:00, 1 Jan.–23:00, 31 Dec. 2002
8740
x_lkwf1_2003(t)
0:00, 1 Jan.–23:00, 31 Dec. 2003
8582
x_lkwf1_2004(t)
0:00, 1 Jan.–14:00, 5 Oct. 2004
6655
Measured data at LONF1.
Series name
Record date and time
L (record length)
x_lonf11_1998(t)
0:00, 3 Nov.–23:00, 31 Dec. 1998
1416
x_lonf1_1999(t)
0:00, 1 Jan.–21:00, 31 Dec. 1999
8757
x_lonf1_2000(t)
0:00, 1 Jan.–23:00, 31 Dec. 2000
8484
x_lonf1_2001(t)
0:00, 1 Jan.–23:00, 31 Dec. 2001
8760
x_lonf1_2002(t)
0:00, 1 Jan.–23:00, 31 Dec. 2002
8760
x_lonf1_2003(t)
0:00, 1 Jan.–23:00, 31 Dec. 2003
8697
x_lonf1_2004(t)
0:00, 1 Jan.–23:00, 31 Dec. 2004
8758
x_lonf1_2005(t)
0:00, 1 Jan.–23:00, 31 Dec. 2005
8750
x_lonf1_2006(t)
0:00, 1 Jan.–23:00, 31 Dec. 2006
8735
x_lonf1_2007(t)
0:00, 1 Jan.–23:00, 31 Dec. 2007
8692
x_lonf1_2008(t)
0:00, 1 Jan.–21:00, 19 Jan. 2008
444
Measured data at SAUF1.
Series name
Record date and time
L (record length)
x_sauf1_1996(t)
0:00, 1 Jan.–14:00, 10 Aug. 1996
5511
x_sauf1_1997(t)
0:00, 25 Feb.–23:00, 31 Dec. 1997
6240
x_sauf1_1998(t)
0:00, 1 Jan.–23:00, 31 Dec. 1998
8736
x_sauf1_1999(t)
0:00, 1 Jan.–23:00, 31 Dec. 1999
8136
x_sauf1_2000(t)
0:00, 1 Jan.–23:00, 31 Dec. 2000
8715
x_sauf1_2001(t)
0:00, 1 Jan.–21:00, 31 Dec. 2001
8758
x_sauf1_2002(t)
20:00, 6 Feb.–23:00, 20 Aug. 2002
4684
Measured data at SMKF1.
Series name
Record date and time
L (record length)
x_smkf1_1998(t)
0:00, 3 Nov.–23:00, 31 Dec. 1998
1416
x_smkf1_1999(t)
0:00, 1 Jan.–23:00, 31 Dec. 1999
7775
x_smkf1_2000(t)
0:00, 1 Aug.–23:00, 31 Dec. 2000
3542
x_smkf1_2001(t)
0:00, 1 Jan.–23:00, 31 Dec. 2001
5776
x_smkf1_2002(t)
0:00, 1 Jan.–23:00, 31 Dec. 2002
8742
x_smkf1_2003(t)
0:00, 1 Jan.–23:00, 31 Dec. 2003
5851
x_smkf1_2004(t)
0:00, 1 Jan.–23:00, 31 Dec. 2004
8439
x_smkf1_2005(t)
0:00, 1 Jan.–23:00, 31 Dec. 2005
8667
x_smkf1_2006(t)
0:00, 1 Jan.–23:00, 31 Dec. 2006
8623
x_smkf1_2007(t)
0:00, 1 Jan.–23:00, 31 Dec. 2007
8702
x_smkf1_2008(t)
0:00, 1 Jan.–23:00, 31 Dec. 2008
8679
x_smkf1_2009(t)
0:00, 1 Jan.–23:00, 31 Dec. 2009
8109
x_smkf1_2010(t)
0:00, 1 Jan.–23:00, 31 July 2010
5074
x_smkf1_2011(t)
0:00, 1 Jan.–23:00, 31 Dec. 2011
8759
Measured data at SPGF1.
Series name
Record date and time
L (record length)
x_spgf1_1996(t)
0:00, 1 Jan.–23:00, 15 Dec. 1996
8616
x_spg1_1997(t)
0:00, 6 Mar.–23:00, 15 Dec. 1997
7080
x_spg1_1998(t)
0:00, 1 Jan.–23:00, 7 Jan. 1998
168
Measured data at VENF1.
Series name
Record date and time
L (record length)
x_venf1_2002(t)
0:00, 1 Oct.–23:00, 31 Dec. 2002
2208
x_ven1_2003(t)
0:00, 1 Jan.–23:00, 31 Dec. 2003
8760
x_ven1_2004(t)
0:00, 1 Jan.–16:00, 7 Jan. 2004
634
x_ven1_2006(t)
14:00, 22 July–23:00, 31 Dec. 2006
3882
x_ven1_2007(t)
0:00, 1 Jan.–23:00, 31 Dec. 2007
8663
x_ven1_2008(t)
0:00, 1 Jan.–23:00, 31 Oct. 2008
7189
3. Methodology
Let B(t) be the standard Brownian motion. Then, B(t) satisfies the following properties.
The increments B(τ+t)-B(t) are Gaussian.
E[B(τ+t)-B(t)]=0 and
(3.1)Var[B(t+τ)-B(t)]=σ2τ,where σ2=E{[B(t+1)-B(t)]2}=E{[B(1)-B(0)]2}=E{[B(1)]2}.
In nonoverlapping intervals [t1, t2] and [t3, t4], the increments B(t4)-B(t3) and B(t2)-B(t1) are independent.
B(0)=0 and B(t) is continuous at t=0.
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
(3.2)f(τ)=Var[x(t+τ)-x(t)]=E{[x(t+τ)-x(t)]2}
is termed serial variation function; see, for example, 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 BH(t). Based on the Weyl’s fractional derivative or integral [163], it is expressed by
(3.3)BH(t)-BH(0)=1Γ(H+1/2){∫-∞0[(t-u)H-0.5-(-u)H-0.5]dB(u)+∫0t(t-u)H-0.5dB(u)}.
If the first item on the right hand of (3.3) is taken as the zero-input response of the system that generates BH(t) for t>0, we may regard the fBm as the convolution of the impulse function tH-1/2/Γ(H+1/2) and dB(t)/dt [169]. Therefore, (3.3) may be rewritten by
(3.4)BH(t)-BH(0)=B0(u)+tH-0.5Γ(H+1/2)*dB(t)dt,
where * is the operator of convolution and
(3.5)B0(u)=1Γ(H+1/2)∫-∞0[(t-u)H-0.5-(-u)H-0.5]dB(t).
It may be interesting to note that tH-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 BH(t) has the following properties.
BH(0)=0.
The increments BH(t+t0)-BH(t0) are Gaussian.
Its structure function is given by
(3.6)Var[BH(t+τ)-BH(t)]=σ2τ2H,
where σ2=E{[BH(t+1)-BH(t)]2}=E{[BH(1)-BH(0)]2}=E{[BH(1)]2}.
In addition, it satisfies the self-similarity expressed by (1.2), which implies that BH(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 Hölder 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,
(3.7)X(t)=1Γ(h(t)+1/2){∫-∞0[(t-u)h(t)-0.5-(-u)h(t)-0.5]dB(u)+∫0t(t-u)h(t)-0.5dB(u)}.
Considering the local growth of the increment process of X(t), one may write a sequence given by
(3.8)Sk(j)=mN-1∑j=0j+k|X(i+1)-X(i)X(i+1)mN-1|,1<k<N,
where m is the largest integer not exceeding N/k. Then, h(t) at point t=j/(N-1) is given by
(3.9)h(t)=-log(π/2Sk(j))log(N-1).
The above is the expression of applying mBm to investigate h(t) of sea level time series, which measures the Hölder roughness of sea level on a point-by-point basis.
4. 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.
4.1. Hölder 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 Hölder 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.
Daily sea level at the station SMKUF1 from January 1 to Jan. 4 in 2008. (a). x_smkf1_2008(t) on Jan. 1, 2008. (b). x_smkf1_2008(t) on Jan. 2, 2008. (c). x_smkf1_2008(t) on Jan. 3, 2008. (d). x_smkf1_2008(t) on Jan. 4, 2008.
Hölder exponents of daily sea level at the station SMKUF1 from Jan. 1 to Jan. 4 in 2008. (a). h_smkf1_2008(t) on Jan. 1, 2008. (b). h_smkf1_2008(t) on Jan. 2, 2008. (c). h_smkf1_2008(t) on Jan. 3, 2008. (d). h_smkf1_2008(t) on Jan. 4, 2008.
Remark 4.1.
The Hölder 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.
4.2. Hölder Roughness at Weekly Time Scale
Four weekly series of sea level at the station SMKUF1 in Jan. 2008 are shown in Figure 3. Their corresponding Hölder 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.
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.
Hölder 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.
Remark 4.2.
The Hölder exponents of sea level at the weekly time scale, that is, 168 hours, may not vary considerably enough.
4.3. Hölder Roughness at Monthly Time Scale
Figure 5 illustrates 4 monthly series of sea level at the station SMKUF1 in 2008. Their corresponding Hölder exponents are indicated in Figure 6. From Figure 6, we see the following.
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.
Hölder exponents of month sea level at the station SMKUF1 in 2008. (a). h_smkf1_2008(t) on Jan. 2008. (b). h_smkf1_2008(t) on March 2008. (c). h_smkf1_2008(t) in July 2008. (d). h_smkf1_2008(t) in October 2008.
Remark 4.3.
The Hölder exponents of sea level at the monthly time scale vary with time significantly.
4.4. Variation of Hölder Roughness at Large Time Scale
We now investigate the Hölder exponents of sea level at large time scale. By large time scale, we mean that the scale is around month or larger. Figure 7(a) indicates the sea level series x_smkf1_2008(t), Figure 7(b) shows its Hölder exponent, and Figure 7(c)the histogram of its Hölder exponent.
Illustrations of h_smkf1_2008(t) and its Hölder exponent. (a). x_smkf1_2008(t). (b) Hölder exponent h_smkf1_2008(t). (c). Histogram of h_smkf1_2008(t).
One thing worth noting is that variances of Hölder exponents of sea level at different stations may be considerably different. For instance,
(4.1)Var[h_smkf1_2008(t)]=1.203×10-3,Var[h_lonf1l2005(t)]=6.425×10-4.
The above implies that the variance of h_smkf1_2008(t) is larger than that of h_lonf1l2005(t) in one magnitude of order. Consequently, comes the following remark.
Remark 4.4.
The variances of the Hölder exponents of sea level at different observation stations may be considerably different.
We summarize the variances of the Hölder exponents of test data in Tables 7, 8, 9, 10, 11 and 12.
Variances of the Hölder exponents at LKWF1.
Series name
Var[h(t)]
x_lkwf1_1996(t)
1.217×10-3
x_lkwf1_1997(t)
1.006×10-3
x_lkwf1_1998(t)
9.499×10-4
x_lkwf1_1999(t)
1.164×10-3
x_lkwf1_2000(t)
5.901×10-4
x_lkwf1_2001(t)
1.169×10-3
x_lkwf1_2002(t)
8.939×10-4
x_lkwf1_2003(t)
9.710×10-4
x_lkwf1_2004(t)
9.361×10-4
Variances of the Hölder exponents at LONF1.
Series name
Var[h(t)]
h_lonf11_1998(t)
3.978×10-3
h_lonf1_1999(t)
4.123×10-4
h_lonf1_2000(t)
1.570×10-3
h_lonf1_2001(t)
1.135×10-3
h_lonf1_2002(t)
1.407×10-3
h_lonf1_2003(t)
2.359×10-3
h_lonf1_2004(t)
9.493×10-4
h_lonf1_2005(t)
6.425×10-4
h_lonf1_2006(t)
1.245×10-3
h_lonf1_2007(t)
2.142×10-3
h_lonf1_2008(t)
8.245×10-5
Variances of the Hölder exponents at SAUF1.
Series name
Var[h(t)]
x_sauf1_1996(t)
1.083×10-3
x_sauf1_1997(t)
1.355×10-3
x_sauf1_1998(t)
8.766×10-4
x_sauf1_1999(t)
1.324×10-3
x_sauf1_2000(t)
7.272×10-4
x_sauf1_2001(t)
6.961×10-4
x_sauf1_2002(t)
3.992×10-3
Variances of the Hölder exponents at SMKF1.
Series name
Var[h(t)]
x_smkf1_1998(t)
9.501×10-4
x_smkf1_1999(t)
1.144×10-3
x_smkf1_2000(t)
1.310×10-3
x_smkf1_2001(t)
1.520×10-3
x_smkf1_2002(t)
1.181×10-3
x_smkf1_2003(t)
1.176×10-3
x_smkf1_2004(t)
1.243×10-3
x_smkf1_2005(t)
1.210×10-3
h_smkf1_2006(t)
1.101×10-3
h_smkf1_2007(t)
1.164×10-3
h_smkf1_2008(t)
1.203×10-3
h_smkf1_2009(t)
1.242×10-3
h_smkf1_2010(t)
1.084×10-3
h_smkf1_2011(t)
1.176×10-3
Variances of the Hölder exponents at SPGF1.
Series name
Var[h(t)]
x_spgf1_1996(t)
1.018×10-3
x_spgf1_1997(t)
8.803×10-4
x_spgf1_1998(t)
2.659×10-4
Variances of the Hölder exponents at VENF1.
Series name
Var[h(t)]
x_venf1_2002(t)
1.069×10-3
x_venf1_2003(t)
1.271×10-3
x_venf1_2004(t)
8.863×10-4
x_venf1_2006(t)
2.268×10-3
x_venf1_2007(t)
2.454×10-3
x_venf1_2008(t)
2.930×10-3
4.5. Discussions
Generally, the Hölder 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 Hölder 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 4.1–4.3 exhibit the self-similarity of sea level at small and large time scales, respectively.
5. Conclusions
We have presented our results in the Hölder 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 Hölder 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 Hölder 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 Hölder scales and roughness of sea level.
Acknowledgments
This work was supported in part by the 973 Plan under the Project no. 2011CB302800, and the National Natural Science Foundation of China under the project Grants nos. 61272402, 61070214, and 60873264. The National Data Buoy Center is highly appreciated for its measured data that makes our research possible.
Geophysics Study CommitteeNational Research Council1990National Academic PressCommittee on Engineering Implications of Changes in Relative Mean Sea LevelMarine BoardNational Research Council1987National Academic PressGornitzV.CouchS.HartigE. K.Impacts of sea level rise in the New York City metropolitan area200132161882-s2.0-003570285910.1016/S0921-8181(01)00150-3JevrejevaS.MooreJ. C.GrinstedA.Sea level projections to AD2500 with a new generation of climate change scenarios201280-811420BeckerM.MeyssignacB.LetetrelC.LlovelW.CazenaveA.DelcroixT.Sea level variations at tropical Pacific islands since 1950201280-818598MeyssignacB.CazenaveA.Sea level: a review of present-day and recent-past changes and variability20125896109LongA. J.WoodroffeS. A.MilneG. A.BryantC. L.SimpsonM. J. R.WakeL. M.Relative sea-level change in Greenland during the last 700 yrs and ice sheet response to the Little Ice Age2012315-31676852-s2.0-7996088292610.1016/j.epsl.2011.06.027EliotM.Sea level variability influencing coastal flooding in the Swan River region, Western Australia2012331428WarnerN. N.TissotP. E.Storm flooding sensitivity to sea level rise for Galveston Bay, Texas2012442332OzyavasA.KhanS. D.The driving forces behind the Caspian Sea mean water level oscillations2012656182118302-s2.0-7995998554010.1007/s12665-011-1163-0HinkelJ.BrownS.ExnerL.NichollsR. J.VafeidisA. T.KebedeA. S.Sea-level rise impacts on Africa and the effects of mitigation and adaptation: an application of DIVA2012121207224GassonE.SiddallM.LuntD. J.RackhamO. J. L.LearC. H.PollardD.Exploring uncertainties in the relationship between temperature, ice volume, and sea level over the past 50 million years2012501RG1005WopelmannG.MarcosM.Coastal sea level rise in southern Europe and the nonclimate contribution of vertical land motion20121171C01007GregoryJ. M.ChurchJ. A.BoerG. J.DixonK. W.FlatoG. M.JackettD. R.LoweJ. A.O'FarrellS. P.RoecknerE.RussellG. L.StoufferR. J.WintonM.Comparison of results from several AOGCMs for global and regional sea-level change 1900—21002001183-42232402-s2.0-0035669373AlbrechtF.WahlT.JensenJ.WeisseR.Determining sea level change in the German Bight20116112203720502-s2.0-7996028757210.1007/s10236-011-0462-zRayR. D.DouglasB. C.Experiments in reconstructing twentieth-century sea levels2011914496515BarberN. F.UrsellF.The generation and propagation of ocean waves and swell1948240824527560Bitner-GregersenE. M.GranS.Local properties of sea waves derived from a wave record1983542102142-s2.0-0020830040VeltchevaA.CavacoP.SoaresC. G.Comparison of methods for calculation of the wave envelope20033079379482-s2.0-003740859410.1016/S0029-8018(02)00069-0SoaresC. G.ChernevaZ.Spectrogram analysis of the time-frequency characteristics of ocean wind waves20053214-15164316632-s2.0-2214446622410.1016/j.oceaneng.2005.02.008BluesteinH. B.A review of ground-based, mobile, W-band Doppler-radar observations of tornadoes and dust devils20054031631882-s2.0-2044448695410.1016/j.dynatmoce.2005.03.004BaxevaniA.RychlikI.WilsonR. J.A new method for modelling the space variability of significant wave height2005842672942-s2.0-3374771134110.1007/s10687-006-0002-2BreakerL. C.Nonlinear aspects of sea surface temperature in Monterey Bay200669161892-s2.0-3364650641310.1016/j.pocean.2006.02.015GowerJ.HuC.BorstadG.KingS.Ocean color satellites show extensive lines of floating sargassum in the gulf of Mexico20064412361936252-s2.0-3384561621810.1109/TGRS.2006.882258SchumannR.BaudlerH.GlassÄ.DümckeK.KarstenU.Long-term observations on salinity dynamics in a tideless shallow coastal lagoon of the Southern Baltic Sea coast and their biological relevance2006603-43303442-s2.0-3364649677210.1016/j.jmarsys.2006.02.007SarkarA.KshatriyaJ.SatheesanK.Auto-correlation analysis of wave heights in the Bay of Bengal200611522352372-s2.0-3374637309310.1007/BF02702037CairesS.SwailV. R.WangX. L.Projection and analysis of extreme wave climate20061921558156052-s2.0-3375142362410.1175/JCLI3918.1WaltonT. L.Jr.Projected sea level rise in Florida20073413183218402-s2.0-3425083998810.1016/j.oceaneng.2007.02.003PirazzoliP. A.TomasinA.Estimation of return periods for extreme sea levels: a simplified empirical correction of the joint probabilities method with examples from the French Atlantic coast and three ports in the southwest of the UK2007572911072-s2.0-3394719767710.1007/s10236-006-0096-8RomanowiczR. J.YoungP. C.BevenK. J.PappenbergerF.A data based mechanistic approach to nonlinear flood routing and adaptive flood level forecasting2008318104810562-s2.0-4704910081810.1016/j.advwatres.2008.04.015CastanedoS.MendezF. J.MedinaR.AbascalA. J.Long-term tidal level distribution using a wave-by-wave approach20073011227122822-s2.0-3454831250110.1016/j.advwatres.2007.05.005KalraR.DeoM. C.Derivation of coastal wind and wave parameters from offshore measurements of TOPEX satellite using ANN20075431871962-s2.0-3384636260310.1016/j.coastaleng.2006.07.001TonnonP. K.van RijnL. C.WalstraD. J. R.The morphodynamic modelling of tidal sand waves on the shoreface20075442792962-s2.0-3394720733810.1016/j.coastaleng.2006.08.005BazarganH.BahaiH.Aminzadeh-GohariA.Calculating the return value using a mathematical model of significant wave height200712134422-s2.0-3394761349710.1007/s00773-006-0234-5GünaydinK.The estimation of monthly mean significant wave heights by using artificial neural network and regression methods20083514-15140614152-s2.0-5114910548910.1016/j.oceaneng.2008.07.008KangJ. W.MoonS.-R.ParkS.-J.LeeK.-H.Analyzing sea level rise and tide characteristics change driven by coastal construction at Mokpo Coastal Zone in Korea2009366-74154252-s2.0-6454910334910.1016/j.oceaneng.2008.12.009ItoT.OkuboM.SagiyaT.High resolution mapping of Earth tide response based on GPS data in Japan2009483–52532592-s2.0-7204908398410.1016/j.jog.2009.09.012LinY.GreatbatchR. J.ShengJ.The influence of Gulf of Mexico loop current intrusion on the transport of the Florida Current2010605107510842-s2.0-7795811119210.1007/s10236-010-0308-0XiaoP.YuZ.LiC. S.Compressive sensing SAR range compression with chirp scaling principle201254122922300http://www.ndbc.noaa.gov/historical_data.shtmlHurstH. E.Long-term storage capacity of reservoirs1951116770799MandelbrotB. B.WallisJ. R.Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence196955967988MandelbrotB. B.WallisJ. R.Some long-run properties of geophysical records196952967988ReaW.RealeM.BrownJ.OxleyL.Long memory or shifting means in geophysical time series?2011817144114532-s2.0-7995181279810.1016/j.matcom.2010.06.007PelletierJ. D.TurcotteD. L.Long-range persistence in climatological and hydrological time series: analysis, modeling and application to drought hazard assessment19972031–41982082-s2.0-003159305810.1016/S0022-1694(97)00102-9ShaoQ.LiM.A new trend analysis for seasonal time series with consideration of data dependence20113961-21041122-s2.0-7865017732310.1016/j.jhydrol.2010.10.040DolgonosovB. M.KorchaginK. A.KirpichnikovaN. V.Modeling of annual oscillations and 1/f-noise of daily river discharges20083573-41741872-s2.0-4704910575210.1016/j.jhydrol.2008.04.023GantiV.StraubK. M.Foufoula-GeorgiouE.PaolaC.Space-time dynamics of depositional systems: experimental evidence and theoretical modeling of heavy-tailed statistics201111622-s2.0-7995745993810.1029/2010JF001893F02011Alvarez-RamirezJ.AlvarezJ.DagdugL.RodriguezE.EcheverriaJ. C.Long-term memory dynamics of continental and oceanic monthly temperatures in the recent 125 years200838714362936402-s2.0-4164912031610.1016/j.physa.2008.02.051BerronesA.Persistence in a simple model for the Earth's atmosphere temperature fluctuations200553L365L3742-s2.0-2564444683310.1142/S021947750500280XHaslettJ.RafteryA. E.Space-time modelling with long-memory dependence: assessing Ireland's wind power resource1989381150FraedrichK.LukschU.BlenderR.1/f model for long-time memory of the ocean surface temperature200470342-s2.0-4274909926010.1103/PhysRevE.70.037301037301SanthanamM. S.KantzH.Long-range correlations and rare events in boundary layer wind fields20053453-47137212-s2.0-964426890010.1016/j.physa.2004.07.012BouetteJ.-C.ChassagneuxJ.-F.SibaiD.TerronR.CharpentierA.Wind in Ireland: long memory or seasonal effect?20062031411512-s2.0-3364483628010.1007/s00477-005-0029-yMonettiR. A.HavlinS.BundeA.Long-term persistence in the sea surface temperature fluctuations20033205815892-s2.0-003744533710.1016/S0378-4371(02)01662-XBlenderR.FraedrichK.SienzF.Extreme event return times in long-term memory processes near 1/f20081545575652-s2.0-47749125362PrassT. S.BravoJ. M.ClarkeR. T.CollischonnW.LopesS. R. C.Comparison of forecasts of mean monthly water level in the Paraguay River, Brazil, from two fractionally differenced models201248513W05502BarbosaS. M.FernandesM. J.SilvaM. E.Long-range dependence in North Atlantic sea level200637127257312-s2.0-3374869978910.1016/j.physa.2006.03.046LiM.CattaniC.ChenS. Y.Viewing sea level by a one-dimensional random function with long memory20112011132-s2.0-7865073597810.1155/2011/654284654284MandelbrotB. B.1982New York, NY, USAW. H. FreemanAdlerR. J.FeldmanR. E.TaqquM. S.1998Boston, Mass, USABirkhäuserAbryP.BorgnatP.RicciatoF.ScherrerA.VeitchD.Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail2010433-41471652-s2.0-7795102289110.1007/s11235-009-9205-6GneitingT.SchlatherM.Stochastic models that separate fractal dimension and the hurst effect2004462269282ZBL1062.600532-s2.0-314271038510.1137/S0036144501394387LiM.Generation of teletraffic of generalized Cauchy type2010812102-s2.0-7714912707410.1088/0031-8949/81/02/025007025007LimS. C.LiM.A generalized Cauchy process and its application to relaxation phenomena2006391229352951ZBL1090.820132-s2.0-3364490293610.1088/0305-4470/39/12/005MuniandyS. V.StanslasJ.Modelling of chromatin morphologies in breast cancer cells undergoing apoptosis using generalized Cauchy field20083276316372-s2.0-4994911411910.1016/j.compmedimag.2008.07.003CeresettiD.MoliniéG.CreutinJ.-D.Scaling properties of heavy rainfall at short duration: a regional analysis2010469122-s2.0-7795770849310.1029/2009WR008603W09531RadziejewskiM.KundzewiczZ. W.Fractal analysis of flow of the river Warta19972001–42802942-s2.0-003157417410.1016/S0022-1694(97)00024-3HirabayashiS.SatoT.Scaling of mixing parameters in stationary, homogeneous, and stratified turbulence2010115992-s2.0-7795758558710.1029/2010JC006254C09023MantillaR.TroutmanB. M.GuptaV. K.Testing statistical self-similarity in the topology of river networks20101153122-s2.0-7795755828010.1029/2009JF001609F03038SuleymanovA. A.AbbasovA. A.IsmaylovA. J.Fractal analysis of time series in oil and gas production2009415247424832-s2.0-6764956633210.1016/j.chaos.2008.09.039RehmanS.SiddiqiA. H.Wavelet based hurst exponent and fractal dimensional analysis of Saudi climatic dynamics2009403108110902-s2.0-6534909993910.1016/j.chaos.2007.08.063ZuoR.ChengQ.XiaQ.AgterbergF. P.Application of fractal models to distinguish between different mineral phases200941171802-s2.0-5824909384110.1007/s11004-008-9191-3GarcíaR. C.GalánA. S.Castrejón PitaJ. R.PitaA. A. C.The fractal dimension of an oil spray20031121551612-s2.0-004157346310.1142/S0218348X03001641BenmehdiS.MakaravaN.BenhamidoucheN.HolschneiderM.Bayesian estimation of the self-similarity exponent of the Nile River fluctuation20111834414462-s2.0-7996028118810.5194/npg-18-441-2011BadulinS. I.PushkarevA. N.ResioD.ZakharovV. E.Self-similarity of wind-driven seas20051268919452-s2.0-30744440892CarboneV.VeltriP.BrunoR.Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws1996342472612-s2.0-0008527580SangoyomiT. B.LallU.AbarbanelH. D. I.Nonlinear dynamics of the Great Salt Lake: dimension estimation19963211491592-s2.0-002966390910.1029/95WR02872OzgerM.Scaling characteristics of ocean wave height time series201139069819892-s2.0-7875158035610.1016/j.physa.2010.11.019ChmelA.SmirnovV. N.AstakhovM. P.The Arctic sea-ice cover: fractal space-time domain20053573-45565642-s2.0-2464443429810.1016/j.physa.2005.04.009BerizziF.BertiniG.MartorellaM.BertaccaM.Two-dimensional variation algorithm for fractal analysis of sea SAR images2006449236123732-s2.0-3374830438810.1109/TGRS.2006.873577FraedrichK.BlenderR.Scaling of atmosphere and ocean temperature correlations in observations and climate models2003901042-s2.0-0038303118108501XuT.MooreI. D.GallantJ. C.Fractals, fractal dimensions and landscapes—a review1993842452622-s2.0-0027805369KorvinG.1992ElsevierLevy-VehelJ.LuttonE.TricotC.1997SpringerMandelbrotB. B.2001SpringerLimS. C.MuniandyS. V.On some possible generalizations of fractional Brownian motion20002662-3140145ZBL1068.825182-s2.0-003469591110.1016/S0375-9601(00)00034-7StanleyH. E.AmaralL. A. N.GoldbergerA. L.HavlinS.IvanovP. C.PengC.-K.Statistical physics and physiology: monofractal and multifractal approaches19992701-23093242-s2.0-003317053010.1016/S0378-4371(99)00230-7MandelbrotB. B.1998Springer1713511HarteD.2001Chapman & Hall10.1201/97814200360082065030TurcotteD. L.19972ndCambridge, UKCambridge University Press1458893Lévy-VéhelJ.LuttonE.2005SpringerGaciS.ZaourarN.BriqueuL.HamoudiM.ChenD.Regularity analysis of airborne natural Gamma ray data measured in the Hoggar area (Algeria)2011InTech93108GaciS.ZaourarN.ChenD.Two-dimensional multifractional Brownian motion-based investigation of heterogeneities from a core image2011InTech109124AfzalP.ZarifiA. Z.YasrebiA. B.Identification of uranium targets based on airborne radiometric data analysis by using multifractal modeling, Tark and Avanligh 1:50 000 sheets, NW Iran2012192283289JouiniM. S.VegaS.MokhtarE. A.Multiscale characterization of pore spaces using multifractals analysis of scanning electronic microscopy images of carbonates2011186941953TeotiaS. S.KumarD.Role of multifractal analysis in understanding the preparation zone for large size earthquake in the North-Western Himalaya region20111811111182-s2.0-7995178890510.5194/npg-18-111-2011SerinaldiF.Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models20101766977142-s2.0-7865011915310.5194/npg-17-697-2010MacekW. M.Multifractality and intermittency in the solar wind20071466957002-s2.0-36248971058BadulinS. I.PushkarevA. N.ResioD.ZakharovV. E.Self-similarity of wind-driven seas20051268919452-s2.0-30744440892IdaY.HayakawaM.AdalevA.GotohK.Multifractal analysis for the ULF geomagnetic data during the 1993 Guam earthquake20051221571622-s2.0-14944362364SeurontL.SchmittF.ShertzerD.LagadetteY.LovejoyS.Multifractal intermittency of Eulerian and Lagrangian turbulence of ocean temperature and plankton fields1996342362462-s2.0-0001553934AriasM.GumielP.SandersonD. J.Martin-IzardA.A multifractal simulation model for the distribution of VMS deposits in the Spanish segment of the Iberian Pyrite Belt2011371219171927Paz-FerreiroJ.VázquezE. V.MirandaJ. G. V.Assessing soil particle-size distribution on experimental plots with similar texture under different management systems using multifractal parameters2010160147562-s2.0-7864942193210.1016/j.geoderma.2010.02.002ChuP. C.Multi-fractal thermal characteristics of the southwestern GIN sea upper layer20041922752842-s2.0-004309384510.1016/S0960-0779(03)00041-9SuN.YuZ.-G.AnhV.BajracharyaK.Fractal tidal waves in coastal aquifers induced both anthropogenically and naturally20041912112511302-s2.0-454432816310.1016/j.envsoft.2003.12.002LiangX. S.RobinsonA. R.Localized multiscale energy and vorticity analysis. I. Fundamentals2005383-41952302-s2.0-1754437762010.1016/j.dynatmoce.2004.12.004PelletierJ. D.Natural variability of atmospheric temperatures and geomagnetic intensity over a wide range of time scales200299supplement 1254625532-s2.0-003713324310.1073/pnas.022582599DaoudiK.Lévy-VéhelJ.Signal representation and segmentation based on multifractal stationarity20028212201520242-s2.0-003688782210.1016/S0165-1684(02)00198-6BedfordT.Hölder exponents and box dimension for self-affine fractal functions19895133482-s2.0-000203625110.1007/BF01889597FalconerK. J.2003John Wiley & SonsKantelhardtJ. W.ZschiegnerS. A.Koscielny-BundeE.HavlinS.BundeA.StanleyH. E.Multifractal detrended fluctuation analysis of nonstationary time series20023161–4871142-s2.0-003711453710.1016/S0378-4371(02)01383-3DaoudiK.Lévy-VéhelJ.MeyerY.Construction of continuous functions with prescribed local regularity19981433493852-s2.0-0039657305WestB. J.Fractal physiology and the fractional calculus: a perspective20101, article 12PeltierR. F.Levy-VehelJ.1995INRIA RR, 2645AyacheA.CohenS.Levy VehelJ.Covariance structure of Multifractional Brownian motion, with application to long range dependenceProceedings of the IEEE Interntional Conference on Acoustics, Speech, and Signal Processing (ICASSP '00)June 2000381038132-s2.0-0033724630AyacheA.The generalized multifractional field: a nice tool for the study of the generalized multifractional brownian motion2002865816012-s2.0-003645239910.1007/s00041-002-0028-zAyacheA.Levy-VehelJ.The generalized multifractional Brownian motion200031-2718GuevelR. L.Levy-VehelJ.2010INRIAFalconerK. J.Le GuévelR.Lévy-VéhelJ.Localizable moving average symmetric stable and multistable processes20092546486722-s2.0-7795144628710.1080/15326340903291321FalconerK. J.Lévy VéhelJ.Multifractional, multistable, and other processes with prescribed local form20092223754012-s2.0-6454908845310.1007/s10959-008-0147-9FalconerK. J.The local structure of random processes20036736576722-s2.0-0037900816DȩbickiK.KisowskiP.Asymptotics of supremum distribution of α (t)-locally stationary Gaussian processes200811811202220372-s2.0-5304909510710.1016/j.spa.2007.11.010CattaniC.PierroG.AltieriG.Entropy and multifractality for the myeloma multiple TET 2 gene201220121419376110.1155/2012/193761MuniandyS. V.LimS. C.MuruganR.Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates20013011–44074282-s2.0-003557581210.1016/S0378-4371(01)00387-9ShangP.LuY.KamaS.The application of Hölder exponent to traffic congestion warning200637027697762-s2.0-3374813513710.1016/j.physa.2006.02.032JinT.ZhangH.Statistical approach to weak signal detection and estimation using Duffing chaotic oscillators2011541123242337LiM.ZhaoW.ChenS. Y.MBm-based scalings of traffic propagated in internet20112011212-s2.0-7925155010510.1155/2011/389803389803ShengH.ChenY.-Q.QiuT.-S.Heavy-tailed distribution and local long memory in time series of molecular motion on the cell membrane2011101931192-s2.0-7955158812410.1142/S0219477511000429ShengH.SunH.ChenY.-Q.QiuT.-S.Synthesis of multifractional Gaussian noises based on variable-order fractional operators2011917164516502-s2.0-7995204356610.1016/j.sigpro.2011.01.010ShengH.ChenY. Q.QiuT. S.2012SpringerMuzyJ. F.BacryE.ArneodoA.Multifractal formalism for fractal signals: the structure-function approach versus the wavelet-transform modulus-maxima method19934728758842-s2.0-614426151310.1103/PhysRevE.47.875BarrièreO.Lévy-VéhelJ.Local Hölder regularity-based modeling of RR intervalsINRIA-00539046, Version 1, 2010GaciS.ZaourarN.HamoudiM.HolschneiderM.Local regularity analysis of strata heterogeneities from sonic logs20101754554662-s2.0-7795694840010.5194/npg-17-455-2010http://www.ndbc.noaa.gov/index.shtmlhttp://www.ndbc.noaa.gov/historical_data.shtmlhttp://www.ndbc.noaa.gov/station_page.php?station=lkwf1http://www.ndbc.noaa.gov/station_page.php?station=lonf1http://www.ndbc.noaa.gov/station_page.php?station=sauf1http://www.ndbc.noaa.gov/station_page.php?station=smkf1http://www.ndbc.noaa.gov/station_page.php?station=spgf1http://www.ndbc.noaa.gov/station_page.php?station=venf1http://www.ndbc.noaa.gov/historical_data.shtml#wlevelGilhousenD. B.A field evaluation of NDBC moored buoy winds19874194104http://www.ndbc.noaa.gov/measdes.shtml#wlevelKolmogorovA. N.Wienersche Spiralen und einige andere interessante Kurven im. Hilbertschen Raum1940262115118MatérnB.19862ndSpringerChilesJ.-P.DelfinerP.1999New York, NY, USAWileyWebsterR.OliverM. A.2007Chichester, UKWileyWackernagelH.2005Dordrecht, The NetherlandsSpringerSchabenbergerO.GotwayC. A.2005Boca Raton, Fla, USAChapman & Hall, CRCRipleyB. D.2004Hoboken, NJ, USAWiley-InterscienceSchlatherM.GneitingT.Local approximation of variograms by covariance functions20067612130313042-s2.0-3364617682110.1016/j.spl.2006.02.002MinasnyB.McBratneyA. B.The Matérn function as a general model for soil variograms20051283-41922072-s2.0-2374447728710.1016/j.geoderma.2005.04.003GorsichD. J.GentonM. G.Variogram model selection via nonparametric derivative estimation20003232492702-s2.0-003411662010.1023/A:1007563809463GentonM. G.The correlation structure of Matheron's classical variogram estimator under elliptically contoured distributions20003211271372-s2.0-003400015210.1023/A:1007511019496MarcotteD.Fast variogram computation with FFT19962210117511862-s2.0-003043515410.1016/S0098-3004(96)00026-XMoninA. S.YaglomA. M.19712Cambridge, Mass, USAThe MIT PressKaganovE. I.YaglomA. M.Errors in wind-speed measurements by rotation anemometers197610115342-s2.0-001687956410.1007/BF00218722Schulz-DuBoisE. O.RehbergI.Structure function in lieu of correlation function19812443233292-s2.0-001955678810.1007/BF00899730WarhaftZ.Turbulence in nature and in the laboratory2002991248124862-s2.0-003713322610.1073/pnas.012580299YaglomA. M.1987SpringerBandelbrotB. B.van NessJ. W.Fractional Brownian motions, fractional noises and applications1968104422437HidaT.1980SpringerNigamN. C.1983The MIT PressPapoulisA.19842ndMcGraw-HillGelfandI. M.VilenkinK.19641New York, NY, USAAcademic PressBiaginiF.HuY.ØksendalB.ZhangT.2008SpringerLiM.Fractal time series—a tutorial review20102010262-s2.0-7795148927610.1155/2010/157264157264MikusinskiJ.1959Pergamon PressAyacheA.JaffardS.TaqquM. S.Wavelet construction of generalized multifractional processes20072313273702-s2.0-34547197313AyacheA.VéhelJ. L.On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion200411111191562-s2.0-164241917410.1016/j.spa.2003.11.002BertrandP. R.HamdouniA.KhadhraouiS.Modelling NASDAQ series by sparse multifractional Brownian motion2012141107124LinZ.ZhengJ.Some properties of a multifractional Brownian motion20077776876922-s2.0-3394716420710.1016/j.spl.2006.11.002LimS. C.TeoL. P.Weyl and Riemann-Liouville multifractional Ornstein-Uhlenbeck processes20074023603560602-s2.0-3424971849810.1088/1751-8113/40/23/003MuniandyS. V.LimS. C.Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type200163472-s2.0-4243216985LimS. C.Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type2001347130113102-s2.0-003593705510.1088/0305-4470/34/7/306StoevS. A.TaqquM. S.How rich is the class of multifractional Brownian motions?200611622002212-s2.0-3034444083410.1016/j.spa.2005.09.007CoeurjollyJ.-F.Identification of multifractional Brownian motion200511698710082-s2.0-3394739202410.3150/bj/1137421637StoevS.TaqquM. S.Path properties of the linear multifractional stable motion20051321571782-s2.0-2114443363910.1142/S0218348X05002775BenassiA.CohenS.IstasJ.Identifying the multifractional function of a Gaussian process19983943373452-s2.0-0032555366NavarroR.Jr.TamanganR.Guba-NatanN.RamosE.GuzmanA. D.The identification of long memory process in the Asean-4 stock markets by fractional and multifractional Brownian motion2006551-26583