Estimation of dimensions and orientation of multiple riverine dune generations using spectral moments
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Abstract
A new spectral analysis technique is proposed for rhythmic bedform quantification, based on the 2D Fourier transform involving the calculation of a set of loworder spectral moments. The approach provides a tool for efficient quantification of bedform length and height as well as spatial crestline alignment. Contrary to the conventional method, it not only describes the most energetic component of an undulating seabed surface but also retrieves information on its secondary structure without application of any bandpass filter of which the upper and lower cutoff frequencies are a priori unknown. Validation is based on bathymetric data collected in the main Vistula River mouth area (Przekop Wisły), Poland. This revealed two generations (distinct groups) of dunes which are migrating seawards along distinct paths, probably related to the hydrological regime of the river. The data enable the identification of dune divergence and convergence zones. The approach proved successful in the parameterisation of topographic roughness, an essential aspect in numerical modelling studies.
Keywords
Bedforms Spectral Moment Alignment Angle Topographic Roughness Large DuneIntroduction
Subaqueous rhythmic bedforms such as ripples and dunes (as in Ashley 1990) are important seafloor features which reflect physical phenomena taking place at the sediment–water interface and, at the same time, modulate these phenomena (Allen 1982). Due to changes in elevation along bedforms, modification of the pressure distribution is observed. It is reflected in a decrease of pressure and consequently in flow acceleration on the stoss face, and an increase of pressure and flow deceleration downstream of the bedform crest (e.g. Vanoni and Hwang 1967; Best 2005). The presence of bedforms therefore strongly influences the form drag, impacting directly on the overall flow resistance as well as the flow velocity structure and sediment transport pattern (e.g. Venditti 2007; Bartholdy et al. 2010; Lefebvre et al. 2016). Detailed characterisation of bedform roughness reflected in bedform parameters of their spatial distribution and orientation is therefore vital for constructing numerical models of flow field, hydrology and sediment transport.
Since 1977 when the first world ocean floor map was compiled by Mary Tharp and Bruce Heezen based on singlebeam echosounder data (see Blondel 2009), methods for bathymetry mapping have improved significantly. Development of highresolution sonar systems and satellite positioning methods led to the availability of precise techniques for seafloor relief data acquisition. Swath bathymetry systems (multibeam echosounders, MBES) have become a widely used tool for highresolution, rapid and costeffective mapping of submerged environments. It has become possible to investigate quantitatively in far more detail important aspects of bottom surface roughness and to verify geomorphological processes responsible for bedform formation through interactions between sediments and water flow (e.g. Bartholomä et al. 2004; Knaapen et al. 2005; Parsons et al. 2005; Knaapen 2008; Barnard et al. 2011).
Many investigators approached the characterisation of bedform roughness based on spectral analysis decomposing the seafloor relief into its major spatial frequency components, which opened possibilities for better identifying and describing morphological features. Bottom roughness description through spectral analysis techniques was first reported by Nordin and Algert (1966), who identified variability in a sand wave bed formed under a unidirectional flow regime based on spectral density function calculations. In turn, Hino (1968) first proposed a “power law” function in representing a bottom roughness spectrum. Later, Fox and Hayes (1985) developed a valid stochastic description of several single bathymetric profiles based on a statistical measure of bottom roughness from the slope of the amplitude spectrum, and showed evidence of powerlaw behaviour for roughness in the 10 cm to 200 m length range. As argued by Briggs (1989), however, the data in the highest frequencies (for lengths from 10 cm to 1 mm) were too scarce to be conclusive. Briggs et al. (2002) split the seafloor roughness spectrum into multiple frequency bands and fitted powerlaw relationships to the measured spectra. In turn, Lyons et al. (2002) highlighted that it is advantageous to model naturally rippled sediment surfaces by a twocomponent spectrum comprised of a noncentred Gaussian component and an isotropic powerlaw component.
Contrary to the onedimensional approach, the earliest successful attempt to quantify seabed roughness based on usage of a twodimensional power spectrum and related autocorrelation function obtained from pairs of stereo microtopography photographs of the bottom was done by Akal and Hovem (1978). This was confirmed and expanded by numerous studies related to bedform characterisation based on the algorithm of the twodimensional discrete Fourier transform (2D DFT) spectral analysis (e.g. Cazenave et al. 2008; Van Dijk et al. 2008; Lefebvre and Lyons 2011; Lefebvre et al. 2011; Lisimenka et al. 2013). Extant quantitative analysis techniques and methodologies for the semiautomated and objective description of primary parameters such as length, orientation, crest and trough position, height, asymmetry direction and asymmetry ratio, which are commonly used to describe bedforms on subaerial and subaqueous roughness surfaces, were adapted and proposed recently by Cazenave et al. (2013). However, one of the disadvantages of current methodologies is the need to apply bandpass filtering in order to quantify secondary and potential further generations (distinctive groups) of rhythmic bedforms.
The present study reports a newly developed concept for quantitative analysis of subaqueous straightcrested undulating bottom roughness using the 2D DFT algorithm, which involves calculation of loworder spectral moments. Based on a twodimensional spectrum estimated from a height field and a set of onedimensional “slices” executed from the 2D spectrum with onedegree step, spectral moments of different orders are calculated in order to determine the primary parameters of the observed bedforms, such as crestline spatial alignment, length and height. Contrary to previous studies, this approach provides a possibility not only to quantify primary generations, i.e. the most energetic component of the undulating seabottom surface, but also to obtain information about secondary and further generations (i.e. groups of dunes with common parameters) without application of a bandpass filter of which the upper and lower cutoff frequencies are a priori unknown. The method was developed to identify and quantify various generations of subaqueous dunes in the Vistula River mouth, Poland.
Physical setting
The main river mouth “Przekop Wisły” is a crosscut artificial channel of ca. 3,000 m length, 400 m width and temporarily up to 10 m depth (Fig. 1c). Due to locks controlling water levels in the old channels of the Vistula (Nogat, Szkarpawa and Martwa Wisła, Fig. 1b), about 95% of the total Vistula water outflows into the Baltic Sea through this channel. The closest gauging station is located in Tczew 31.2 km upstream of the Vistula channel mouth (Fig. 1b). It represents 99.92% of the catchment area (Augustowski 1982). Based on operational data obtained from this gauging section (IMGWPIB 2015), the longterm average annual water discharge reaches 1,042 m^{3}/s. The average daily water discharge, however, varies from 266 m^{3}/s in the dry season to 6,360 m^{3}/s during floods. The river plays a crucial role also in sediment delivery processes as the Vistula carries 0.6–1.5×10^{6} m^{3} of sediments annually (Pruszak et al. 2005), thereby expanding the delta front (Wróblewski et al. 2015).
Hydrological regime of lower Vistula
Materials and methods
In situ measurements
The measurement campaign in the Vistula River mouth was carried out on 4–6 June 2012. The bathymetry was mapped using a Reson SeaBat 7101 multibeam echosounder (MBES) operating at 240 kHz. The sonar provides 511 discrete sounding beams across the wide 150° swath with 1.5° alongtrack transmit beam width, 1.8° acrosstrack receive beam width and 0.0125 m depth resolution. The sound velocity probe Reson SVP70 was fixed to the MBES head, and the portable sound velocity profiler Reson SVP15 was used to obtain the sound speed at the depth of the MBES draft and through the water column. The positioning system DGPS RTK Trimble SPS 851 together with the Ixsea Hydrins inertial navigation system were integrated with the MBES and SVPs using the QINSy data acquisition software package. Full coverage of the area of interest was achieved. The postprocessing of the MBES raw data was performed in the QINSy Processing Manager according to standard procedures widely used in hydrography.
Bedform parameters
A digital terrain model (DTM) of the riverbed was obtained from the highresolution multibeam echosounder data. The bathymetric data were gridded with a cell size of 0.1 m. This was justified by the high density of individual soundings—in 99.7% and 83.2% of cases, at least 100 soundings per m^{2} were collected in shallow waters (shallower than –3 m depth) and deep waters (deeper than –7 m depth) respectively.
The entire riverbed elevation dataset was subsequently divided into subdomains using a sliding square window of 200×200 m size with a 75% overlap between adjacent windows. The chosen size of the window was a result of compromise between the overall area extent and the observed bedform lengths.
Within each of the test boxes, the mean depth was subtracted from each data sample and then the bathymetric surface was detrended in 2D by fitting and removing a thirdorder polynomial regression model. In order to minimize spectral leakage due to the finite length of the dataset, data in each subdomain were premultiplied by a single orthogonal taper belonging to a family of functions known as discrete prolate spheroidal sequences (DPSS or Slepian sequences; Percival and Walden 1993). Among numerous existing window functions, the DPSS is a unique taper function which offers the best sidelobe suppression (maximization of energy concentration in the main lobe of the window function frequency response). In fact, “windowing” reduces bias but, on the other hand, it causes a reduction of resolution in the spectral estimate which becomes smoother (the variance of the periodogram increases). Therefore, according to Lyons et al. (2002), it is desirable to apply the widest data taper which reduces bias to an acceptable level for estimation of accurate frequency components.
Subsequently, spectral analysis based on the 2D DFT (performed with the Matlab® fft2 function) was executed, yielding a twodimensional power spectrum function which describes the seabed height field in the space frequency domain. In order to minimize any false components in the spectra associated with the border of the data coverage effect, the 2D DFT was calculated only in subdomains with data coverage greater than 75%. In this way, 331 subdomains were analysed.
At the next step, several spectral parameters were derived from the spectral function based on the calculation of spectral moments and their combinations. One of the wellknown advantages of spectral moments, which retrieve information directly from the Fourier spectrum, is their insensitivity to signal phase changes. These can be often distorted by “usual” filtering, thereby falsifying the amplitude spectrum. Moreover, by conducting complete analogy between the power spectrum density and the probability density function, they can also be viewed as statistical descriptors of spectral function.
The physical meaning of the m _{2} can be interpreted as a weighting of each frequency content in the spectrum by the second power of the frequency, thereby “amplifying” a highfrequency tail, the spectral power of which is usually significantly lower in comparison with the more energetic lowfrequency part of the spectrum. Consequently, it enables extraction of information about the presence of smaller dune lengths in the spectral curve, which can be of particular interest as well.
The choice of f _{thresh} is based on a compromise between the size of the analysed subdomains (200×200 m) and the minimization of space alignment angle and dune length uncertainties. An effect of varying length as a function of domain size on both alignment and length uncertainties was examined by Cazenave et al. (2013). Using the approach of those authors, the alignment angle and length uncertainties do not exceed 26% and 35% respectively for the largest dune length of interest in the present case, i.e. λ _{thresh}=75 m, which is 37.5% of the subdomain size. In the case of the calculated median length λ _{median}=30 m (15% of the subdomain size), the alignment angle and length uncertainties decrease significantly to less than 10% and 12.5% respectively. In the general case, since the uncertainties are a function of the subdomain selected for the analyses, choosing a larger subdomain would decrease those uncertainties.
Physically, the mean frequency plays a role as a frequency centroid (spectral centre of gravity) of the onesided power spectrum function (Michaelov et al. 1999). Thus, in the case of a single distinctive sharp maximum in the roughness spectrum (such as the one caused by a single dune generation), it can be used to assess the spatial distance between rhythmic bedforms, i.e. dune length. In the case of a real seabed, however, it is often observed that a dune height field with a particular spatial alignment may be narrowband at one location and broadband at another. Furthermore, primary (most energetic) bed surface roughness can be accompanied by secondary bedforms with the same alignment but different lengths. In such cases, the predominant mean frequency may shift and would not reflect dune length in the most reliable way.
Results
One dominating dune generation
Two and more dune generations
Morphology of Vistula River mouth
The present bathymetric data confirm earlier findings of Staśkiewicz et al. (2010)—the bottom relief of the coastal sector of the Vistula River mouth is characterised by a sandbar along the western bank of the river and a deep trough along the eastern bank. The sandbar elevated above the –4 m isoline is elongated towards the middle of the river valley in the most upstream sector of the study area. There is another shoal shallower than –4 m in the middle of the channel in the most northern sector. It breaches the water surface in the north as islands visible in aerial photographs (Fig. 1c). The deepest trough bottom reaches below –7 m.
Discussion
Method performance
The most suitable probability density functions (with the smallest error estimated based on Eq. 1 in Van der Mark et al. 2008) were fitted to the appropriate geometric variables distributions. Thus, it was found that the Weibull and Gamma distributions yield the best approximation for bedform height and wavelength respectively, in agreement with the results of Van der Mark et al. (2008). In addition, the significant bedform height was calculated as the mean wave height value of the highest third of the waves, and H _{s}=0.65 m was obtained.
Conducting a direct analogy with the significant wave height concept widely used within sea surface wave theory, the characteristic bedform height η _{ch} can also be estimated from the zero spectral moment (spectral analysis gives an energybased significant wave height, Eq. 10). However, it should be noted here that, based on the results obtained by the spectral method proposed herein and the traditional statistical approach, the characteristic height η _{ch} of 0.8 m seems to be overestimated in comparison with the significant bedform height H _{s} of 0.65 m. The hypothetical assumption that bedforms were approximated as symmetrical sinusoidal waves (multiplication coefficient \( 2\sqrt{2} \) in Eq. 10), which is rarely the case for real seabed roughness, may partly explain this overestimation. Conversely, Van der Mark et al. (2008) showed that, if the ratio of river width W to hydraulic radius R is larger than approximately 10 (in the case of the Vistula River mouth, W/R>>10), a linear relation exists between the standard deviation and mean value of bedform height and, for field conditions, a constant coefficient of variation C _{Δ}=σ/μ=0.47 can be applied. Considering the above, the result obtained here is in reliable agreement with the conclusions of Van der Mark et al. (2008).
Generally, the most appropriate azimuth for extracting onedimensional crosssections from bathymetric surfaces is a priori unknown, so that application of the zerocrossing method can give even larger differences when analysing domains of more complicated shape than a canalized river. Although the zerocrossing method seems to be irreplaceable in determining the locations of crest and trough points and then the geometric properties of individual bedforms, it can fail in an environment of multiple generations of dunes. By contrast, application of the proposed procedure allows to identify and parameterise various generations of rhythmic bedforms more efficiently.
Morphodynamics of Vistula River mouth
Discharge values of the Vistula River recorded in Tczew suggest that there existed a certain bimodality of the riverine regime during 18 months before the present measuring campaign. Hydrological forces acting during the most common dry period of low discharge of 700 m^{3}/s and less should produce smaller bedforms than discharge higher than 1,100 m^{3}/s related to spring snow melt and precipitation floods. In May and the first days of June 2012 shortly prior to the present bathymetric measurements, the discharge only twice exceeded 1,000 m^{3}/s (in toto for 28 h, 4% of all records), whereas in March and April it was generally higher than 1,200 m^{3}/s (80% of records, Fig. 2a). This suggests that supposedly larger bedforms produced in April were not entirely washed out in the next 2 months.
With respect to the global mean trend of Flemming (1988), the bedforms of the Vistula River mouth fall below the average relationship between dune length and height (Fig. 12). If one assumes that the mean global trend reflects dunes which stay in equilibrium with the driving force of flow, one concludes that the bedforms of the Vistula River mouth are underdeveloped. This suggests that the dunes may be sedimentstarved due to a lack of material for constructing fully developed dune bodies under peak velocities, or that peak velocities occur for too short a time to be able to amass the dune body and reach equilibrium. There are, however, five exceptions concerning the steepest dunes in the study area (Fig. 12, in red), which evidently reached their equilibrium form.
Conclusions

The approach enables the quantification of bedform parameters such as spatial alignment, length and height, also providing the possibility of describing multiple generations of undulating bedform structures without applying any bandpass filtering.

By calculating the zero spectral moment m _{0} (Eq. 5), which plays the role of spectrum energy, it is possible to obtain the roughness spectrum energy angular distribution and determine the space alignment angle of primary bedforms (most energetic). Distinguishing secondary bedforms is possible on the basis of analysis of the second spectral moment m _{2} (Eq. 6) angular distribution.

In the case of a real seabed, the mean frequency f _{mean}, defined as the ratio of the first spectral moment to the zero one (Eq. 8), is not a suitable parameter to obtain information about the dominant bedform length. In relation to the determination of the spectrum peak frequency, which is of more interest, the characteristic frequency f _{ch} (Eq. 9), defined by using a weighted integral of power 4 of the spectral function proposed by Young (1995), seems to be the most reliable approach.

The overall result of “scanning” the study area automatically in 331 predesigned 200×200 m subdomains shows the complex character of the Vistula River mouth bed covered by dunes with distinct crestline alignments, lengths and heights, highlighting the diversity of bedform geometric dimensions from one subdomain to the other. The analysis revealed two distinct paths of two dune generations. The largest dunes (at least 70 m in length and 1 m in height) are migrating north along the deeper eastern bank, whereas smaller bedforms (less than 20 m long and 0.5 m high) migrate along the shoal and western bank. Moreover, areas of dune divergence in the south and dune convergence in the north parts of the Vistula River mouth were identified.

The approach considers only periodic bottom structures with space frequency components which fall within a limited frequency range. A choice of cutoff lowfrequency f _{thresh} (Eq. 7) in order to filter the “red noise” frequency components out of the spectrum was a compromise between the size of the analysed subdomains (200×200 m) and minimization of crestline alignment and length uncertainties (see Cazenave et al. 2013), and was to a certain extent arbitrary. In order to describe large datasets on rhythmic seabed surface roughness in a fully automated way, an improved method which considers the presence of bedforms with larger lengths is required.

The method requires highprecision bathymetry and as a result provides a quick and efficient tool for estimating topographic roughness, vital especially for constructing numerical models of hydrology and sediment transport.
Notes
Acknowledgements
The authors are grateful to the National Centre for Research and Development (Poland) for funding this work by grant VISTULA No PBS1/A2/3/2012. We wish to thank to scientists of the Department of Operational Oceanography of the Maritime Institute in Gdańsk for their help in preparing and carrying out the bathymetric survey. The authors would like to express their gratitude to Pierre W. Cazenave, one anonymous reviewer and the journal editors for their constructive criticism and valuable comments.
Compliance with ethical standards
Conflict of interest
The authors declare that there is no conflict of interest with third parties.
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