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Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology

César M. Alpaca-Chacón

^1,2,*,

Bismarck Jigena-Antelo

^1,*

César A. Quispe-Gonzales

²,

Douglas D. Sarango-Julca

²,

Antonio Contreras-de-Villar

and

Juan J. Muñoz-Perez

CASEM (Andalusian Higher Marine Studies Center), University of Cadiz, 11510 Puerto Real, Spain

Professional School of Fluid Mechanics Engineering, Universidad Nacional Mayor de San Marcos, Lima 15001, Peru

ETSIA (Algeciras Higher Technical School of Engineering), University of Cadiz, Avda. Ramón Puyol s/n, 11202 Algeciras, Spain

Authors to whom correspondence should be addressed.

J. Mar. Sci. Eng. 2024, 12(9), 1660; https://doi.org/10.3390/jmse12091660

Submission received: 5 August 2024 / Revised: 7 September 2024 / Accepted: 12 September 2024 / Published: 16 September 2024

(This article belongs to the Special Issue Recent Experiences and Monitoring in Coastal, Fluvial and Marine Hydrography)

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Abstract

This review updates the different categories and formulations of the calculation of longshore sediment transport (LST) and summarizes their advantages and disadvantages. Most of these methodologies require calibration for areas different from those studied by their authors. Thus, a method of validation and calibration is presented here by processing satellite images with CoastSat software (release v 2.7) to determine accretion and erosion volumes. This low-cost methodology was applied to Salaverry Beach (Peru) to compare the results of the different formulations. A range of variation between −96% and +68% was observed concerning the error, with van Rijn’s formula being the most accurate for this particular case.

Keywords:

erosion; accretion; CERC; low-cost methodology; Salaverry Beach; Perú

1. Introduction

One of the most frequently needed variables in coastal engineering is the total longshore sediment transport (LST) rate. This is essential for addressing issues related to filling dredged channels, spreading beach fills, disposing of dredged material, and the morphodynamic reaction of coastal areas for engineering projects. In order to obtain adequate management of coastal engineering for port design or beach maintenance applications with the inclusion of structures such as breakwaters and others, it is imperative to have a solid knowledge of LST.

There are currently different methods for dealing with the issue of LST. In LST studies, energy, current power, and dimensional analysis methods are used, as they are simple and easy to calculate, as opposed to computational numerical models, which are more complex and require greater computational expenditure. When it comes to energy methods, we have two subclassifications. While the current energy-method approach is more broadly applicable to any sediment transport circumstance, the energy-flux method was specifically developed for coastal sediment transport. The method of balancing forces is more related to the shear stresses on the seabed, and the dimensional analysis method employs a set of dimensionless parameters for the study of LST [1].

Recent improvements in numerical simulations depicting coupled physical systems have demonstrated that the interaction of currents, waves, and winds can have a significant impact on nearshore processes [2,3,4].

These investigations all demonstrate how strongly waves and currents interact. A correct explanation of coastal hydrodynamics requires an understanding of both waves and tidal processes. Currents and tides are observed to affect wave breaking and sea turbulence, according to Ardhuin et al. [4].

The three primary categories of the LST prediction technique are energy methods, which are further divided into energy-flux approaches and the current energy method, dimensional analysis, and balancing forces.

A brief summary of the state of the art should begin with USACE [5], Tomasicchio et al. [6], and Swart [7], who introduced mean grain size as a variable of LST in sands. On the other hand, Nichols and Wright [8] and Chadwick [9] studied coarser sediments, combining data from sedimentation traps and electronic and aluminum tracers. Moreover, Chadwick [9] and Brampton and Motika [10] obtained other coefficients for dimensionless calibration parameters.

Later, by using the H_s (significant wave height, in meters) for the 46 data points that most closely matched their selection criteria for D₅₀ < 1 mm, Schoonees and Theron [11,12] fitted an energy-flux expression. The coefficient of determination (R²) for D₅₀ > 1 mm was significantly poor.

As for the energy-flux method, the CERC equation [5] (created by the Corps of Engineers, US Army) is available. It was developed for sandy beaches and is widely used around the world to this day. There are also the formulas of Inman and Bagnold [13], Komar [14]. Several authors have proposed values for the calibration constant K, including Greer and Madsen [15], Watts [16], and Caldwell [17], which were of little use.

Regarding the current power method, Bailard [18] and Bagnold [19,20] propose an LST equation derived from a locally time-averaged equation. Bailard [18] derived a K value for CERC by integrating the LST rate averaged over local time, used in conjunction with H_rms (root-mean-square wave height, in meters). He introduced the median diameter variable D₅₀ through the velocity of particle fall. Bagnold [19,20] indicated that a percentage of the current is reversed in suspended and background LST.

The LST method of balancing forces is connected to the bed’s shear stresses, derived from the currents induced by the waves, and, therefore, is an evolution towards energetic methods and dimensional analysis. Frijlink [21] proposed the first such formula, divided into two parts: one for the calculation of currents and the other for LST. Bijker [22,23] used hydrodynamic methods averaged by their integration into the wave zone. Damgaard and Soulsby [24] also used this method based on a formula created by Soulsby [25], in which the authors used simplifications, such as a uniform beach, waves in shallow water, constant breakage, and no refraction, among others.

The dimensional analysis method was developed thanks to laboratory experiences, calibrated with oceanographic parameters, and corroborated with LST rates. Its expressions are very similar to those of the energy-flux method. Evidently, like any dimensionless method, it was proposed based on dimensionless physical parameters. van Hijum and Pilarczyk’s formula [26] was derived from their laboratory experiments and those from van Hijum [27]. The LST formula by Kamphuis et al. [28] was created with consideration for beach slope and grain size, specifically for usage on sandy beaches. This came about as a result of numerous lab experiments and an extensive collection of field data. Kamphuis refined this formula himself [29] by using an additional set of hydraulic model tests.

In addition, van der Meer [30] proposed an alternative LST equation for the dimensional analysis method, modifying the threshold parameters and the calibration constant. This equation is also valid for coarse-grained beaches, given its comparison with experimental data from van Hijum and Pilarczyk [26], who concluded that gravel beaches absorb more energy from percolation when compared to Schoonees and Theron sandy beaches [31].

According to van Wellen [1], equations based on energy methods give good results for coarse-grained beaches, whereas dimensional analysis equations overpredict LST results. Thus, it should be noted that the former models do not give the same results, as they cannot be calibrated for all wave conditions and beach types. For example, in LST studies on pebble beaches, calibration is mainly performed using results from the use of traps or shoreline profilers (all of them high cost) [1]. On the other hand, tracers are the best technique for calibrating LST models, according to Muñoz-Perez et al. [32], rather than expensive traps. However, the results of the LST calculation will remain uncertain and require further validation [33].

For all these reasons, and after numerous investigations and the collection of field and laboratory data, a general formula has not yet been found for all types of beaches and all places because of the different geographical and environmental conditions of the littoral and the ocean.

Due to being faced with this reality, this paper aims to present an update of the different LST formulas created to date. In addition, their advantages and disadvantages are highlighted, and an estimate of the error percentage by applying the different formulations to a particular case is found. As a reference value, an easy and low-cost method is proposed, namely the calculation of the erosion/accretion volume by comparing satellite photos.

Study Site

The geographical coordinates of the study area (the port of Salaverry in Peru) are −8.1650139° S and −79.0108019° W (Figure 1). Swell waves come from the SW direction (85% probability) and, to a lesser degree, from the S and SSW directions. The mean annual breaking wave height is H_b = 2.23, while the significant wave height of the swell is H_s = 2.1 m. Its peak period (T_p) is 14.21 sec, and its predominant deepwater direction is 213.5°. The sediments are mostly very fine sand D₅₀ = 0.16 mm. The tidal range on average is 0.80 m, and the tidal range in syzygies is 1.28 m. The LST calculation will be carried out using the CoastSat method as validation for the Port of Salaverry (Peru). The data were obtained from private studies by the company H&O Ingenieros S.A., the National Port Company (ENAPU), and the Directorate of Hydrography and Navigation of the Peruvian Navy (DHN).

Hydrodynamic forcing used (wave parameter, water depth, and D₅₀) in situ data. Numerical models were not used for current velocity prediction, as they are not necessary for the LST calculations. Hydrodynamic variables are included as constant values in the LST calculations.

2. LST Formulations

In order to compile the different formulas developed over the years for the calculation of sediment transport, it was decided to group them into three categories according to the criteria contemplated for their conception and elaboration. The first group is based on energy methods and the second group on the balance of forces. The third group of equations is based on dimensional analysis.

2.1. Energy Methods

The methods based on energy are subdivided into energy flux and current energy approaches. While the current energy approach is more broadly applicable to any sediment transport condition, such as LST in rivers (estuaries), the energy-flux method was especially established for LST [2].

2.1.1. Energy-Flux Method

Waves travel from the deep ocean to the coast and dissipate their energy in different ways, including the resuspension of sedimentary material and the net movement of water masses in the form of currents (transverse and longitudinal). The angle of incidence of the wave fronts and their energy gradient along the coast determine a parallel net mass flow, which forms a longitudinal current (Figure 2). This component is mainly responsible for the morphological alterations of beaches and the transport of sediments [34].

According to Komar and Inman [35], the foundation of the LST energy-flux approach is based on the fact that wave power, P (W/m), is proportional to the submerged sediment transport rate, I (kg/s). The CERC (Coastal Engineering Research Center) equation, which was put forth by the USACE (US Army Corps of Engineers), is the most well-known formula for this category [5]. The formula includes background and suspended LST, expressed as the following:

I = K \cdot P

(1)

where K is a coefficient that was determined empirically. Through the use of the root-mean-square wave height in the breaking (b) zone, the K value used for H_b,rms (root-mean-square breaking wave height, in meters) is 0.77 [15]. By using significant wave height, H_s, instead of H_rms, a few more datasets were added to the CERC [5] equation, updating the value to K = 0.39. However, K = 0.92 is the comparable result using H_b,rms, according to Rosati et al. [36].

P = {(E \cdot C_{g})}_{b} \cdot \sin θ_{b} \cdot \cos θ_{b}

(2)

Here,

E = \frac{1}{8} \cdot ρ \cdot g \cdot H_{b}^{2}

is the energy density of waves (in J/m²), ρ is the density of seawater (in kg/m³), gravity accelerates at an average speed of g = 9.81 m/s², the wave’s height in the breaking zone is expressed in meters (or H_b), and θ_b is the wave’s angle at the breaking zone (the distance between the wave’s crest line and the coastline). Finally, C_g is the velocity of the group celerity (m/s), given by

C_{g} = \frac{c}{2} \cdot (1 + \frac{2 k h}{s i n h 2 k h})

(3)

where c is the velocity of the wave, the wave number is k (m⁻¹), depth is represented by h (m), and the volumetric LST (Q in m³/s) is

Q = \frac{I}{Γ} = \frac{K \cdot P}{Γ}

(4)

with

Γ = \frac{(ρ_{s} - ρ) \cdot g}{1 + e}

(5)

ρ_{s}

is the sediment’s density (2650 kg/m³ approx.), e is the void ratio (0.6 approx.), and

Γ

is the ratio used to convert volumetric transport rates from mass ones.

Equation (4) gave acceptable results with the calibrated K applied to the break zone. The calibrated K was lower than the theoretically used version and has a strong influence on transport velocity in the breaker zone [37].

Swart [7] introduced the variable D₅₀ within K for medium-sized sediments.

Q = 0.116 {l o g}_{10} (\frac{0.0146}{D_{50}}) \cdot \frac{P}{Γ}

(6)

Chadwick [9] included two dimensionless parameters, D₉₀/H and L/D₉₀, in the CERC equation, where the wavelength is represented by L (m). H is the representative height of the waves (m).

Q = \frac{K \cdot P}{Γ} {(\frac{L}{D_{90}})}^{ε_{1}} {[1 - \frac{8.1 D_{90}}{H}]}^{ε_{2}}

(7)

In this Equation, it is necessary to know the ε₁ and ε₂ for each location. Chadwick [9] used Equation (7), making ε₁ = 0 and ε₂ = 1, resulting in

Q = \frac{K \cdot P}{Γ} [1 - \frac{8.1 D_{90}}{H}]

(8)

When Chadwick [9] applied Equation (8) compared to the normal CERC equation (4) that is, without the threshold and particle size terms of Equation (7), the K value produced was 7%. However, applying the extended field for data, Shoreham data in Bray et al. [38] resulted in K = 0.07 (measuring wave height with H_rms), which was only 9% of the original K value.

Chadwick [9] adapted the information from van Hijum [27] and van Hijum and Pilarczyk [26] in terms of the breaking wave condition.

Q = 0.0013 (g D_{90}^{2} T_{s}) \cdot w \cdot (w - 8.3) \cdot \sin θ_{b}

(9)

w = \frac{H_{o b} \sqrt{\cos θ_{b}}}{D_{90}}

(10)

Here, T_s is the significant wave period in s, H_ob is the significant wave height in wave breaking in m, and θ_b is the wave angle in wave breaking (Figure 3).

Subsequently, Schoonees and Theron [11,12] found a correlation for D₅₀ < 1 mm and K = 0.41 by using a correlation value of R² = 0.77.

I = 0.41 P (R^{2} = 0.77)

(11)

Tomasicchio et al. [39] proposed a general longshore transport (GLT) procedure that belongs to the typology based on an energy-flux approach and is combined with an empirical relationship between wave-induced forcing and the number of moving beach sediment units. To be more precise, the GLT approach takes a suitable mobility index into account and assumes that the material (unit) moves with the same obliquity of breaking and reflected waves at the breaker depth during both up- and down-rush. If, and only if, a unit is removed from an updrift area of an extension equal to the longitudinal component of the displacement length, l_d sin ϑ_d, where l_d is the displacement length, and ϑ_d is its obliquity, then the unit passes through a particular control section. This methodology will not be expanded during this review due to the extension of the calculations; we are thinking about developing a fascinating approach in a subsequent article.

2.1.2. Current Energy Method

Water flows from coastal systems are accompanied by sediment transport, which requires the work of energy of waves and wind (effort at the bottom). The potential energy of the water (due to its elevation) is transformed into kinetic energy, which it uses to move the sediment. Another part dissipates to overcome the friction at the boundaries and in the bed. The rate of potential energy expenditure over time is the power of the current, a fundamental parameter for the evolution of a coastal system.

Bagnold [19] pointed out that a percentage of the wave power produces suspended and background LST, as the energy of the current is able to transport deposit particles in the current’s direction.

Based on Bagnold [20], Bailard [18] presented an equation for LST averaged over local time, which applies to the K from the CERC formula for H_rms.

K = ε_{b} K_{1} + ε_{s} K_{2} + ε_{s}^{2} K_{3}

(12)

Equation (12) considers the load at the bottom, as well as the suspended load. Additionally, he notes that on coarse-grained beaches, suspended sediment transfer is unlikely to happen because the LST can be overestimated.

Bailard [18] proposed ε_b = 0.13 and ε_s = 0.32. The K₁, K₂, and K₃ values, which are consistent with eight field and two USACE laboratory datasets [5], are given by the following relationships:

K_{1} = 0.385 + 20 \sin^{2} (2 θ_{b}); K_{2} = 0.228 \frac{u_{m b}}{w_{s}}; K_{3} = 0.123 t a n β (\frac{u_{m b}}{W_{s}})

The last term in Equation (12) represents the suspended load, and the first term can be neglected because its input to the formula is small. Bailard [18] concluded that the modification of the coefficient K allows us to broaden the CERC equation’s use when considering the velocity of fall, w_s, for representative grains.

With these considerations, the K value is obtained from

K = 0.05 + 0.007 \frac{u_{m b}}{w_{s}} + 2.6 \sin^{2} (2 θ_{b})

(13)

where u_mb is the maximum orbital speed below the surface of the sea (m/s), w_s is the speed at which sediment falls (m/s), and tanβ is the average beach slope. The original meaning of u_mb is the maximum orbital speed on the bottom, though the maximum orbital speed below the surface equals the maximum orbital velocity on the bottom because Bailard [18] used the shallow water wave theory.

Morfet [40,41,42] proposed a new formula:

Q = \frac{K_{M} {(ρ u_{*}^{3} - ρ u_{+ c r}^{3})}^{3 / 2} {(\sin θ_{b})}^{3 / 4}}{g (ρ_{s} - ρ) D_{90}^{2}}

(14)

where K_M is a calibration coefficient equal to 2.84 × 10⁻⁵,

ρ u_{*}^{3} - ρ u_{+ c r}^{3}

is the strength of the virtual wave, u_+cr is the velocity in critical or threshold wave conditions for sediment movement,

θ_{b}

is the angle between the wave front and the coastline in breaking, and

u_{*}

is the dissipation velocity, shown as

u_{*} = {(\frac{D_{d}}{ρ})}^{1 / 3}

(15)

with D_d being the energy dissipation rate of waves, which can be used as a factor of LST instead of current power and can be found by using

D_{d} = \frac{ρ g^{3 / 2} H^{5 / 2}}{4 L}

(16)

where L is the wavelength (m), and H is the representative wave height (m) at the breaking zone.

2.2. Methods of Balancing Forces

This method indicates that LST is connected to the lowest shear stresses. It requires a prior hydrodynamic formulation for radiation-induced currents caused by waves. Consequently, in comparison to the energy technique and the dimensional analysis method, it is more intricate (and difficult to solve). Out of all the calculations that were not evaluated using Shoreham’s data, the balance of forces equation is the most precise.

Subsequently, Bijker [22,23] improved the use of hydrodynamic methods by numerically integrating the rate of transportation in the surfing area. It should be noted that they did not propose a new LST equation.

Damgaard and Soulsby [24] combined the hydrodynamic method with the balance of forces method to determine total LST. In order to do this, they introduced simplifications and assumptions, such as (a) homogeneous beach circumstances, (b) a roller in a shallow sea, (c) a regular rupture indicator, (d) no refraction in the surfing area, and (e) the gradient of radiation stress balanced by the shear force at the bottom. They obtained a resultant analytic expression for Q (total longshore transport rate), which is a combination of current-dominated transport, Q₁, and wave-dominated transport, Q₂. It also considers the current angle, θ_m, and the wave angle, θ_w:

Q = s i g n \{θ_{b}\} \cdot m a x \{|Q_{1}|, |Q_{2}|\}

(17)

θ_{m a x} = \sqrt{{(θ_{m} + θ_{w} \cos ϕ)}^{2} + {(θ_{w} \sin ϕ)}^{2}}

(18)

ϕ = \frac{π}{2} - θ_{b}

(19)

where the threshold condition is Q = 0 for

θ_{m a x} \leq θ_{c r}

θ_{m a x}

is the direction of the shear stress vector maximum, which is the result of a component mean,

θ_{m}

, and an oscillatory part,

θ_{w}

, resulting from the incoming wave, and

θ_{c r}

is the threshold angle for Q = 0.

The methodology of Damgaard and Soulsby [24] will not be included in the present study due to the high complexity of its formulation and the requirement of data that are not available for its treatment.

Wilson [43] used the following wave-induced friction factor (f_w) for moving bed and laminar flow conditions as a parameter for the calculation of wave-dominated transport Q₂:

f_{w, s f} = 0.0655 {(\frac{φ_{b} \cdot H}{g})}^{1 / 5} {(π \cdot (s - 1) \cdot T)}^{- 2 / 5}

(20)

where H represents the depth (m), s is the relative density, T is the representative wave period (s), and

φ_{b}

is the surf zone’s constant ratio of wave height to depth.

2.3. Dimensional Analysis Methods

The set of dimensional formulas was created from multiple regression analysis, and some dimensionless parameters are considered important in LST. These equations are similar in shape to the energy methods but were calibrated and validated experimentally without considering the van Wellen energy approach [1].

Damgaard and Soulsby’s [24] and Soulsby’s [25] formulas, based on laboratory data from 1994 and input from van Hijum and Pilarsick [26], showed reliable results on coarse-grained beaches. In this case, Hs was used in the equations.

These equations were derived from dimensionless mathematical parameters rather than physical principles. From the data obtained from laboratory tests carried out by van Hijum and Pilarsick [26] and van Hijum [27], the following expression was obtained:

\frac{Q}{g D_{90}^{2} T_{s}} = 7.12 \times 10^{- 4} \frac{H_{s d} {(\cos θ_{b})}^{1 / 2}}{D_{90}} [\frac{H_{s d} {(\cos θ_{b})}^{1 / 2}}{D_{90}} - 8.3] \frac{\sin θ_{b}}{\tanh (\frac{2 π d}{L})}

(21)

where H_sd is the significant wave height, and d is the water depth, and T_s is the wave period (in s).

The term B =

\frac{H_{s d} {(\cos θ_{b})}^{1 / 2}}{D_{90}} - 8.3

of Equation (21) must be greater than zero. Otherwise, Q would be zero or negative. Likewise, Brampton and Motika [10] pointed out that this term should be raised to a power greater than one to represent LST with high levels of wave energy.

For sandy beaches, Kamphuis et al. [28] developed an LST equation that includes the slope of the beach (

\tan α)

and D₅₀:

Q = 1.28 \frac{\tan α H_{s b}^{\frac{7}{2}}}{D_{50}} \sin 2 θ_{b}, [\frac{k g}{s}]

(22)

Equation (22) can also be applied to coarse-grained beaches, according to extensive laboratory data, with H_sb being the significant wave height in the breaking condition.

Van der Meer [30] reanalyzed the information from van Hijum and Pilarsick [26] and proposed the following equation:

Q = 0.0012 g D_{n 50} T_{p} H_{s} \sqrt{\cos θ_{b}} (\frac{H_{s} \sqrt{\cos θ_{b}}}{D_{50}} - 11) \sin θ_{b} [k g / s]

(23)

where

D_{n 50} = {(\frac{M_{50}}{ρ_{s}})}^{1 / 3}

(24)

M_{50}

is the average weight of the sediment, and T_p is the peak wave period.

Kamphuis [29] refined the equation from additional experiences in hydraulic models.

Q = 2.27 H_{s b}^{2} T_{p}^{1.5} m_{b}^{0.75} D_{50}^{- 0.25} \sin^{0.6} 2 θ_{b} [k g / s]

(25)

Subsequently, Kamphuis [29] found Equation (26) to be valid for LST on sandy beaches when contrasted with the field data and laboratory experiments. However, for coarse-grained beaches, the results can exceed this by two to five times. This is because gravel beaches absorb a greater amount of wave energy as they filter through the grains.

The annual LST is estimated by using the following:

Q = 63433 χ_{K a m p h u i s}, [m^{3} / y e a r]

(26)

where

χ_{K a m p h u i s} = \frac{1}{(1 - p) ρ_{s}} \frac{ρ}{T_{p}} L_{o}^{1.25} H_{s b}^{2} m_{b}^{0.75} D_{50}^{- 0.25} \sin^{0.6} 2 θ_{b}

(27)

with p representing the porosity index and L_o representing the deepwater wavelength. m_b is the beach slope.

While considering the precision of the expected rates of transportation during hurricanes, the following alternate rendition is proposed:

Q = 50000 χ_{K a m p h u i s}, [m^{3} / y e a r]

(28)

Van der Meer and Veldman [44] indicated that Equation (23) can be simplified to

Q = 0.0012 π H_{s} C_{o p} \sin 2 θ_{b}, (g r a v e l a n d s a n d b e a c h e s)

(29)

where C_op is the peak deepwater celerity wave =

g T_{p} / 2 π

. For breakwater barriers and angles of 15–40°, we have

\frac{H_{s}}{Δ D_{n 50}} < 10

, with Δ being the relative specific weight of the submerged solid material.

Q = 0 for H_{o} T_{o p} < 105

Q = 0.00005 {(H_{o} T_{o p} - 105)}^{2}

(30)

Here, H_o is the deepwater wave height (m), and T_op is the deepwater peak wave period (s).

Mil-Homens et al. [45] and Kamphuis [29] concluded that, to reduce the bias to zero and the RMSE value to around 0.40 (close value to the noise in the statistics present from validated information), this could be achieved by using multiple regression and varying the exponents of Kamphuis’ formula, obtaining the following equation:

Q = 0.15 \frac{ρ_{s}}{ρ_{s} - ρ} T_{p}^{0.89} m_{b}^{0.86} D_{50}^{- 0.69} H_{s, b r}^{2.75} \sin^{0.5} 2 θ

(31)

where H_s,br is the significant wave height in the breaking condition.

Then, by gathering field data, van Rijn [46] proposed a formula for LST known as the CROSMOR framework, which applies to sandy, as well as gravelly, beaches.

Q = 0.00018 ρ_{s} g^{0.5} m_{b}^{0.4} D_{50}^{- 0.6} H_{b}^{3.1} \sin 2 θ_{b}

(32)

where H_b is the wave height in the breaking condition and m_b is the beach slope.

The CROSMOR model gives slightly increased findings (by a factor of 1.5) about the storm circumstances’ (high-energy circumstances) measurement values. Additionally, in low-wave circumstances, it produces lower results (by a factor of 2). When dealing with swell-type wave situations, the quasi-regular wave technique yields the greatest results. The lack of coastal transport in the oscillation zone could be the reason for the underprediction for low-wave situations.

2.4. Summary of the Different Formulations

In order to summarize the various methodologies cited in this work, the methods with the most important hypotheses and the field of validity of each of these are described in Table 1. Column 1 shows the LST calculation method type, column 2 the author(s) and reference, column 3 the hypotheses, and column 4 the formula validity field.

2.5. CoastSat Software Method

The free software CoastSat was used for the calibration and validation of this methodology. It is a Python program (using the Anaconda 3 environment manager). This allows users to obtain a time series of shorelines for any coast in the world, starting 39 years ago (and increasing) from the Sentinel 2 and Landsat missions. With these satellite images and the CoastSat software Vos et al. [48], the evolution of the coastline is determined computationally, Figure 4.

With this digital information on the temporal evolution of the coastlines, the area of advance or retreat of all of them is calculated given a baseline defined by the researcher. With this same baseline, closed polygonal areas are generated for all the coastlines of the different images used by CoastSat. Then, with the temporal evolution of the areas and with the help of the Birkemeier closure depth [49], the volumes of accretion or erosion are calculated, and these volumes are divided by the elapsed time interval of each of them, finally obtaining the average LST in m³/s. The equation used is the following:

Q = \sum Q_{k} a n d Q_{k} = \frac{∆ A_{k}}{∆ t_{k}} D_{s} (33)

where

A = \sum A_{k} = \sum ∆ x_{k} \cdot ∆ y_{k}

, and thus, A is the variation in the total surface of the beach.

Figure 4. The conservation equation LST (Pernald-Considere) [50] used with the images downloaded from CoastSat.

Moreover,

D_{s} = h_{B} + h_{C}

is the sum of the height of the emerged beach or berm (h_B) plus the Birkemeier closure depth (h_C) (see Figure 4).

Finally, the area A_k is the area between the current coastline k and a base coastline (selected by the researcher, but behind the entire group of coastlines of the beach). N = the quantity of coastlines discharged between the years 2016 to 2022. Thus, we have n-1 areas ∆A_k = A_k+1 − A_k in each interval of time ∆t_k.

Q_k is the LST in m³/sec for each ∆A_k, obtaining Q_n−1 LST flows in total. And to obtain the average annual flow, all the Q_k Δt_k are added together and this final volume of sediments is divided by the interval of time between the downloaded images (2016–2022). This process included the dredging volumes carried out during the years under study.

As for the boundary of the artificial structure (breakwater), the CoastSat method is true in that Q_L = 0, and Equation (33) only determines the advance or retreat (sedimentation or erosion) that accumulates in the entire southern area of the breakwater.

As a deficiency of this CoastSat method, we can mention that the calculation depends on the sub-pixel resolution of the images (10 m approx.) and that the LST can only be determined a posteriori, according to the availability of existing historical images of the study area. The hydrodynamic variables to estimate the LST calculations were obtained from the study [51].

3. Results and Discussion

A comparison of LST in the port of Salaverry was made according to the calculations of the different methodologies with the results of the processing of the CoastSat software. The main advantages and disadvantages of each method are pointed out. The results are shown in Table 2. Column 1 shows the type of method, and column 2 shows the author(s). Columns 3 and 4 show the advantages and disadvantages observed during the use of the various calculation methodologies for LST.

The data in Table 3 depend on the temporal availability of the LandSat and Sentinel 2 missions in the Salaverry port area. The sum of all these areas gives us the LST area in the entire available time range (2550 days). With this data, the volume of accumulated sediments (+) or erosion (−) was calculated thanks to the closing depth D_s.

From Table 3, it is verified, according to LST calculations and sediment volumes, that the El Niño and La Niña phenomena on the northern coast of Peru (Salaverry beach), increased these quantities in the years 2019, 2021, and 2022, according to reports of its presence in these years by the National Meteorological and Hydrological Service of Peru—SENAMHI.

Given these natural events presented in Peru, the values of standard deviation of the mean are so high in the main variables determined in Table 3.

The results of the calculations carried out with the different methodologies for the LST are shown in Table 4 and are compared with the results obtained when using the CoastSat software, obtaining the percent relative error. Column 1 shows the type of method, and column 2 shows the author(s). Column 3 shows the calculation of each author’s LST, and column 4 shows the percent relative error based on the CoastSat method.

The calibration and validation methodologies used were carried out by using the free software CoastSat with satellite images, which determines the evolution of the coastline computationally through remote sensing. These images correspond to the years from 2016 to 2022. These results are used to determine the forward or backward areas over a given time interval from a baseline. Then, with these determined areas, the accretion or erosion volumes are calculated with the help of Hallermeier and Birkemeier’s depth of closure [52]. Finally, this volume of accretion or erosion, divided by the elapsed time analyzed, results in the average LST in m³/s.

The study of the shorelines by using the CoastSat methodology resulted in a quantity of 0.025 m³/s for LST, which represents an average transport rate between multiple temporal shorelines along the beach.

3.1. Energy Methods

Energy-based methods are subdivided into energy-flux method and current energy approaches.

3.1.1. Energy Flux

Regarding the CERC [5] equation, it must be highlighted that it does not include the variables m_b (beach slope) and D₅₀ (median diameter). However, T_p (wave period) is considered through C_g (group celerity). This could give different results on beaches not yet calibrated compared to the CoastSat model, resulting in a negative relative error of 84%.

Schoonees and Theron [12] stated that the Swart formula [7], which includes D₅₀, does not provide correct results for fine sand beaches, which is a disadvantage. However, the equation of Schoonees and Theron [11,12] gives a result far from that calculated with the CoastSat method, with a relative error of −92%. While using the Swart formula, which is very simple, its result is close to that calculated by the CoastSat method, with a relative error of only 4%.

3.1.2. Current Energy

Bailard’s Equation (13) considers the grain size of the sediment, included in the particle fall velocity w_s. In the present case, the relative error is 52% compared to that calculated with CoastSat. This calculation is simple, and the data needed to determine the LST are easy to obtain. As a disadvantage, it can be indicated that this equation does not consider variables such as depth or current velocity.

3.2. Balance of Forces Method

Regarding the balance of forces methods, we can indicate that they use previously calculated hydrodynamic formulations for wave-induced velocity currents and bottom stresses. A disadvantage of these methods is that they are complex and difficult to calculate, since they contain parameters that are difficult to specify in reality, such as impulse variables.

3.3. Dimensional Analysis

It should be noted that most dimensional analysis-type formulas depend largely on the quality and representativeness of the field data used for their development and validation. It also presents difficulties to accurately predicting coastal transport under high-energy wave conditions (surf zones), where field data are scarcer.

Brampton and Motyka [10] modified the formula of van Hijum and Pilarczyk [26], simplifying this equation, which includes laminar flow and a coefficient of friction (f_ws,f) for bed motion. In the Port of Salaverry, the results show an error of 68% compared to that calculated by CoastSat. For more details on this method, it is recommended to see their article.

The Kamphuis [29] formula considers key parameters that affect the LST, such as grain size, beach slope, and wave characteristics. It shows good applicability to sandy beaches, which are the most common. For our particular calculation case, applied to a fine sand beach, an error of −52% is obtained. As a disadvantage, we can indicate that it is not well-validated for coarse-grained and pebble beaches.

According to van Rijn [46], his Equation (32) provides good results between the calculated and measured values in moderate wave conditions and fine sediments. It also includes key parameters that influence the LST, such as grain size, beach slope, and wave height. Therefore, it is applicable to a variety of coastal sites and climatic conditions. For our particular case, it is observed that the error, compared to that calculated by the CoastSat method, is 20%. As a disadvantage, it can be pointed out that the formula is not always accurate in extreme conditions, such as very high waves or very coarse sediment size.

3.4. General Summary

As we can see in Figure 4, the results of the methods that come closest to the results obtained with CoastSat are van Rijn [46], Swart [7], and van der Meer [30], with 20%, 4%, and 0%, respectively. The methods of CERC [5], Schoonees and Theron [11,12], and van Hijum and Pilarczyk [26] are very far from the result obtained with CoastSat (−84%, −92%, and −96%, respectively), Figure 5.

Energy-flux methods for calculating LST are predicated on the principles of wave energy dynamics as they interact with coastal environments. The most prominent approach, the energy-flux method (CERC), leverages the parallel component of wave energy flux to establish a linear correlation between sediment transport and energy input, encapsulated in the equation (Q = K

\cdot

P_ls), where (Q) represents LST and (P_ls) is energy flux.

The current energy methods for estimating LST are grounded in the interaction between hydrodynamic forces and sediment dynamics along coastal environments. These methodologies primarily focus on the energy imparted by currents, particularly those generated by wave action and tidal influences, to predict sediment movement.

Regarding dimensional analysis methods, it can be stated that these are derived from dimensionless parameters and use physical variables calibrated through laboratory and field experiments.

The formulation of van Hijum and Pilarczyk [26] and van Hijum [27], following Brampton and Motyka [10], indicated that their term defined in Equation (21) as A should be used for the high energy levels of the waves producing the LST. The results obtained with this equation generate an error of −96% compared to that obtained with CoastSat. At the same time, this formulation introduces numerous variables, complicating its calculation.

Van der Meer [30] reanalyzed the data of van Hijum and Pilarczyk [26], generating a new formula introducing D₅₀ instead of D₉₀. This is a simple formula to calculate and has an error of 0% with respect to that obtained with CoastSat. Furthermore, van der Meer and Veldman [44] proposed Equation (29), which can be used for gravel and sand beaches, although the results generate a higher error, on the order of 46%.

Van Rijn [46] collected field data from 22 experiences, and proposed his Formula (32) for LST, giving rise to the CROSMOR model, which is applicable to sand and gravel beaches. This is a simple formula and results in a variation of only 20% in the calculation compared to that obtained by CoastSat.

It should be noted that, in this work, several calibration procedures of the cited LST methods are summarized, for example, the creation of sedimentation pools, littoral drift roses, gravel traps, tracers, samplers, tip growth, or breakwater accretion, among others (Walton and Dean [53]). Regarding the cost of implementing these calibration methods, it is noted that this can vary considerably. This depends on several factors, such as the scale and complexity of the study, the geographic location, and the equipment and technology used, as well as the costs associated with specialized personnel, logistics, and duration of the study. In the present validation carried out with CoastSat, the costs are much lower, since it would not be necessary to have workers or civil works, personnel, and equipment displacement. In our proposal, only a good computer and the CoastSat software are needed, which is freely available and considering that satellite images are also free. Furthermore, compared to traditional calibration methods, this low-cost methodology does not experience a significant loss of accuracy.

It is worth mentioning that there are still places where not many coastal satellite images are available. Not all available images have the same temporal and spatial resolution, which limits the ability to obtain more accurate results in these coastal areas. Both considerations make the calculation of LST difficult. In addition, global sea-level rise is a critical issue in coastal areas, requiring accurate estimation of coastal retreat and sand volume to address the sediment deficit. However, a specific methodology has not yet been developed [54].

Even so, the use of satellite data through comparison software (e.g., CoastSat) is recommended as a validation method for LST studies.

Finally, from the analysis carried out, it can be observed that all the formulations of the three LST categories present results with a wide variation, even greater than 90%. This is possibly due to the fact that the equations use very diverse parameters and variables.

With the results obtained from Table 4, an average LST value of 0.029 m³/s was determined, with a relative standard deviation of 46.63% and a standard deviation of 0.0135. However, this average value was obtained for a wide range of LSTs and may not be significant. The value calculated through the CoastSat method was 0.0254 m³/s for Salaverry Beach in Peru.

4. Conclusions

CoastSat uses real satellite images with good resolution at a sub-pixel level, which provides a real image of the variation of the coastline applied to many coastal environments. In addition, it is widely used to determine shorelines by many researchers, due to its very low computational complexity and high data availability for its calculation in most cases. The uncertainty in the parameters is minimal given the scaling factor of the images and their good resolution. In this work, the LST calculation was carried out using the CoastSat method, which only requires freely accessible satellite images. The result was an LST of 0.025 m³/s.

The energy flow methods were analyzed, observing that the Swart equation has a good agreement, with an error of 4% with respect to the calculation with the CoastSat method. While the CERC requires less data and is very simple, errors increase when not including important variables in the determination of the LST, such as the slope of the beach and the wave period. For the present case, the error was −84%.

While formulas such as CERC, Swart, Chadwick, Schoonees, and Theron offer a low computational complexity and acceptable data availability, their predictive accuracy benefits from high-quality in situ measurements. However, inaccurate LST predictions for the design and maintenance of coastal structures imply higher economic investments and a pressing need for cost-effective calibration methodologies.

Using the current energy method proposed by Bailard [18], an error of 52% was obtained with respect to that calculated by CoastSat. This equation is very simple, requiring data that are easy to obtain. But, it does not consider variables such as depth or current speed.

As for the force balance methods, they have the disadvantage of using complex formulations, and they require parameters that are difficult to quantify in reality.

Dimensional analysis methods for predicting longshore sediment transport (LST) are significantly influenced by the quality and representativeness of the field data, particularly under high-energy wave conditions, where data acquisition is challenging. Among these, van Rijn’s method demonstrates a 20% error relative to the CoastSat calculations and incorporates essential factors, such as beach incline, sediment density, median diameter, and breaker wave height, making it suitable for sandy and gravel beaches. In contrast, the van der Meer formula yields a 0% error but is not recommended for gravel and pebble beaches. Other methodologies exhibit substantial errors, ranging from 20% to over 90%, compared to CoastSat results.

The main disadvantage of the CoastSat method is that it can only be used a posteriori after having satellite images available. This study proposes to improve the existing approaches by integrating digitized satellite imagery using remote-sensing techniques to improve LST calibration while addressing the economic constraints faced by many countries.

Author Contributions

Conceptualization: J.J.M.-P., B.J.-A., C.M.A.-C., C.A.Q.-G.; Methodology: B.J.-A., J.J.M.-P., C.M.A.-C., C.A.Q.-G., D.D.S.-J., A.C.-d.-V.; Software: C.M.A.-C., A.C.-d.-V.; Validation: B.J.-A., J.J.M.-P.; Formal Analysis: C.M.A.-C., A.C.-d.-V., C.A.Q.-G., D.D.S.-J.; Investigation: B.J.-A., J.J.M.-P., C.M.A.-C.; Resources, B.J.-A., J.J.M.-P.; Data Curation: C.M.A.-C., A.C.-d.-V.; Writing—Original Draft Preparation: C.M.A.-C., C.A.Q.-G., D.D.S.-J.; Writing—Review and Editing: B.J.-A., J.J.M.-P., A.C.-d.-V.; Visualization: C.M.A.-C., A.C.-d.-V.; Supervision: B.J.-A., J.J.M.-P., C.A.Q.-G.; Project Administration: B.J.-A., J.J.M.-P.; Funding Acquisition: J.J.M.-P., B.J.-A.. This research is part of the Doctoral Thesis of PhD Student Cesar Alpaca Chacón. All the authors contributed to the writing and supervision of the manuscript, as well as discussing the results. Each writer has examined and accepted the published version. All authors have read and agreed to the published version of the manuscript.

Funding

This research was partially funded by: Junta de Andalusia and European Union (Recovery, Transformation and Resilience Plan), grant number PCM_00124, and RNM912 Research Group belonging to the University of Cádiz.

Acknowledgments

To H&O Ingenieros Consultores S.A. and DHN for their greater willingness to support this study by providing relevant information.

Conflicts of Interest

The authors declare no conflict of interest.

Notation

c	Wave celerity (m/s)
C_g	Group celerity (m/s)
C_op	Peak deepwater celerity wave (m/s)
d	Depth of water (m)
D_(50/90)	(50/90%) Grain diameter representative (m)
D_d	Rate of wave energy dissipation (W/m²)
e	Ratio of voids
E	Density of waves (kg/s²)
f_w,t	Friction factor for rough turbulent flow induced by waves
f_w,sf	Friction factor for wave-induced rough turbulent flow
g	Acceleration due to gravity (m/s²)
h	Depth (m)
H	Wave height representative (m)
H_rms	Root-mean-square wave height (m)
H_s	Significant wave height (m)
I	Total mass rate of coastal transport (Kg/s)
I_ls	Coastal transport rate for total submerged weight (N/s)
k	Wave number (m⁻¹)
K_1,2,3	Coefficients of proportionality
L	Wavelength (m)
L_o	Deepwater wavelength (m)
M₅₀	50% of a unit’s medium mass is indicated on the mass distribution curve (kg)
m_b	Beach slope (m/m)
n	Quantity of observations
p	Porosity index
P_*	Power from virtual waves (kg/s³)
P_ls	Coastal wave power flow (W/m)
P_ls0	Coastal wave energy threshold value flow (W/m)
Q	Total volumetric rate of coastal transport (m³/s)
Q₁	Integrated long-distance coastal transport rate under current conditions (m³/year)
Q₂	Integrated coastal transport rate along the coast in circumstances where waves predominate (m³/year)
q	Rate of volumetric load transport (m³/(m s))
R	Coefficient of correlation
s	Comparative density
T	Typical wave duration (s)
T_op	Deep water peak wave period (s)
T_p	Peak wave period (s)
T_s	Significant time of wave (s)
T_z	Zero cross wave period (s)
u_*	Rate of dissipation (m/s)
u_+cr	Velocity in critical (threshold condition of wave for sediment movement, m/s)
U_mb	Maximum speed in orbit at the bottom (m/s)
w_s	Sediment fall speed (m/s)
b	Subscript indicating values sampled at the wave’s breaking point
C_r	Subscript signifying important values
d	Sampled data at the point when the water depth equals d are indicated by a subscript
m	Subscript representing mean values
max	Subscript indicating the highest values
sb	Subscript indicating significant wave height in breaking
sd	Subscript indicating significant wave height in the water depth d
w	A subscript representing values caused by waves
α	Beach gradient (rad)
Δ	Relative specific weight of the submerged solid material
ε_b, ε_s	Efficiency terms for suspended and bottom loading
ε₂, ε₁	Coefficients of calibration
γ_b	Wave’s angle break index between the direction of the waves and the contours of the beach (°)
$φ_{b}$	Constant ratio of the surf zone of wave height to depth.
$ϕ$	Complementary angle of wave breaking angle (°)
Г	Factor of conversion between volumetric and mass transport rates (kg/m² s²)
θ_b	Angle between the wave front and the coastline (°)
ρ	Fluid density (kg/m³)
ρ_s	Sediment density (kg/m³)
σ	Error in relative standard of estimation

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Figure 1. Port of Salaverry (Perú). Fuente: Google Earth Pro (2024).

Figure 2. Sketch showing LST based on wind direction and coastal waves.

Figure 3. Wave angle in wave breaking in LST.

Figure 5. The error of the different formulas used related to the CoastSat results for Salaverry Beach (Peru).

Table 1. Summary of the main formulas of LST.

1. Type	2. Method	3. Hypothesis	4. Field of Validity
Energy Flux	CERC [5]	The longshore wave power per beach length is directly related to the LST - immersed weight sediment transport rate, or I.	The CERC formula overpredicts transport rates for particle sizes and beach slopes, and significant tidal current velocities prevent its application.
	Swart [7]	Addition of a variable, K, as a function of D₅₀.	Despite its reliance on the grain diameter, [12] discovered that the equation by Swart was unable to provide a dependable K, even within the range 0.1 mm < D₅₀ < 1 mm.
	Chadwick [9]; Brampton and Motyka [10]	A significantly lower value for K, Brampton and Motyka [47], limiting the significant breaking wave height to 0.5 m or less, and treating ε₁ and ε₂ as zero.	The normal CERC equation yielded a K value of 7% for sand, contrasting with a 9% K value obtained using Equation (8) with threshold and particle size terms.
	Schoonees and Theron [11,12]	The study found a low correlation coefficient for D₅₀ > 1 mm in an energy-flux expression, suggesting that shingle transmission becomes more efficient when an energy threshold is surpassed.	Electronic shingles showed higher transport rate than aluminum tracers, suggesting a potential underestimate of transmission efficiency due to lack of transport data in high-energy situations.
Current Energy	Bailard [18]	It considers both bed load and suspended load, since coarse-grained beaches are unlikely to experience suspended sediment transfer.	It does not take into consideration the influence of the movement threshold. Therefore, applying it to beaches with coarse grains could lead to an overestimation of the overall longshore transport rate.
Dimensional Analysis	van Hijum and Pilarczyk [26]	The Delft coastal transport equation, based on laboratory tests, suggests sediment movement starts when square brackets exceed one, but higher wave energy levels require a higher threshold term.	This Equation (21) presents challenges in predicting LST due to wave characteristics varying between offshore and beach locations, causing errors in inshore measurements.
	Kamphuis et al. [28]	The methodology, developed through field data and laboratory experiments, may be applicable to beaches with coarse-grained sediments, considering beach slope and grain size impact.	The LST formula of Kamphuis et al. It was created for use on sandy and coarse-grained beaches.
	van der Meer [30]	This equation reanalyzed the data from former studies [27] after realizing how difficult it was to obtain the necessary parameters for the original Delft equation.	Equation Chadwick (1989) [9] and this one are very similar. Primarily, this one is different in that the threshold term and the constant’s value have been slightly altered.
	Kamphuis [29]	Kamphuis refined his formula using hydraulic models, validating it for field and laboratory sand transport rates, and comparing it to coarse-grained beaches.	Grain size and beach slope are included in Equation (25). These two components tend to cancel each other out as grain size increases on beaches with reduced slopes.
	van der Meer and Veldman [44] for H_s/(ΔD₅₀)	A statement for the longshore transportation of gravel and rock beaches. The kind of structure (sand, shingle, rock, or berm breakwater) and the wave climate affect longshore transit.	According to [43], Equation (29) should only be used for rock/gravel beaches (berm breakwater) at angles between 15 and 40° and only within 10 < H_s/D₅₀.
	Mil-Homens et al. [45]	The study modified the coefficients of Kamphuis’ formula to reduce bias to zero and RMSE to 0.40, resulting in Equation (31).	The Mil-Homens formula overlooks the impact of submerged sand bars near the coast, which can significantly influence wave breaking and LST calculation, unlike a representative Dean profile.
	van Rijn [46]	The CROSMOR model, a formula for longshore transport (LST) based on 22 field-data gatherings, analyzes wave height, wave incidence angle, and longshore transport.	The data set was used to analyze wave period, profile shape, and particle size, resulting in a common formula for gravel and sand LST.

Table 2. The application of the main formulas of LST to the case of the Port of Salaverry in Peru. The main advantages and disadvantages of each method are pointed out.

1. Type	2. Method	3. Advantages	4. Disadvantages
CoastSat Value	CoastSat applied to satellite images	Real data	This method can only be used a posteriori, after having satellite images available.
Energy Flux	CERC [5]	Flexibility: the CERC formula is adaptable to various conditions and can address problems also solvable by the Bijker formula; Accuracy: it effectively predicts sedimentation in narrow channels without breaking waves, aiding in hydraulic engineering project planning; Simplification: the CERC formula streamlines the calculation of longitudinal sediment transport, enhancing its practical applicability.	Limitations: the CERC formula solely considers the sign of function values in its approximations, neglecting their magnitude; Complexity: implementing the CERC formula can be challenging, particularly for individuals lacking experience in mathematical modeling; The accuracy of the CERC formula is contingent upon the quality and quantity of data used for calibration, which may pose challenges in certain situations.
	Swart [7]	Simplicity: it is easy to apply and only requires measurements of wave height and wave incidence angle; General applicability: it has been used in a wide range of coastal conditions, including sand and gravel; 3. Flexibility: it can be used to predict transport in different wave and sediment conditions.	The Swart formula has limitations, including limited precision, reliance on data quality, and not considering all processes affecting longshore sediment transport, such as current velocity and water depth.
	Chadwick [9]; Brampton and Motyka [10]	Precision: the Chadwick formula has been verified through field measurements and numerical modeling, indicating good precision in estimating longshore transport; General applicability: the formula has been applied to different wave and sediment conditions, making it useful for a wide range of situations; Flexibility: The formula can be used to predict transport in different wave and sediment conditions.	Limited precision: the Chadwick formula may inaccurately estimate transport in specific conditions, especially in high wave or coarse sediment scenarios; Dependence on data quality: the accuracy of results is contingent upon the quality and quantity of calibration data; Does not consider all processes: the formula does not include all processes that influence longshore sediment transport, such as longshore current velocity or water depth.
	Schoonees and Theron [11,12]	This formula, based on 270 global data sets, offers superior accuracy compared to the Shore Protection Manual, achieving a standard error of 0.515.	This formula, despite its potential, has limitations for coarse sediments due to the variability of the calibration coefficient.
Current Energy	Bailard [18]	The formula is simple, precise, and generally applied, requiring minimal complex data, and is verified through field measurements and numerical modeling for estimating longshore transport.	The formula’s precision is limited due to potential underestimation in high wave or coarse sediment situations, insufficient consideration of longshore current velocity, or water depth.
Dimensional Analysis	van Hijum and Pilarczyk [26]	Precision: the formula has been verified through field measurements and numerical modeling; General applicability: the formula has been applied to different wave and sediment conditions, making it useful for a wide range of situations; Flexibility: the formula can be used to predict transport under different wave and sediment conditions.	The formula has limited precision, may underestimate transport in high wave or coarse sediment situations, does not consider all processes, and may not be as precise as other formulas.
	Kamphuis et al. [28]	The formula offers superior precision in fitting field data compared to the Shore Protection Manual, incorporating critical parameters affecting longshore transport, primarily validating for sandy beach conditions.	The Kamphuis formula, while effective for gravel and pebble beaches, has limitations due to low complexity and reliance on low-quality field data, making it challenging to predict alongshore transport under high-energy wave conditions.
	van der Meer [30]	Precision: the formula has shown a good fit to field data in moderate wave and fine sediment conditions; Flexibility: the formula can be used to predict transport in different wave and sediment conditions; Applicability: the formula has been validated in a wide range of coastal sites and climatic conditions.	The formula’s precision is limited in high wave and coarse sediment conditions, and its accuracy is influenced by data quality, making it less precise than alternative models.
	Kamphuis [29]	Accuracy: under moderate wave and fine sediment conditions, the formula has demonstrated an excellent fit to field data; Flexibility: predicting movement under various wave and sediment conditions is possible using this approach; Applicability: a variety of coastal locations and climates have been used to validate the algorithm.	The formula’s precision is limited, underestimating longshore transport in high wave and coarse sediment conditions, and its accuracy depends on data quality, making it less reliable than alternative models.
	van der Meer and Veldman [44]	Precision: in situations with moderate waves and fine silt, the formula has demonstrated a strong fit to field data; Flexibility: using the method, one can forecast transport under various wave and sediment circumstances; Applicability: the formula has been verified in a variety of coastal locations and climates.	The van der Meer and Veldman formula for predicting coastal performance has limitations, including inaccuracies, limited applicability, complexity, and neglect of long-term processes, requiring further research.
	Mil-Homens et al. [45]	The formula improves precision and applicability across wave and sediment conditions, including sandy and gravel beaches, and is validated using a comprehensive field data set.	The Mil-Homens et al. (2013) formula for longshore sediment transport has limitations, such as empirical nature, calibration sensitivity, and data requirements, requiring further refinement or alternative models.
	van Rijn [46]	The van Rijn formula for longshore sediment transport is user-friendly, versatile, and accessible, incorporating wave height and sediment characteristics, and useful for coastal management and engineering projects.	The van Rijn formula for longshore sediment transport has limitations due to its empirical basis, limited sediment size range, uniform conditions, and data-intensive nature.

Table 3. Calculation of sedimented (+) or erosion (−) area data for each date from the available images extracted with CoastSat.

Year	Days	Dotation (m³/m)	Sedimentation (m³)	Dredged Volume (m³)	Total Volume (m³)	LST (m³/s)
2016	360	33	36,618	163,204	199,823	0.0064
2017	370	31	11,073	177,806	188,880	0.0059
2018	375	54	331,390		331,390	0.0102
2019	365	198	1,209,564		1,209,564	0.0384
2020	355	135	822,967		822,967	0.0268
2021	365	234	1,428,009		1,428,009	0.0453
2022	360	228	1,395,205		1,395,205	0.0449
∑	2550	913	3,839,622	341,011	4,180,632	0.1779
Average	365	130	747,832		796,548	0.0254
St. Dev.		88	605,590		558,290	0.0179

Table 4. Application of the main formulas of LST to the case of the Port of Salaverry in Peru. Moreover, the calculation of the relative error committed by comparing the results of the different formulas and taking as reference the value of CoastSat (assisted by the satellite images).

1. Type	2. Method	3. Calculated LST Rate (m³/s)	4. Relative Error Regarding Coastsat Value
Real Value	CoastSat applied to satellite photos	0.025	0%
Energy Flux	CERC [5]	0.046	−84%
	Swart [7]	0.024	4%
	Chadwick [9]; Brampton and Motyka [10]	0.008	68%
	Schoonees and Theron [11,12]	0.048	−92%
Current Energy	Bailard [18]	0.012	52%
Dimensional Analysis	van Hijum and Pilarczyk [26]	0.049	−96%
	Kamphuis et al. [28]	0.03	−20%
	van der Meer [30]	0.025	0%
	Kamphuis [29]	0.037	−48%
	van der Meer y Veldman [44]	0.012	52%
	Mil-Homens et al. [45]	0.03	−20%
	van Rijn [46]	0.02	20%

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Alpaca-Chacón, C.M.; Jigena-Antelo, B.; Quispe-Gonzales, C.A.; Sarango-Julca, D.D.; Contreras-de-Villar, A.; Muñoz-Perez, J.J. Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology. J. Mar. Sci. Eng. 2024, 12, 1660. https://doi.org/10.3390/jmse12091660

AMA Style

Alpaca-Chacón CM, Jigena-Antelo B, Quispe-Gonzales CA, Sarango-Julca DD, Contreras-de-Villar A, Muñoz-Perez JJ. Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology. Journal of Marine Science and Engineering. 2024; 12(9):1660. https://doi.org/10.3390/jmse12091660

Chicago/Turabian Style

Alpaca-Chacón, César M., Bismarck Jigena-Antelo, César A. Quispe-Gonzales, Douglas D. Sarango-Julca, Antonio Contreras-de-Villar, and Juan J. Muñoz-Perez. 2024. "Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology" Journal of Marine Science and Engineering 12, no. 9: 1660. https://doi.org/10.3390/jmse12091660

APA Style

Alpaca-Chacón, C. M., Jigena-Antelo, B., Quispe-Gonzales, C. A., Sarango-Julca, D. D., Contreras-de-Villar, A., & Muñoz-Perez, J. J. (2024). Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology. Journal of Marine Science and Engineering, 12(9), 1660. https://doi.org/10.3390/jmse12091660

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Updated Review of Longshore Sediment Transport: Advantages, Disadvantages, and Comparisons Using a Satellite Image Methodology

Abstract

1. Introduction

Study Site

2. LST Formulations

2.1. Energy Methods

2.1.1. Energy-Flux Method

2.1.2. Current Energy Method

2.2. Methods of Balancing Forces

2.3. Dimensional Analysis Methods

2.4. Summary of the Different Formulations

2.5. CoastSat Software Method

3. Results and Discussion

3.1. Energy Methods

3.1.1. Energy Flux

3.1.2. Current Energy

3.2. Balance of Forces Method

3.3. Dimensional Analysis

3.4. General Summary

4. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

Notation

References

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