This section deals with spectral modeling of the impact between a sphere and a slab (Akay and Hodgson, 1978; Hunter, 1957), because the main assumption of this study is that the acoustics of gravel is described by impacts between bedload sediments and the river. To prove the validity of our model, the study includes some comparative facts with the sphere–sphere spectral model of Thorne and Foden (1988).

As a brief introduction, the collision between bed particles radiates energy. Such a rigid body radiation phenomenon is due to both vibrations and accelerations. These processes are very well separated with respect to their dominant frequencies, such that the spherical mode vibrations generate much higher frequencies than the acceleration-based sound (Barton, 2006; Thorne and Foden, 1988). The acoustic effect of accelerating rigid bodies is physically modeled by Kirchhoff (1883). A framework was constructed by Goldsmith (2003), Hertz (1882) and Hunter (1957) to model acceleration profiles from elastic impacts between two solid rigid bodies like two spheres or a sphere and a slab. In a mathematical sense, the acoustic pressure field generated from the acceleration of a rigid body is evaluated by the integral convolution from Eq. () (Akay and Hodgson, 1978; Koss and Alfredson, 1973; Thorne and Foden, 1988). The integral consists of the convolution between Kirchhoff's impulse response pI and an acceleration profile A. In the case of elastic (Hertzian) impacts the acceleration occurs during the impact, and so the integral is evaluated by intervals with respect to a contact duration Tc. The contact duration Tc is modeled by Hertz's law and it is put in a simplified form in Eq. (), for both sphere–sphere and sphere–slab impact models. 1p(t)=∫0χpI(t′)⋅A(t-t′)dt′,2Tc=ϑ(1)ζρs0.4aUimp-0.2, where χ is the time interval of convolution, with χ=t, if 0 ≤τ≤Tc, and χ=Tc, if τ>Tc, with τ a delayed time due to sphere geometry, τ=t-(r-a)/c, r is the distance between the observation point and the impact (see also Fig. 1a and b), a is the radius of the sphere, c is the sound celerity, ρs is material density and Uimp is the impact velocity. The parameter ϑ(1) is a constant, ϑ(1)=9.229, for the impact between two spheres of the same radii and ϑ(1)=10.601, for the impact of a slab and a sphere. The parameter ζ=(1-ν2)/(πElong) is a material parameter, and it contains Young's modulus (Elong) and the Poisson ratio (ν).

The general form of the acceleration profile is provided by Goldsmith (2003) and it is rewritten in a unified form for both sphere–sphere and sphere–slab impact models; see Eq. (). 3A(t)=ϑ(2)UimpTcsin⁡πtTc,0≤t≤Tc0,t>Tc where the constant ϑ(2)=1.5708, for sphere–sphere impact, and ϑ(2)=3.353 for sphere–slab impact.

The first important observation from Eq. () is the half-period sinusoidal form of the Hertzian acceleration. The two modeled acceleration laws show close frequencies as the constant ϑ(1) from Tc's formula is not dramatically different from one case to another. If the frequency of acceleration of sphere–sphere impact is 1000 Hz, then the frequency of acceleration of sphere–slab impact is 909 Hz, which is almost only 10 % of the deviation. The maximum amplitude of acceleration for the impact between two spheres of radius a is almost 2 times less than the impact between a sphere of radius a and a slab, considering the same Uimp and Tc.

The integral convolution in Eq. () is transformed into multiplication in the complex Fourier space. Thus, the analytical magnitude spectrum of the noise from the rigid body acceleration, Facc, is given in Eq. (4a). 4aFacc(ω)=F(pI(t))⋅F(A(t)), where F(pI) is the Fourier transform (FT) of Kirchhoff's impulse response pI (Koss and Alfredson, 1973), for a sphere of radius a, defined in Eq. (4b), F(A) the FT of Hertzian acceleration due to elastic impact between two identical radii and the same material spheres, defined in Eq. (4c), and ω is the angular frequency which is a measure of rotation rate, in radians per seconds, and it is equal to 2πf; f is the linear frequency, a measure of number of occurrences per second. 4bF(pI)=ρa3cr2c+jωr2c2-(ωa)2-j2ωac2c2-(ωa)22+2ωac2cos⁡θ4cF(A)=ϑ(3)Uπ2-ωTc2e-jωTc+1 j is an imaginary unit and ϑ(3)=±π2/2 for sphere–sphere impact, and ϑ(3)=1.067π2 for sphere–slab impact.

As we know, the nature of Hertzian sound is the oscillation of a rigid solid along a particular direction, and so each of the two objects in collision will give rise to a dipole acoustical source, shown in Fig. 1a. The case of sphere–slab impact is treated below. Hence, the amplitude of an oscillating sphere is dependent on the cos⁡θ term and the phase of the acoustic pressure field changes by 180∘ at θ=90∘; i.e., the rarefaction wave changes into a compression wave or vice versa. In the case of the sphere–slab impact shown in Fig. 1b, the total pressure field is modeled as the addition between the compression wave and the slab-reflected rarefaction wave of the acoustic dipole. Thus, the addition becomes a subtraction as the reflected rarefaction wave keeps its sign (does not shift in phase), so there are two waves (compression and rarefaction) arriving at the sensor almost at the same time (Akay and Hodgson, 1978). This acoustic process is modeled by the so-called method of images by which one considers a mirrored sphere replacing the slab and being responsible for the rarefaction wave generation.

The same subtraction is applied in the case of complex spectra to obtain the total spectrum Fim, Eq. (5). In this formula, the first term of the right member is attributed to the impacting sphere, whereas the second term pertains to the mirror. The time delay Td of sound arrival due to distance of measurement and the sphere's geometry make the two terms not perfectly cancel out or not arrive at the same time at the sensor. 5Fim(ω)=Facc(ω)-Facc(ω)⋅e-jωTd

Introducing Eqs. (), () and (4a–c) into Eq. (5), one obtains the complex magnitude spectrum of the impact between a sphere and a slab. The spectrum contains complex numbers, so one applies the multiplication of the spectrum and its conjugate to compute the magnitudes of the energy spectrum, Eq. (6). 6|Fim|2=Fim⋅Fim∗, where Fim∗ is the complex conjugate of Fim.

The quantity |Fim|2 from Eq. (6) is noted as in Eq. (7) by E and its unit of measurement is Pa2 sHz-1. Thus, the analytical model of impact used in this paper is an energy spectral density and it will be used to inverse acoustic spectra measured in the field. 7E(ω)=|Fim|2 An example of an analytical model computed in time by Akay and Hodgson (1978) and reformulated in Appendix B is presented in Fig. a. The impacting sphere is 20 mm in diameter, the material is granite and the impacting velocity is 1 ms-1. The shape of the waveform is an approximately one and a half period sinusoid. The subtraction of the two pressure fields, the rarefaction and the reflected compression wave fields are observed. It is also important to notice that the first arrival at the sensor is the compression wave. Thereafter, the other part of the acoustic dipole (the rarefaction wave) arrives with delay Td at the sensor. The power spectrum density modeled by Eq. (7) is shown in Fig. 2b. Here, the spectrum has a principal lobe and numerous side lobes. The principal lobe has a peak at the frequency of approximatively 1/(1.1⋅Tc) and the side lobes are approximatively associated with the term cos⁡(ωTc) also observed by Thorne and Foden (1988).

In Fig. c it is shown that the frequency peaks of spectra from both types of impact model decrease with the sphere's diameters (from 1 to 150 mm), as experimentally observed by Thorne (1986b). Frequency peak as a function of diameter D, in the case of sphere–slab impact, fpeak(D)=a⋅Db, is given in the case of three impact velocities, Uimp={0.01;0.1;1}ms-1. The exponents of the regression laws prove the exact inverse proportionality between fpeak and D. Besides, the power peaks and peak frequencies increase, for a certain diameter, when the impact velocity increases. There is only a doubling of fpeak when Uimp changes by an order of magnitude. This is also proved by the formula of Eq. () of Tc (almost the reciprocal of fpeak) where the parameter Uimp is raised to a weak exponent of -0.2.

The fpeak in the case of sphere–sphere impact, modeled for impact velocity Uimp=1 ms-1, is higher than in the case of the sphere–slab impact. Here, the analytical model of the sphere–sphere spectrum was computed using Eqs. (4a) and (5), with the two pressure fields auditioned instead subtracted. This gives the same results as the spectral model reported by Thorne and Foden (1988). To give an idea, a 150 mm diameter particle in sphere–sphere impact has a spectrum of fpeak=1700 Hz (see detail of Fig. 2c), whereas sphere–slab impact has fpeak=1500 Hz, so the 200 Hz represents cca. 15 % of variation between the cases.

It is worth mentioning that Uimp greatly influences the power peak, if the former is changed by 1 order of magnitude. On the other hand, the power peak of the sphere–sphere impact is slightly weaker than the sphere–slab impact. In this paper, we choose to use a slab model to model bedload SGN as it simplifies the inverse problem. Indeed, the task of determining the dimensions of impacted particles is skipped. Therefore, we consider that the riverbed could be modeled as a slab. This hypothesis could be supported when the riverbed is armored or paved, but may be false when the riverbed is totally mobile and when the impacts between particles of different diameters are very common.

In the previous section, the analytic energy spectral density (ESD) was defined for the impact between a sphere and a slab. In this section, we model the power spectral density (PSD) of a sediment mixture using these analytic ESDs and the impact rate of each class of diameter, or the number of impacts per second. Assuming that particle collisions are random and independent noise sources, the model of the PSD of a mixture, denoted by P, can be expressed as a linear summation of the elementary ESD, denoted by Ei (Johnson and Muir, 1969; Jonys, 1976; Thorne, 2014) weighted by the impact rate Ii. The acoustic bedload model under discussion is defined in the scalar form in Eq. () and the matrix form in Eq. (). 8P=∑i=1KEi⋅IiP1P2⋮PNFFT︸P=E1,1︷E1E1,2︷E2…E1,K︷EKE2,1E2,2⋯E2,K⋮⋮⋱⋮ENFFT,1ENFFT,2…ENFFT,K︸Δ⋅I1I2⋮IK︸I 9a⇒P=Δ⋅I where Δ is the dictionary of elementary ESD of impacts between spheres and slab and I is the vector of impact rates per diameter class or, basically, a histogram. Class i takes integer values, from the lowest limit, 1 mm, to the highest one, K mm, where K is the largest diameter considered in modeling. Here, we consider K equal to 150 mm. The parameter NFFT is the number of values contained in the spectrum or the number of Fourier transform points on which the spectrum is modeled.

The histogram I can be transformed in the probability mass function γ by normalizing it by its sum of elements. The cumulative form of γ will be denoted by Γ. Thus, one of the main assumptions is that Eq. () can be written in terms of probabilities γ, as in Eq. (): 9bP=Δ⋅γ, where γi represents the probability of having a number of impacts of particles per second for size class i.

Therefore, the random variable here is I and γ is the probability of impacts, and so the quantity γ (I=Ii) is a discrete probability, given that we operate on size classes of 1 mm in diameter. This probability is computed from a histogram of the number of impacts per second, so one needs to transform it into a histogram in mass of sediments M, to be compatible with the measured GSD by physical sampling. In consequence, γ (I=Ii) will be scaled by Di3, as in Eq. (), in order to obtain γm(M=mi). Finally, the grain size distribution (GSD), or the cumulative distribution form of γm, will be Γm(M≤mi), expressing the probability of sediments finer than Di, as defined in Eq. (). 10γm(M=mi)=κ⋅γ(I=Ii)⋅Di3,i=1,…,K11Γmj(M≤mi)=∑i=1jγmi(M=mi),j=1,…,K where Di is the diameter in meters and κ is a constant which is the rest of the mass-volume formula coefficient including the material density.

This section gave a formal definition of the PSD of a bedload size mixture defined by its GSD. The proportions are considered to be a probability mass function (PMF) of the rate of impacts. The size classes concerned in this study are integer numbers, from 1 to K, with a resolution of 1 mm per size class.

This analysis was done to determine the importance of input parameters for the shape of the PSD modeled with Eq. (). The parameters are defined in Table 1. Global sensitivity analysis (GSA) is done to assess the impact of input parameters on the model output, which in our case is the peak frequency fpeak of spectra modeled by Eq.(). We use the Fourier amplitude sensitivity test (FAST) (Cukier et al., 1973) to compute the first order indices of sensitivity Si for each input parameter. The coded version of the FAST algorithm is presented in Cannavó (2012).

The flowchart of the GSA is presented in Fig. . The FAST analysis uses the typical range of parameters found in rivers, defined in the second column of Table 1. The input log-normal distributions (GSD) have median diameter values in the range from 1 to 150 mm and SDs σ from 0.01 to 10. All other input parameters needed for the model in Eq. () are given in Table 1. As the output model analyzed is the peak frequency fpeak of the simulated PSD curves, the analysis does not claim to completely describe the model, but pertinent ideas could be drawn on the model's behavior.

The results in terms of first-order indices are presented in Table 2. The SD σ of log-normal GSD has the greatest influence on the PSD shape. This is because σ affects the values of all percentiles of the GSD curve. The median D50 is almost 2 times less important than σ. The third-greatest parameter as a degree of influence on output is surprisingly Young's modulus, but this is due to a very wide range of values (here Elong=1010…7×1010 Pa, i.e., from quartz to granite materials). Such variation is not possible at the reach scale, where the sediments are of the same material. The impact velocity Uimp comes shortly after Young's modulus and confirms the conclusion of the previous local analysis, that Uimp has ca. 10 % of importance in the output. Other relatively important parameters are Poisson's ratio and the density of sediments, which means that the type of material also plays a role in the dynamics of the fpeak. The distance of measurement, r, also plays also a role in the fpeak variation. The angle of the point of observation with respect to the impact, θ, and the propagation medium properties, ρ and c, are considered of little influence on the values of fpeak.

In conclusion, the first-order global sensitivity analysis of the peak frequency shows a comprehensive view of its dynamics with input parameter variation. It is found that the peak frequency is mainly affected by two parameters, the distribution's SD and the median diameter, together making up ca. 65 % of output variation, whereas the material properties (i.e., density, Poisson's ratio, Young's modulus) have almost ca. 20 %, and the impact velocity Uimp has ca. 10 %. In conclusion, the acoustic model is quite complex and care must be taken regarding the recording of the power spectra on the field, as their shape heavily affects the estimation of GSD. Also, the impact velocity is regarded as a minor factor of uncertainty and, because it is almost impossible to be measured for each grain size class, the 10 % uncertainty in peak frequency is almost unavoidable. The material properties should not be a problem with the condition that the sediments are the same. For a complete GSA, the computation of high-order sensitivity indices can be made using Sobol's methodology (Sobol, 2001), but this type of analysis is beyond the scope of this article.

Modeling of single impacts requires definition of parameters typical for river environments, in Table 1. Using the global sensitivity model, it has been shown that PSD shapes are essentially influenced by four parameters: the shape of the GSD curve, the median diameters of the colliding particles, the impact velocities and the material. Grain sizes are estimated later using the inversion algorithm presented in Sect. 3. Concerning the other model parameters, as they do not affect the PSD shape, they will be fixed for the inversion process, using realistic values. These parameters are listed in the third column of Table 1. The main assumptions of the SGN spectrum model are the following. i.

The geometry of the channel and of the material: the river bed is considered a massive slab and moving particles are considered spherical.

A simulation case is proposed here to test the robustness of the LS inverse method. The simulated PMF, or grain size distribution, γm is uniformly distributed between 10 and 50 mm. The uniform distribution means that 1 kg of D=10 mm has the same probability of producing impact noise as 1 kg of D=11 mm, and so on. To obtain γ, the simulated PMF γm is converted back to impact rates by dividing by D3, D in meters. Using an impact velocity of 1 ms-1 and the rest of the input parameters defined in Table 1, the simulated PSD P is shown in Fig. a. Here, the dictionary Δ contains spectra from 1 to 150 mm and the grain size distribution has 1 mm resolution. Applying Eq. (12) on the simulated spectrum and considering exactly the same parameters in modeling and in simulation, it is found that the estimated γm is exactly the same as the simulated γm, as is expected; see Fig. b.

However, if the impact velocity used in modeling the dictionary is set to a value (Uimp=0.1 ms-1) which is different than the one used in simulation (Uimp=1 ms-1), then high instabilities are observed on the estimated γm; see Fig. c. This is explained by the fact that there is a high similarity between the elementary spectra E, especially for the larger size classes. Thus, the matrix Δ is ill-conditioned and the problem is ill-posed. Ill-conditioning is linked to the high condition number of the normal matrix (Δt.Δ). It is defined as the ratio between the largest and smallest eigenvalues of a matrix. A well-conditioned algebraic system requires that the normal matrix should have a condition number as close as possible to 1 (Strang, 2006). In these tests, Δ's condition number reaches huge values on the order of 1012–1020. In consequence, the similar spectra from the matrix Δ produce high instability in solution.

To avoid the instability in the LS solution, the non-negative least squares (NNLS) algorithm (Lawson and Hanson, 1974) is proposed to solve the LS problem. This optimization algorithm, Eq. (13), casts non-negative constraints on solution γ. The non-negative factorization is widely used, for example, in various domains like image processing or chemometrics. The side-effect of using this algorithm is the strong regularization of solutions. The regularization aims to keep the sum of components in γ constant. The solution of the NNLS algorithm (see Fig. c) shows that the instabilities are completely removed off. Besides, it is important to note that the estimated diameters are inside the simulated interval of diameters. 13γ^(I=Ii)= minimize(Δ+⋅P-γ),γ^≥0

The signal processing tools in this paper refer to using the PSD as the method of spectral representation of bedload signal. The use of PSD is worthwhile because the type of bedload signal is a stationary random one. Random stationary signals are signals varying in time but whose average and SD of amplitude values over some fixed periods are constant.

A particular concern for the signal processing of random processes is the minimization of the variance on the PSD. This work makes use of the periodogram algorithm for the PSD estimation, which means the Fourier transform is applied on local portions (windows) of the random signal, with an overlap of 50 %, and then the local results are averaged in narrow bandwidths (Oppenheim and Verghese, 2010). The averaging is useful because it mitigates the variance on the PSD. In this work, the quality of spectra is vital for accuracy of estimations. The uncertainty principle tells us that the smaller the temporal window, the greater the uncertainty in locating two very close frequencies on the spectrum, so a trade must be made between the PSD variance and its spectral resolution. If the bedload signal is too short, the quality of spectra toward the low-frequency bands is worsened because in one single bandwidth of the Fourier transform there is spectral information of impacts from multiples grain sizes. Finally, the longer the signal the better the spectral resolution and the less the variance on the PSD curve.

The NNLS algorithm will be tested on three simulated spectra which have different degrees of variance. The simulated γ used is identical to that in Sect. 3.1. The simulated signal is obtained by convolving a realization of a white noise with a transfer function being the modeled spectrum shown in Fig. a. The simulated noised PSD is shown in Fig. 4d. The results of the inversion using the NNLS algorithm show that, even for the worst scenario of variance on a spectrum, the inversion method correctly reconstructs the simulated GSD; see Fig. e.

Finally, we conclude that the NNLS algorithm is robust with respect to PSD noise and fits to this kind of inversion problem. The inversion procedure will now be tested on in situ measurements.

The Isère River is a piedmont gravel river bed located in southeastern France, and it is one of the main tributaries of the Rhône River, which reaches the Mediterranean Sea. The monitoring section is located in the city of Grenoble (45∘11′52.8′′ N, 5∘46′14.88′′ E); see Fig. 5a. In this reach, the mean slope is about 0.06 %, the area of the watershed is 5500 km2 and the annual average flow rate is 180 m3s-1. At the time of experiments, 29–30 June 2016, the monitored discharge was on average 300 m3s-1. The measurement section has a rifle-pool morphology with riprap-protected embankments. Two different types of instrument were used: SGN measurements using hydrophone and direct sampling using a pressure-difference sampler, shown in Fig. 5b. All these measurements were carried out from a suspension bridge (Fig. 5c).