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Seismic modelling and observations of rainfall

Maarten Bakker, Cédric Legout, Florent Gimbert, Guillaume Nord, Brice Boudevillain, Guilhem Freche

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Maarten Bakker, Cédric Legout, Florent Gimbert, Guillaume Nord, Brice Boudevillain, et al. "Seismic modelling and observations of rainfall." Journal of Hydrology, 2022, 610, pp. 127812. 10.1016/j.jhydrol.2022.127812. hal-04416034

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Distributed under a Creative Commons Attribution 4.0 International License
# 1 SEISMIC MODELLING AND OBSERVATIONS OF RAINFALL

2 Maarten Bakker, Cédric Legout, Florent Gimbert, Guillaume Nord, Brice Boudevillain and Guilhem Freche

3 University Grenoble Alpes, CNRS, IRD, Grenoble INP, IGE, Grenoble, France

4

# 5 ABSTRACT

6 Rainfall is a key driver of geomorphological processes ranging from impacting drops that lead to the 7 small-scale dislodgement of soil particles to large-scale morphogenic floods and rainfall-induced hillslope 8 processes. Although rainfall has been identified in seismic records, the associated power spectral density 9 and its quantitative relation to the underlying physical processes have not yet been studied. Here, we 10 analyze nearly 2 years of combined seismic and optical disdrometer measurements, where the latter 11 enables the drop-based quantification of rainfall physical properties. Our measurements confirm the 12 broadband observation of ground velocity power spectral density due to rainfall, allowing the seismic 13 identification of rainfall at intensities as low as 1 mm/h. Seismic power shows a power-law scaling with 14 rainfall intensity, *I*, and kinetic energy, *E*: $P \propto I^{2.1}$ and $P \propto E^{1.6}$. The observed scaling relations are 15 consistent between the three monitored sites although there are absolute differences in seismic power 16 of about 1 order of magnitude, which are likely due to variability in landcover and subsurface seismic 17 properties. With a physical model, we demonstrate that the observed power-law relations are set by an 18 underlying linear relation between seismic power and rainfall impulse power, and that the associated 19 exponent values of *I* and *E* are due to the covariance of the raindrop size distribution with the total 20 number of drops. The largest raindrop fractions, whose relative contribution increases with rainfall 21 intensity, dominate the seismic signal where, in our case, 90% of the seismic power is attributed to drops 22 larger than 3 mm. Using our model, we estimate the contributing area of rainfall to seismic observations 23 to be within a radial distance of ~5-25 meters. The spatially integrated nature of the seismic 24 measurements and its sensitivity to large raindrops, which control the disaggregation and the 25 mobilization of soil particles, make seismic records well-suited for the investigation of soil erosion 26 processes. More generally, our work provides a basis for the temporally-resolved seismic quantification 27 of rainfall that drives the dynamics of various hydro-geomorphological processes.

# 28 KEYWORDS

29 Rainfall, raindrop impulse, seismic power, drop size distribution, soil erosion
30 **1. INTRODUCTION**

31 Rainfall is a key driver of hydro-geomorphological processes at the Earth's surface. At a small scale,
32 raindrops that hit the ground can lead to the breakdown of soil aggregates (e.g., Ghadiri & Payne, 1977;
33 Legout et al., 2005) and the mobilization of particles (e.g., Angulo-Martínez et al., 2012; Legout et al.,
34 2005). At the same time, the drop impacts may also lead to changes in surface microtopography
35 (Planchon & Mouche, 2010) and near-surface compaction and clogging, which affect the capacity of
36 rainwater to infiltrate into the ground (e.g., Assouline & Mualem, 1997; Helalia et al., 1988). These
37 processes condition the availability and size of particles at the ground surface which can subsequently be
38 entrained by surface runoff (Gabet & Dunne, 2003; Neave & Rayburg, 2007; Zhou et al., 2013). At large
39 scales, high-magnitude rainfall events may lead to morphogenic floods (e.g., Hooke, 2016) and induce
40 mass movements (e.g., Chen et al., 2015). Such extreme events are expected to play an even more
41 prominent role in the transfer of water and sediment in watersheds under ongoing climate change
42 (Naylor et al., 2017).

43 A wide range of in-situ instruments have been developed to measure the physical properties of
44 raindrops at a given location, i.e., their size and velocity, using mechanical, optical, and acoustic
45 principles (for review see Lanza et al., 2021). These instruments allow various physical measures of
46 rainfall to be derived which may subsequently be applied to study small-scale rainfall-induced processes
47 at the ground surface. Rainfall kinetic energy and momentum have typically been related to the capacity
48 of rainfall to erode soil (Carollo et al., 2018; Furbish et al., 2007; Rose, 1960; Sharma & Gupta, 1989),
49 although various other measures have also been proposed, e.g., the ratio of kinetic energy to raindrop
diameter (Ghadiri & Payne, 1977; Riezebos & Epema, 1985), the ratio of momentum to drop diameter
50 (Goebes et al., 2014), or the product of momentum and drop diameter (Salles et al., 2000). At the same
51 time, the impact mechanics of the raindrop also depend on ground properties including landcover (e.g.,
52 Bochet et al., 2002), microtopography (e.g., Roth & Helming, 1992), moisture content (e.g., Zambon et
53 al., 2021), soil texture (e.g., Bradford et al., 1987), etc. In this study, we propose to use the ground as a
54 medium to directly study the seismic vibrations caused by impacting raindrops.
55

56 Rainfall has been observed in seismic measurements from various studies that were devoted to the
57 investigation of fluvial (e.g., Burtin et al., 2011; Roth et al., 2016) and hillslope processes (e.g., Burtin et
58 al., 2013; Dietze et al., 2017a). Rainfall was identified within the general frequency range of 10-100 Hz
59 (Burtin et al., 2011; 2013; Dietze et al., 2017a; Roth et al., 2016), going as high as 500 Hz (Roth et al.,
60 2016) and as low as 2 Hz (Díaz et al., 2014). Whether rainfall was actually detected depended on the
local landcover (Dietze et al., 2017a) and substrate (Burtin et al., 2011; 2013), but also on the relative contributions of other sources of ground vibrations, including anthropogenic noise (e.g., D' Alessandro et al., 2021) or river flow (Gimbert et al., 2014; Roth et al., 2016). However, to the best of our knowledge, there have been no studies devoted to evaluating the relation between seismic power spectral density and rainfall physical properties. This is the objective of the present study.

We present temporally resolved measurements of seismic power and raindrop size and velocity distributions, which we obtain using an optical disdrometer (Löffler-Mang & Joss, 2000). We relate observed seismic power to rainfall intensity, kinetic energy, and impulse power, and explain the obtained scaling relations using a seismic physical framework adapted from Tsai et al. (2012). Through applying this framework, we quantitatively assess the seismic power spectral density generated by raindrops that hit the ground surrounding a seismic instrument. With this work, we seek to provide a fundamental basis for the application of seismic records, ranging from dedicated field experiments to readily-available, long-term network station measurements, to study rainfall dynamics, soil erosion and rainfall-induced hydro-geomorphological processes.

## 2. THEORY

### 2.1 Physical measures of rainfall

Bulk physical measures of rainfall can be derived from the observation of individual raindrops using a disdrometer. Typical measures are rainfall intensity, $I$ in mm/h, and rainfall kinetic energy, $E$ in J/m²/h, which are quantified based on the number of drops that are registered for a range of drop diameter and velocity categories, which we refer to with subscripts $i$ and $j$, respectively (e.g., Angulo-Martínez et al., 2018):

$$I = \frac{4}{3} \pi c_I \sum_i N_i(D_i) (0.5 D_i)^3, \qquad [1a]$$

$$E = \frac{2}{3} \pi \rho_w c_E \sum_{i,j} N_{i,j}(D_i) (0.5 D_i)^3 v_j^2, \qquad [1b]$$

where $D_i$ is the mean equivalent drop diameter of category $i$ and $v_j$ is the mean vertical drop velocity for category $j$ and $N_{i,j}(D_i) = \frac{n_{i,j}}{A(D_i)T}$ is the number of drops per category, $n_{i,j}$, divided by the effective surface area, $A(D_i)$, and temporal resolution, $T$, of monitoring. Lastly, $c_I$ and $c_E$ are unit conversion
factors of respectively 3.6×10⁶ mm/m s/h and 3600 s/h, and ρ<sub>w</sub> is the density of water (1000 kg/m³).

Note that N<sub>i,j</sub>(D<sub>i</sub>) is a flux-based measure and not a volume-based measure (as often used in studies addressing radar reflectivity; e.g., Uijlenhoet & Stricker, 1999) and I and E are, effectively, measures for the volume flux density, where 4/3π(0.5D<sub>i</sub>)<sup>3</sup> is the volume, V, of a spherical drop, and kinetic energy flux density, since they are quantified per unit area and time.

When parametrizing rainfall, the relevance of raindrop diameter D is evident from equations 1a and 1b, even more so, when considering that v typically depends on D (e.g., Beard, 1976). If we assume terminal velocity is reached, v = v<sub>t</sub>(D), and the absence of (vertical) wind effects (e.g., Erpul et al., 2003; Montero-Martínez & García-García, 2016), then v may be quantified as a function of D using a simple power-law relation (Atlas & Ulbrich, 1977):

$$v_t(D) = 3.78(10^3 D)^{0.67} . \qquad [2]$$

We now introduce the raindrop flux density per unit drop diameter, N<sub>D</sub>(D), that is quantified as N<sub>D</sub>(D) = N<sub>tot</sub>p(D), where N<sub>tot</sub> is the total number of drops (n<sub>tot</sub>) per unit area and time, and p(D) is the drop size distribution. For the entire distribution: Σ<sub>i,j</sub> N<sub>i,j</sub>(D<sub>i</sub>) = ∫<sub>D</sub> N<sub>D</sub>(D)dD. Various functions have been proposed to describe drop size distributions, e.g., Weibull, log-normal, etc. (for review see Serio et al., 2019) that typically vary with geographical setting (continental vs. maritime and orography; e.g., Hachani et al., 2017; Rosenfeld & Ulbrich, 2003) and vary with weather systems (typically stratiform vs. convective rainfall; e.g., Kirsch et al., 2019; Tokay & Short, 1996). The raindrop size distribution depends on mechanisms of drop formation and the subsequent interaction between drops, i.e., the coalescence and breakup of drops (Assouline & Mualem, 1989; Hu & Srivastava, 1995; Low & List, 1982), where p(D) may be expressed as a power-law function of N<sub>tot</sub> (Sempere-Torres et al., 1994). For the scaling analysis in section 2.3, we express rainfall intensity and kinetic energy as a function of the total raindrop flux density, the drop size distribution and the drop diameter: I ∝ N<sub>tot</sub>p(D)D<sup>3</sup> and E ∝ N<sub>tot</sub>p(D)D<sup>4.33</sup> (e.g., Uijlenhoet & Stricker, 1999).

## 2.2 Seismic framework of raindrop impacts

To assess ground vibrations due to raindrop impacts, we adopt a seismic framework that was developed for the quantification of bedload transport (Gimbert et al., 2019; Tsai et al., 2012). We consider a force time series, F(t, D), due to the impact of a single raindrop with diameter D on a bare ground surface. Assuming an instantaneous impact, the force integrated over the impact duration corresponds to a change of momentum which is equal to the drop impulse, J(D) (Johnson, 1987; Tsai et al., 2012):
$$F(t, D) = J(D)\delta(t) \quad , \quad J(D) = \frac{4}{3}\pi\rho_w\gamma(0.5D)^3v \quad [3a, 3b]$$

where $\delta$ is a Dirac delta function, for which $\int_t \delta(t)dt = 1$. We assume a fully inelastic impact, $\gamma=1$, where the drop kinetic energy is expended during the splash (e.g., Guigon et al., 2008). In equation 3b, we express $J(D)$ as a function of $D$ because we can estimate $v$ with $v_t(D)$ using equation 2. The assumption of an impact that is instantaneous relative to our seismic observations may be justified as we typically consider ground vibrations in the range 5–200 Hz, which corresponds to time intervals of 5-200 milliseconds that are longer than the expected drop impact duration on a rigid surface which is typically of the order of 1 millisecond or less (e.g., Li et al., 2014; Nearing et al., 1986; Schmid et al., 2021). For simplicity, here we do not account for different types of surface properties (e.g. soil type, near-surface vegetation, etc.) or the development of a thin water layer that may develop at the surface (Nouhou Bako et al., 2016; Yu & Hopkins, 2018), which will affect both the impact duration and mechanics.

At any location, the absolute vertical ground velocity in the frequency domain, $|\tilde{u}(D, f, r)|$, where $f$ is frequency, can be related to the force spectrum of an impact, $\tilde{F}(D)$, which is the Fourier transform of $F(t)$, and the Fourier transform of a displacement Green's function, $\tilde{G}(f, r)$, which converts the force into a displacement and describes the frequency-dependent attenuation of the displacement with radial distance, $r$, from the impact location (Aki & Richards, 2002; Tsai et al., 2012):

$$|\tilde{u}(D, f, r)| = 2\pi f \tilde{F}(D) \tilde{G}(f, r) \quad [4]$$

In order to obtain the power spectral density of the vertical ground motion, $P(f) = |\tilde{u}(f)|^2 T^{-1}$, we first need to calculate the force power spectrum of raindrop impacts per unit area and time, $\tilde{F}_A^2$, by integrating $\tilde{F}_A^2(D)$ over the impacts from drops of all sizes. When we assume the temporally random occurrence of impacts, we can derive (Gimbert et al., 2019; Tsai et al., 2012):

$$\tilde{F}_A^2 = \int_D N_D(D) J^2(D) dD \quad [5]$$

Substituting equation 3b into equation 5 and discretizing drop diameter into categories i and j, we define the impulse power density flux, $J_p$, that can be calculated from disdrometer measurements as:

$$J_p = \left(\frac{4}{3}\pi\rho_w\right)^2 \gamma \sum_{i,j} N_{i,j}(D) (0.5D_i)^6 v_j^2 \quad [6]$$
Similar to rainfall intensity and kinetic energy, we can express the impulse power, $J_p$, as a function of the total raindrop flux density, the drop size distribution, and the drop diameter as $J_p \propto N_{tot}p(D)D^{7.34}$.

To obtain the power spectral density, $P(f)$, at any location, we subsequently integrate the impulse power contributions over all source-to-receiver distances $r$ using $\tilde{G}^2(f, r)$. Here, we assume uniformly distributed raindrop impacts over an infinitely large rain field which allows us to derive:

$$P(f) = 8\pi^3 f^2 \int_r J_p \tilde{G}^2(f, r)rdr \quad [7]$$

In section 5.2, we will discuss the displacement Green's function for Rayleigh waves, $\tilde{G}(f, r)$, i.e., waves that travel along the surface and lead to vertical (and longitudinal) ground motion, and how this function affects the magnitude of the observed seismic power and the spatial contribution of rainfall to seismic power.

## 2.3 Scaling of seismic power with rainfall

Based on our seismic framework, we expect a linear scaling relation between rainfall-induced seismic power and the impulse power due to impacting drops $P \propto J_p \propto N_{tot}p(D)D^{7.33}$. This scaling should always be obtained, regardless of whether $p(D)$ varies (systematically or not) with $N_{tot}$. For the relation of $P$ vs. $I$ and $P$ vs. $E$, which are more common bulk measures for rainfall, no unique scaling relation can be given as there is no unique relation between $p(D)$ and $N_{tot}$. Nonetheless, we explore the scaling of $P$ vs. $I$ and $P$ vs. $E$ based on three idealized cases following Uijlenhoet et al. (2003):

1. 'Number-controlled' rainfall: In this case, we assume that all changes in $I$ and $E$ are due to changes in the number of drops. Thus, we consider a constant drop diameter $D$, and therefore also a constant $p(D)$ and $v(D)$, which leads to $I \propto N_{tot}$, $E \propto N_{tot}$ and $P \propto N_{tot}$. This implies that seismic power would scale linearly with rainfall intensity but also with kinetic energy or other measures such as momentum: $P \propto I$ and $P \propto E$.

2. 'Size-controlled' rainfall: In this case, we assume that all changes in $I$ and $E$ are due to changes in drop size and consider a constant drop density, $N_{tot}$, and drop distribution, $p(D)$. This leads to a scaling of $I \propto D^3$, $E \propto D^{4.33}$ and $P \propto D^{7.33}$. Through combining the former two scaling relations with the latter, we obtain the power-law scaling relations: $P \propto I^{2.4}$ and $P \propto E^{1.7}$.

3. 'Intermediate' rainfall: In this case, we assume equal proportions of the endmember cases of size- and number-controlled rainfall. To cope with the dependency of $p(D)$ on $N_{tot}$, we follow the method of Uijlenhoet et al. (2003) which is based on a scaling-law framework developed by
Sempere-Torres et al. (1994; 1998). For details on the method and derivation see Appendix A. This allows us to obtain the power-law scaling relations with exponents that are only slightly smaller than for 'size-controlled' rainfall, namely: $P \propto I^{2.1}$ and $P \propto E^{1.6}$.

## 3. METHODS

### 3.1 Study area

We study the seismic record of rainfall in a mountainous mesoscale catchment, the Galabre in the southeast of France (679–1909 m; Figure 1a), which is part of the Draix-Bléone long-term critical zone observatory (Legout et al., 2021). The mean annual precipitation is approximately 1000 mm and falls mostly during the spring and autumn, with occasional high-intensity and short-duration (1–2 hours) convectional thunderstorms in the summer and autumn (Navratil et al., 2012). The catchment is covered mostly with forests and shrublands but also contains relatively small but highly-erodible badlands (nearly 10% of the surface area; visible in the top center of Figure 1d) that yield large amounts of suspended sediment at the basin outlet (Esteves et al., 2019; Legout et al., 2013; Poulenard et al., 2012).

<caption>FIGURE 1</caption>

Two locations within the Galabre catchment are studied: Ainac, in the upstream part of the catchment, and Robine-sur-Galabre (R-s-G), in the downstream part of the catchment (Figure 1a). We first address the rainfall-induced seismic power using a dedicated set of colocated measurements for rainfall events at Ainac (1170 m) in the period March–April 2021 (Figure 1e). We subsequently verify our findings with a longer dataset, February 2019–December 2020, that encompasses year-round rainfall events for two geophones near R-s-G: upstream (731 m; Figure 1f) and downstream (679 m; Figure 1g). These seismic measurements are not colocated with the disdrometer in the village of Robine-sur-Galabre (Figure 1d) but lie at a distance of respectively ~500 m and ~2.15 km from the disdrometer.

### 3.2 Meteorological measurements

Meteorological data were obtained from the meteorological stations at Ainac and R-s-G (Figure 1a). Details on the data acquisition and availability are provided in Legout et al. (2021). In this study, we use 1-min resolution data of rainfall intensity and the distributions of equivalent drop diameter and vertical fall velocity obtained by OTT-Parsivel² optical disdrometers (e.g., Angulo-Martínez et al., 2018; Tokay et
al., 2014). The disdrometers function based on a horizontal plane of light that is set up between a transmitter and receiver (A is schematically added to the image in Figure 1b). The instrument registers the reduction in transmitted light as a drop passes through this plane, which is a measure for the drop size; and the duration of the reduced transmission, which is a measure for the vertical speed of the drop (Löffler-Mang & Joss, 2000). To account for edge effects where (particularly large) drops are only partly registered (Battaglia et al., 2010), the effective sampling area to determine $N_{i,j}(D_i)$ is estimated as $A(D_i)=0.18×(0.03-D_i/2)$ following Tokay et al. (2014). The number of drops per minute are counted for 32 diameter classes and 32 velocity classes (1024 classes in total) that progressively increase in bin size, e.g. the larger the drop diameter, the larger the range of diameters that are included in the class. We consider only 20 diameter classes, excluding drop counts for $D_i<0.25$ mm, as drops of this size are near or below the limit of instrument detection (Tokay et al., 2014), and drop counts for $D_i>7$ mm, as they exceed the maximum expected size of raindrops (Pruppacher & Klett, 2010) and are prone to erroneous detection (Hachani et al., 2017; Jaffrain et al., 2011). We present $I, E$ and $J_p$ values that we calculate using equations 1a, 1b and 6, rather than, for $I$ and $E$, the values that are directly provided by the Parsivel² disdrometer following an undisclosed proprietary routine. We note that the values marginally differ between the two approaches, with differences of less than 0.1 mm/h for more than 95% of the non-zero rainfall measurements.

## 3.3 Seismic measurements

We performed seismic measurements with geophones that were installed for the long-term monitoring of flow and bedload transport near the upstream (Esteves et al., 2019; Navratil et al., 2011) and downstream measurement station (Nord et al., 2020) at R-s-G. The geophones are located approximately 30-40 meters from the Galabre River (Figure 1f and 1g). In addition to river-generated seismic power, we observe the contributions of traffic as the geophones are located at a distance of ~10 meters from the road (Figure 1f and 1g). For this study, we also placed a geophone at the meteorological station Ainac (Figure 1c and 1e) where anthropogenic and river contributions are considered minimal.

We installed the 3 Sensor PE-6/B geophones at a depth of approximately 25 cm into the ground (Figure 1c) which consists of Quaternary sedimentary deposits although bedrock is present in the near-subsurface. The geophones measure the vertical ground velocity which is recorded on a DiGOS DATA-CUBE logger. We recorded the data with a temporal resolution of 400 Hz for the R-s-G upstream and downstream sites, and a resolution of 800 Hz at Ainac (at this site we measure only in the vertical direction, not in the horizontal directions as in R-s-G, allowing a higher acquisition rate). We derived the
power spectral density, $P(f)$, in [m/s]$^2$/Hz, for 3-second time periods using the method of Welch (1967), for frequencies from 5 Hz (lower limit associated with instrument response) to 200 Hz for the R-s-G sites and in the case of Ainac up to 400 Hz (upper limit associated with the temporal resolution of the measurements). We aggregated the 3-second $P(f)$ values to 1-minute periods (the temporal resolution of the rainfall data) by determining the 10th percentile to eliminate short-lived but potentially large-magnitude seismic power contributions from anthropogenic noise (operation of machinery, passing cars, human footsteps, etc.; e.g., D' Alessandro et al., 2021) or due to the occasional occurrence of small earthquakes (during the study period, 4 earthquakes of ML 1.6-2.3 were registered in the region). For more details on the seismic data processing, we refer to Bakker et al. (2020).

## 3.4 Scaling analysis

To assess the power-law scaling of seismic power with bulk rainfall parameters (theoretical scaling relations in section 2.3), we use major axis regression that accounts for uncertainties in both axes through minimizing the sum of squares of the perpendicular distances between each point and the regression line (e.g., Bakker et al., 2020). This typically leads to a steeper scaling than ordinary least-squares approaches (McArdle, 1988). Further, we use Spearman's rank correlation to assess the correlation strength because we use non-linear axes ($10\log_{10}$ scale with seismic power in dB).

To constrain the uncertainties in the scaling analysis we discard data for time steps where: (1) temperatures are smaller than or equal to 0 °C, thus avoiding, albeit but not entirely excluding, precipitation in the form of snow; and (2) the calculated rainfall intensity based on the drop size distribution (equation 1a) is not within a factor 2 of the value directly provided by the disdrometer; we verified that the magnitude of this factor does not affect the scaling results. For the non-collocated datasets at R-s-G, we also remove the first and last 5 minutes of each rainfall period to avoid outliers associated with the presence or absence of rainfall.

# 4. RESULTS

## 4.1 Rainfall physical parameters from disdrometer measurements

Before comparing seismic power with rainfall parameters, we first establish the relation between raindrop diameter, $D$, and velocity, $v$, the relation between drop size distribution, $p(D)$, and rainfall intensity, $I$, and the relations amongst the integral rainfall parameters intensity, $I$, kinetic energy, $E$, and impulse power, $J_p$, for the period February 2019 to December 2020 at R-s-G. Ainac station shows similar
trends (these are not shown because of the limited amount of data, only April-May 2021), which leads us to consider that the meteorological conditions at the two sites, with ~5 km distance and ~430 m elevation difference between them, did not differ significantly during the study period.

Consistent with previous studies (see section 2.1), we find that vertical raindrop velocity strongly depends on raindrop diameter (Figure 2). The observed velocities are accurately predicted using the terminal velocity power-law model of Atlas and Ulbrich (1977), equation 2, up to a maximum velocity of approximately 9.5 m/s (Carollo & Ferro, 2015) near D=4 mm. As the drop diameter increases, the bin size of the measurements increases and velocities become more scattered (velocities smaller than 4 m/s may be due to the presence of snow at temperatures just above 0° C) which results in larger uncertainties in the derived physical parameters I, E and $J_p$.

FIGURE 2

The raindrop size distribution evolves with rainfall intensity at a 1-minute resolution, where the average drop diameter increases and the distribution becomes wider at higher rainfall intensities (Figure 3a). This coarsening trend is most prominent for rainfall intensities up to circa 60 mm/h when the drop size distribution stabilizes, which is in line with observations from Carollo et al. (2016), Kinnell (1981) and Rosewell (1986). The coarse tail of the distribution, where D is larger than ~2-3 mm which corresponds to less than 1% of the raindrops, shows a scaling comparable to $p(D) \propto D^{-7.33}$. Given that $J_p \propto N_{tot}p(D)D^{7.33}$ (section 2.3), this would imply for $J_p(D)$ that a decrease in the number of drops ($N_{tot}p(D)$) with increasing D may be (more than) compensated by the increase in impulse of those drops due to the increase in D. The drops in the large size classes may therefore provide the largest contribution to $J_p$, despite their relatively infrequent occurrence (Figure 3b). This effect is prominent for rainfall intensities from 1 mm/h and higher where the largest drops, D of circa 5 mm, contribute the most. Unfortunately, the registration of these large drops in disdrometer measurements is subject to considerable uncertainties. The potential implications for seismic measurements are considerable where circa 90% of the impulse power, which is expected to set the seismic power ($J_p \propto P$), is due to drops of $D>3$ mm which represent only 0.2% of the total number of drops and ~15% of the total rainfall volume (inset Figure 3b).

284
<caption>FIGURE 3</caption>

As a result of deviations in the proposed raindrop diameter vs. velocity scaling (above and below D=4 mm, Figure 2) and a changing raindrop distribution with rainfall intensity (above and below I=60 mm/h, Figure 3a), it may not be possible to capture the scaling of $J_P$ with I and E in a single power law. We would rather expect a change from partly 'drop-size' dominated rainfall, with scaling exponents associated with $J_P$ vs. I and $J_P$ vs. E that are well above 1 but below respectively 2.4 and 1.7 for rainfall intensities below 60 mm/h, to 'number-dominated' rainfall at higher rainfall intensities, where $P \propto I$ and $P \propto E$ (section 2.3). In practice, however, the observed scaling exponents are relatively constant. The $J_P$ vs. I dataset shows a scaling with an exponent of circa 2.1, while the $J_P$ vs. E dataset, which exhibits considerably less scatter, shows a scaling exponent of circa 1.6 (Figure 4). These values correspond to the predicted values for 'intermediate' rainfall (section 2.3). A decrease in the scaling exponents at rainfall intensities above 60 mm/h is not evident although the number of observations at these high intensities are limited. For low amounts of rainfall, a change in scaling is observed near 0.3 mm/h and 0.6 J/m²/h (Figure 4), but this is likely associated with limitations of the disdrometer to adequately register these low numbers of small raindrops. Following our seismic framework where $P \propto J_P$ (section 2.2), the same scaling exponents as found for impulse power, $J_P \propto I^{2.1}$ and $J_P \propto E^{1.6}$, are expected for seismic power: $P \propto I^{2.1}$ and $P \propto E^{1.6}$.

<caption>FIGURE 4</caption>

## 4.2 Colocated seismic and disdrometer measurements of rainfall events

For the colocated measurements at Ainac, rainfall events can be identified in the spectrogram (that shows the seismic power per frequency over time) at frequencies above ~50 Hz (Figure 5). The registered seismic power due to rainfall spans multiple orders of magnitude reaching values up to ~140 dB (the color scale in Figure 5a is limited to -150 dB for visualization purposes). The rainfall events can be readily distinguished from intermittent noise effects, at this site presumably due to human activities or livestock, that are typically lower in frequency (prominent in the range ~20-35 Hz but present up to 80 Hz) and lower in seismic power (Figure 5a). The onset, phases of low- and high-intensity rainfall, and the end of rainfall events are directly captured in the seismic signal, showing no evidence of lag effects
315 (example in Figure 5c and 5d). For a detailed view of all rainfall events in the period 10-23 April, see
316 rain_events_Ainac.avi in the Supplementary Materials.

317

318

FIGURE 5

319

320 There is a strong correlation between seismic power, on one hand, and the disdrometer-based measures
321 of rainfall intensity, kinetic energy, and impulse power on the other hand (Figure 6). At this particular site
322 (Ainac), rainfall can be detected at a seismic power as low as circa -175 dB, given that the contributions
323 of background noise sources are limited as indicated with the mean (μ) and standard deviations (±σ) of
324 seismic power in the absence of rainfall (Figure 6). We estimate the lower detection limit for rainfall
325 intensity to be ~0.7 mm/h, which is slightly higher than the break in scaling in Figure 4. The
326 corresponding detection limits for kinetic energy and impulse power are respectively ~5.8 J/m²/h and
327 ~5×10⁻⁹ N²s/m².

328 To describe seismic power as a function of the rainfall physical parameters at Ainac, we fit power-law
329 scaling relations to nearly 10 hours of minute-based data where *P* is larger than one standard deviation
330 above the mean noise level and where the rainfall parameters are larger than the limit of detection (data
331 included in the boxed areas in Figure 6, which is 43% of the total dataset). We find the linear scaling of
332 seismic power with impulse power, *P* ∝ *J*<sub>P</sub> (Figure 6c) as expected based on our seismic framework
333 (section 2.2). The scaling exponents of *P* vs. *I* and *E* are very similar to the values for ‘intermediate’
334 rainfall (section 2.3) and the exponents we established based on nearly 2 years of disdrometer data at R-
335 s-G (Figure 4). For values below the identified thresholds of the rainfall parameters, the data deviates
336 from the established trends due to both the relatively large contributions of noise effects and limitations
337 of the disdrometer measurements (Figure 4).

338

339

FIGURE 6

340

341 The scaling relations in Figure 6 are derived for seismic power for a chosen frequency range of 80–120 Hz
342 and may vary with frequency (Figure 7a). For lower frequencies, the seismic power associated with
343 rainfall is relatively low and noise effects are dominant (Figure 5), which leads to lower and sometimes
344 negative scaling exponents (not shown) and weaker correlation, particularly for *f*<40 Hz (Figure 7). For
higher frequencies, $f>100$ Hz, the effect of rainfall is clearly present although the scaling exponent slowly reduces with frequency which we relate to residual noise effects (these are not entirely excluded when using the mean plus 1 standard deviation threshold) that are more prominent at high frequencies where there is a lower signal-to-noise ratio (Figure 5c). At these higher frequencies, however, good correlation values are maintained (Figure 7b). Differences in Spearman's $r$-values obtained with *I*, *E* and $J_p$ are relatively small (Figure 7b) and do not reflect the differences in scatter that we observed with the disdrometer (Figure 4).

FIGURE 7

## 4.3 Non-colocated seismic and disdrometer measurements of year-round rainfall

At the R-s-G upstream and downstream sites, the effect of rainfall on seismic power is visible for frequencies in the range ~50–175 Hz (Figure 8). The rainfall-induced power spectral density is similar for both sites, indicating that the differences in rainfall intensity between the sites are relatively small and that the seismic wave propagation characteristics are comparable. Directly following rainfall events, river-generated seismic power is observed in the frequency range 10-50 Hz (Figure 8e-8g) and, in the case of the upstream site, also at higher frequencies in the range 100-150 Hz; this is possibly due to contributions of plunging flow, aeration, etc. and/or lower seismic attenuation that are specific to the bedrock nature of the channel. For these sites, the spectrogram could only be determined up to 200 Hz (the rate of acquisition is lower than at Ainac), but we do expect that at higher frequencies, up to 400 Hz, the seismic power is strongly dominated by rainfall (similar to Ainac, Figure 5) and that the contribution of river flow is strongly attenuated (Gimbert et al., 2014). Besides rainfall- and river-generated seismic power, we observe a relatively constant diurnal pattern in seismic noise at both sites for frequencies up to ~60 Hz. The observed seismic power during the day is typically ~5 dB higher than at night (Figure 8a and 8b), most likely due to daytime traffic and human activities (Figure 1f and 1g).

FIGURE 8

The scaling relations between seismic power and the rainfall physical measures found for the R-s-G upstream and downstream sites (Figure 9) are very similar to those observed at Ainac (regression lines
derived in Figure 6 are added as a reference to Figure 9). In the fitting we only consider data with river flows below 0.2 m³/s to eliminate the contribution of river-generated seismic power (a 40-minute time delay is considered for the application of the upstream discharge data to the downstream station).

Alternatively, if we were to only consider data with river flows above 0.2 m³/s, this would lead to a very minor change in the *P* vs. *J*<sub>P</sub> scaling exponents of less than 0.1. Although the effect of river flow may be ~15 dB at the limit of rainfall detection in the case of the upstream site, this effect is no longer noticeable at rainfall intensities of 10 mm/h, while at the downstream site, the effect of flow on the observed seismic power is insignificant in the considered frequency range (Figure 9).

The absolute values of seismic power associated with the fitted power-law functions are circa 5 dB (upstream) and 10 dB (downstream) lower than at Ainac. This applies to both rainfall-induced seismic power as background noise, indicating that these values are site-specific rather than related to the rainfall (i.e., the drop size distribution). The differences in seismic power between the sites may be related to differences in the nature and structure of the subsurface that affect seismic wave attenuation.

The geophones were installed in a shallow soil at Ainac (Figure 1b) and in thicker alluvial deposits at the R-s-G sites, which are particularly coarse (cobble/gravel) at the downstream site (see section 5.2).

FIGURE 9

The scaling exponents at frequencies higher than 80 Hz, are more-or-less constant for the three sites, signifying the extensive frequency range over which seismic observations are sensitive to rainfall (Figure 10a and 10c). For these frequencies, the strength of the correlation is also relatively constant and varies little amongst the rainfall physical parameters (Figure 10b and 10d). For the R-s-G upstream and downstream sites, rainfall can be observed in the seismic measurements down to frequencies as low as ~65 Hz; at lower frequencies, the correlation rapidly weakens. At Ainac, this transition is lower, near 40 Hz, likely due to lower contributions of noise from anthropogenic sources and from the river, indicating that these effects could not be entirely eliminated at R-s-G (despite the filtering of seismic data, section 3.3, and only considering data where $Q<0.2 \text{ m}^3/\text{s}$). The 80-120 Hz range that we use to illustrate the observations in Figures 6 and 9 is suitable for the set of sites that we studied (note that this range is not unique and that slightly lower or higher values could have also been considered), but this may not generally be the case, making it important to consider the scaling across the entire frequency range as done in Figure 7 and 10. For the sites that we consider, we also observe a general decrease in
406 (maximum) correlation with distance between the geophones and disdrometer; an r-value of circa 0.8
407 for the colocated instruments at Ainac, a value of 0.6-0.7 with a distance of 500 m for the upstream
408 geophone, and a value of 0.5-0.6 with a distance of 2.15 km for the downstream geophone.

409

410

FIGURE 10

411

412

413 **5. DISCUSSION**

414 **5.1 Scaling of seismic power with rainfall physical measures**

415 In this study, we directly relate observations of seismic power spectral density to independently
416 measured rainfall intensity, kinetic energy, and impulse power, detecting rainfall at intensities as low as
417 1 mm/h (Figure 5). The large bandwidth of rainfall-induced seismic vibrations allows these to be readily
418 distinguished from other sources such as diurnal fluctuations in anthropogenic noise or local river
419 processes that follow rainfall events (Figure 8). At the sites we investigated, we observed a strong
420 correlation between seismic power, *P*, and impulse power, *J*<sub>P</sub> (Figure 6 and 9). More critically, we find
421 that *P* scales linearly with *J*<sub>P</sub>, providing convincing evidence that the seismic modelling framework
422 adapted from Tsai et al. (2012) is applicable, i.e., that ground vibrations are set by the impulse power of
423 instantaneous and randomly occurring raindrop impacts. In addition, our dataset provides an
424 independent confirmation of the same physical principles that are extensively applied to study bedload
425 transport (Bakker et al., 2020; Chao et al., 2015; Lagarde et al., 2021) and debris flows (Allstadt et al.,
426 2020; Farin et al., 2019; Kean et al., 2015; Lai et al., 2018), where field measurements of particle size
427 distributions and fluxes are more difficult.

428 The impulse power of raindrops depends strongly on the drop diameter, $J_p \propto N_{tot}p(D)D^{7.33}$, and it is
429 therefore expected that the seismic power is very sensitive to the largest impacting drops despite their
430 relatively small abundance. In our case, on average approximately 90% of the impulse power that sets
431 the seismic power was attributed to drops larger than 3 mm, corresponding to ~15% of the rainfall
432 volume, but these large drops occur considerably more frequently with above-average rainfall intensities
433 (Figure 3b). A similar size dependency is also found for the seismic power due to bedload transport (e.g.,
434 Lagarde et al., 2021), where $P \propto N_{tot}p(D)D^{6.5}$ and $D$ in this case reflects the diameter of a saltating bed
435 particle (Bakker et al., 2020), and radar reflectivity factor $Z$, where $Z \propto N_{tot}p(D)D^{5.33}$ (e.g., Uijlenhoet
436 & Stricker, 1999; Wexler & Atlas, 1963). For the application of all these measurements, it is therefore
437 critical to constrain p(D) and changes in p(D) when studying fluxes such as rainfall intensity. For rainfall
438 observations, the sensitivity to the largest drops (Figure 3b) and strong correlation with kinetic energy
439 (Figure 4b) make seismic measurements more suitable to study soil erosion. In particular, to study soil
440 particle detachment and transport by raindrop splash on a dry surface, i.e., during the onset of the storm
441 and in interrill areas, and, to a lesser extent, where raindrops penetrate surface water and dislodge soil
442 particles (we do not account for the effect of surface water in our model) that may then be readily
443 transported in the flow (Kinnell, 2005).

444 In addition to the linear scaling of seismic power, *P*, with impulse power, *J*<sub>P</sub>, we found relatively constant
445 power-law scaling relations with the more commonly used measures of rainfall intensity, *I*, and kinetic
446 energy, *E*, for rainfall intensities up to circa 50-100 mm/h. Due to temporal changes in drop-size
447 distributions during rainfall events (e.g., Chapon et al., 2008; Lee & Zawadzki, 2005) and on seasonal
448 timescales (e.g., Seela et al., 2018; Smith et al., 2009), the correlations of *P* vs. *I* and, to a lesser extent, *P*
449 vs. *E* are expected to be weaker than that of *P* vs. *J*<sub>P</sub> (Figure 4). In practice, the observed differences
450 were not discernable (Figure 7 and 10) and likely fall within the general uncertainty range associated
451 with the measurements. The exponents associated with the *P* vs. *I* and *P* vs. *E* power-law relations are
452 not unique and depend on the covariance of the total drop density flux, *N*<sub>tot</sub>, and the drop size
453 distribution, *p*(*D*). The scaling relations we obtained correspond to the ‘intermediate’ rainfall case (see
454 section 2.3), where *P* ~ *I*<sup>2.1</sup> and *P* ~ *E*<sup>1.6</sup>. When combining available records of seismic power and
455 rainfall intensity (from a rain gauge, disdrometer, or any other type of instrument), scaling analysis may
456 provide new insights into the (spatio)temporal variability of rainfall drop-size distributions, through
457 changes in the scaling exponent, and thus changes in meteorological conditions. In addition, such a *P*
458 vs. *I* scaling analysis will indirectly also provide insight into the scaling of *P* vs. *E* through their relative
459 dependencies on *D*.

460 If we make the parallel between seismic power and disdrometer-based estimates of radar reflectivity
461 factor *Z*, the observed variability in scaling relations from various settings, *Z* ~ *I*<sup>1.4±0.2</sup> (e.g., Hazenberg
462 et al., 2011; Tokay & Short, 1996; Uijlenhoet & Stricker, 1999) and *Z* ~ *E*<sup>1.2±0.15</sup> (e.g., Assouline, 2020;
463 Uijlenhoet & Stricker, 1999; Yu et al., 2012), may provide an indication for the expected variability of
464 scaling relations with *P* (*P* vs. *I* and *P* vs. *E*). Smaller exponents for *I* and *E* may be expected for
465 convective events where *Z*-based studies indicate more ‘number-controlled’ rainfall (Hazenberg et al.,
466 2011; Kirsch et al., 2019; Sempere-Torres et al., 1998), although inverse trends have also been observed
467 (Tokay & Short, 1996), which may be due to concurrent orographic effects and the continental vs. 468 maritime character of rainfall (Hazenberg et al., 2011). Whereas we found negligible differences in 469 scaling exponents between our sites (Figure 9), there are uncertainties associated with the measurement 470 techniques and fitting and we may expect P vs. I and P vs. E scaling exponents to vary per climatic 471 setting with at least ±0.2, with maximum values of respectively 2.4 and 1.7 as set by ‘size-controlled’ 472 rainfall.

473 At rainfall intensities higher than 50-100 mm/h, the obtained P vs. I and P vs. E scaling exponents are 474 expected to decrease due to the emergence of a relatively constant drop size distribution (Figure 3), 475 although we did not find evidence for such a transition in our seismic data (Figure 6 and 9). Based on our 476 scaling analysis, the scaling exponent decreases and may approach 1 in ‘number-controlled’ rainfall, 477 similar to the radar reflectivity factor Z (Assouline, 2020; Uijlenhoet et al., 2003). In theory, scaling 478 exponents of less than 1 and potentially less than 0 (i.e., inverse scaling) may be conceivable where 479 raindrop breakup becomes dominant over raindrop coalescence, leading to smaller drop sizes at (very) 480 high rainfall intensities (e.g., Assouline & Mualem, 1989). In any case, seismic records will provide 481 relatively accurate measurements at such high rainfall intensities, i.e., a high signal-to-noise ratio, as 482 compared to the disdrometer-based measures, where more and larger drops lead to larger uncertainties 483 in drop detection (e.g., Park et al., 2017).

484 5.2 Rainfall surface area and subsurface seismic properties affecting seismic power

485 Seismic records provide an integrative measure of the raindrop impulse power over the ground surface 486 area near the instrument (equation 7). Here, we explore the spatial contribution of rainfall to seismic 487 power and the relative magnitude of observed seismic power based on a displacement Green’s function 488 for Rayleigh waves (Aki & Richards, 2002):

$$
\tilde{G}(f, r) = \frac{f}{8\rho_s w_p^2(f) w_g(f)} \left(1 + \left(\frac{\pi^2 f r}{w_p(f)}\right)^3\right)^{-\frac{1}{6}} \exp\left(\frac{-\pi f r}{w_p(f)K}\right), \quad [8]
$$

489 where ρs is the density of the subsurface, wₚ(f) and wg(f) are the frequency-dependent wave phase
490 and group velocity, and K is a dimensionless quality factor. In particular, we focus on the quantification
491 of the source-to-receiver decay of seismic power, Ĝ²(f, r), due to geometric spreading and inelastic
492 attenuation associated with a uniform subsurface. The geometric spreading is represented with the term
493 (1 + (π²f r/w)³)⁻¹/₆ that was proposed by Gimbert et al. (2014) as an approximation of amplitude decay
for all values of *r* and, in particular, accounting for raindrop impacts in the near-field (*r* of the order of one wavelength or smaller), in the close vicinity of the seismic instrument. This function is used instead of the commonly used far-field approximation $\sqrt{\frac{w_p}{\pi^2 f r}}$ (e.g., Tsai et al., 2012) which grows to infinity as *r* goes to 0, while it should be bounded by unity (Aki & Richards, 2002), as accounted for in the function of Gimbert et al. (2014). The inelastic attenuation in equation 8 is quantified with a function that describes the exponential decay with distance *r*. To explore the extent of the surface area that contributes to the measured seismic observations, we calculated the seismic power associated with the fundamental wave mode in a uniform, generic bedrock subsurface following Tsai et al. (2012); details on the derivation of $P_{rel}$ and the used values of $w_p$, $w_g$ and *K* are provided in Appendix B.

Our calculations indicate that the largest seismic power contribution may be expected at a radial distance of the order of 0.1-1 m, where lower frequencies will be susceptible to drop impacts from larger distances (Figure 11a). The cumulative seismic power reaches about half of the total expected power within a distance of the order of 1-10 m (Figure 11b), while 90% of the total is expected to come from within ~5-25 m from a seismic instrument. Due to the spatial distribution of rainfall impacts, all around the instrument and at all distances from the instrument, we expect rainfall to be prominent in the power spectral density over a large frequency range and, in particular, also at high frequencies. This is also generally visible in the spectrograms (Figure 5 and 8) and the observed scaling with frequency (Figure 7 and 10). The spatial distribution of rainfall and the associated large frequency range for seismic observation contrast with more localized and distant sources such as river turbulence- or bedload-induced contributions (in our case the river is at ~30-40 m distance) which are prominent at low frequencies due to the stronger attenuation at high frequencies (near 20 m in Figure 11a).

FIGURE 11

We evaluate the sensitivity of the estimated seismic attenuation on ground properties by considering seismic wave propagation values derived for river gravel bed deposits (Bakker et al., 2020) and overbank deposits (Lagarde et al., 2021); for details on the underlying values see Appendix B. The expected seismic power varies per frequency between a factor of 0.4 to 4 (Figure 11c) relative to that derived following Tsai et al. (2012). These differences are consistent with the differences in seismic power that we observed between our sites (Figure 9). The sensitivity of the areal contribution to the observed seismic
power, indicated with the radial distance within which 90% of the total seismic power is expected, $r(P90%)$, varies with a factor of 0.25 to 2 (Figure 11d). Based on this sensitivity analysis, differences in seismic power between sites with different ground properties, or at a site due to temporal changes, e.g., the evolution of moisture conditions (Illien et al., 2021), are expected to be within an order of magnitude. In practice, these differences may be larger due to assumptions associated with the application of a simple Green's function at high frequencies (>50 Hz), where we only consider the fundamental wave mode (Brissaud & Tsai, 2019) and do not account for (small-scale) heterogeneities in the subsurface (e.g., Allstadt et al., 2020).

If we consider a maximum contributing area corresponding to $r=25$ m around our geophones (Figure 1e-1g), the sites we studied have similar landcover that is characterized by open grassland, roads and isolated bushes or trees. Differences in power spectral density are expected per landcover due to the different drop impact mechanics, e.g., more efficient impulse transfer is expected for a smooth, bare road surface than for a grass-covered surface. In the case of trees, raindrops are intercepted by the tree canopy and subsequently released as relatively large uniform drops that lead to enhanced total kinetic energy (e.g., Brandt, 1988) and seismic power (Dietze et al., 2017a). Seasonal variability is expected for tree-canopy interception of deciduous trees and during the growth season of grass and crops. An additional factor that may contribute to the differences in power spectral density observed between our sites is relief (e.g., Köhler et al., 2012). Ainac lies on an inclined hillslope, R-s-G upstream on a flat surface and R-s-G downstream directly beside a steep road embankment. Considering all potential site-specific differences, the order-of-magnitude differences in seismic power we observed amongst the sites (Figure 9) may be considered relatively modest. Although such order-of-magnitude differences in seismic power may lead to absolute differences of up to 2 orders of magnitude when deriving rainfall kinetic energy and intensity from the seismic record, these may be easily constrained with independent measures of rainfall amounts and, most importantly, the characteristic scaling relations are maintained (Figure 9). In the future, individual effects of surface and subsurface properties on the seismic observation of rainfall will need to be addressed using specifically-designed field experiments.

Our Green's function modelling exercise provides a general spatial constraint on the contribution of rainfall impacts on seismic records. The monitored surface area of ~100-2000 m² is up to 4-5 orders of magnitude larger than the surface area monitored with the disdrometer (0.0054 m²), where the spatial variability in rainfall may be insufficiently captured (Jaffrain & Berne, 2011; Tapiador et al., 2017). The use of temporally-resolved seismic field observations is therefore expected to be suitable for the
555 monitoring of spatiotemporally varying soil erosion processes (Dunkerly, 2021; Kinnell, 2020). With 556 seismic measurements, rainfall impacts can be sampled at a plot scale but this may be, depending on e.g. 557 rainfall type, representative for the scale of hillslopes to small/mesoscale watersheds.

558

559 **CONCLUSIONS**

560 Our study provides a foundation for the seismic observation of rainfall, whose effects have been 561 previously identified but which have not been studied in detail. We adapted a seismic framework for 562 bedload transport (Tsai et al., 2012) to assess the seismic power generated by impacting raindrops. This 563 enabled us to explain the observed linear relation between measured seismic power, typically at 564 frequencies (well) above 50 Hz, and the impulse power of raindrops calculated from disdrometer 565 measurements: $P \propto J_p$ (equation 8; Figures 6 and 9).

566 The scaling of seismic power with rainfall intensity and kinetic energy we observed, $P \propto I^{2.1}$ and $P \propto E^{1.6}$, were consistent between the studied sites (Figure 9) and conform to ‘intermediate’ rainfall 567 predictions (equal proportions of size- and number-controlled rainfall; Uijlenhoet et al., 2003). We could 568 not discern systematic temporal changes in scaling, e.g., event- or seasonally-based, but do expect 569 scaling exponents to vary with geographical setting and weather systems. Of particular importance is the 570 contribution of the largest raindrop fractions, where in our case 90% of the seismic power may be 571 attributed to drops larger than 3 mm that correspond to only ~15% of the rainfall volume (Figure 3b). 572

573 Seismic measurements are most suitable to study rainfall with high intensities and relatively abundant 574 large drops where high signal-to-noise ratios ensure reliable seismic measurements, while disdrometer 575 measurements are prone to uncertainty. Under such conditions, the application of seismic 576 measurements is particularly promising in soil erosion studies where they may provide a record of 577 rainfall erosivity and splash erosion that is temporally resolved (e.g. Figure 5c and 5d) and spatially 578 integrated over a surface of ~100-2000 m² (Figure 11). At a larger scale, seismic measurements may be 579 used to study catchment hydrological response to rainfall events and subsequent river runoff processes 580 (e.g., Figure 8; Chmiel et al., 2021) and the mechanisms with which seismically-monitored landslides (Kuo 581 et al., 2018), rockslides (Helmstetter & Garambois, 2010) and rockfalls (Dietze et al., 2017b) may be 582 triggered or induced.

583
584 ACKNOWLEDGMENTS

585 This study was supported by the French National Research Agency (ANR) grants 856ANR-18-CE01-0019-
586 01 (DEAR project) and ANR-17-CE01-0008 (SEISMORIV project), and the Draix-Bléone observatory,
587 funded by the National Institute for Earth Sciences and Astronomy (INSU/CNRS) and the Observatoire
des Sciences de l'Univers de Grenoble (OSUG). We thank Marco Borga and Francesco Marra for their
588 dedicated editorial work, and Velio Coviello and an anonymous reviewer for their constructive
589 comments on a previous version of this manuscript. We acknowledge Romain Biron for his help with
590 data collection. Disdrometer data, including drop diameter and velocity distributions, is available online
591 for Ainac (https://doi.org/10.17178/DRAIXBLEONE_GAL_ROB_DSD_1420) and Robine-sur-Galabre
592 (https://doi.org/10.17178/DRAIXBLEONE_GAL_AIN_DSD_1920) meteorological stations. For further
593 details and complementary meteorological and Galabre river flow data see Legout et al. (2021). Seismic
594 data (minute-based) will be made available on the OSUG data center upon acceptance of this
595 manuscript.
596

597

598 NOTATION

599 $A(D_i)$ effective surface area [m²];

600 $c_E$ unit conversion factor, $c_E=3600$ s/h;

601 $c_I$ unit conversion factor, $c_I=3.6 \times 10^6$ mm/m s/h;

602 $D$ raindrop diameter [m];

603 $D_i$ mean equivalent raindrop diameter for raindrop diameter category $i$ [m];

604 $E$ kinetic energy flux density [J/m²/h], see equation 1b;

605 $f$ frequency [Hz]

606 $F(t)$ force time series of an impact [N], see equation 3;

607 $\tilde{F}(D)$ Fourier transform of $F(t)$ [Ns];

608 $\tilde{F}_A^2$ force power spectrum of raindrop impacts per unit area and time [N²s/m²];

609 $g(x)$ raindrop size distribution that is scaled to the unit value of reference variable $\Psi$, see Appendix A;

610 $\tilde{G}(f, r)$ Fourier transform of a Green's function [m/N], see equation 8;

611 $I$ rainfall intensity [mm/h], see equation 1a;

612 $J(D)$ raindrop impulse [Ns], see equation 3b;

613 $J_P$ rainfall impulse power flux density, see equation 6 [N²s/m²];
614 K dimensionless seismic quality factor;

615 $N_D(D)$ number of raindrops per unit area, drop size and time [m$^{-3}$s$^{-1}$];

616 $n_{i,j}$ number of raindrops for raindrop diameter category i and velocity category j (dimensionless);

617 $N_{i,j}(D_i)$ number of raindrops in raindrop diameter category i and velocity category j per unit area and time [m$^{-2}$s$^{-1}$];

619 $n_{tot}$ total number of raindrops (dimensionless);

620 $N_{tot}$ total number of raindrops per unit area and time [m$^{-2}$s$^{-1}$];

621 $p(D)$ drop size distribution [m$^{-1}$];

622 $P$ seismic power [(m/s)$^2$];

623 $P(f)$ power spectral density, see equation 7 [(m/s)$^2$/Hz];

624 $Q$ river discharge (at the upstream station) [m/s$^3$];

625 $r$ source-to-receiver distance [m];

626 $T$ temporal resolution of measurements [s];

627 $\tilde{u}(f)$ Fourier transform of vertical ground velocity [m];

628 $v$ vertical raindrop velocity [m/s];

629 $V$ volume of a raindrop [m$^3$];

630 $v_t$ terminal vertical raindrop velocity [m/s];

631 $w_g$ wave group velocity [m/s];

632 $w_p$ wave phase velocity [m/s];

633 $Z$ radar reflectivity factor [mm$^6$ m$^{-3}$];

634 $\alpha$ dimensionless exponent scaling law, see Appendix A, equation A1;

635 $\beta$ dimensionless exponent scaling law, see Appendix A, equation A1;

636 $\delta$ Dirac delta function [s$^{-1}$];

637 $\mu$ mean noise power spectral density [dB rel. (m/s)$^2$];

638 $\gamma$ dimensionless impact elasticity factor; $\gamma=1$;

639 $\rho_s$ subsurface density, $\rho_s=2650$ kg/m$^3$;

640 $\rho_w$ water density, $\rho_w=1000$ kg/m$^3$;

641 $\sigma$ standard deviation of noise power spectral density [dB rel. (m/s)$^2$];

642 $\Psi$ reference variable scaling law, see Appendix A;

643
644 APPENDIX A

645 Raindrop size distributions may be parameterized as a function of both drop diameter, *D*, and a
646 reference variable, $\Psi$ (for which we use rainfall intensity $I$ and kinetic energy $E$), according to the
647 scaling-law framework developed by Sempere-Torres et al. (1994; 1998):

$$N(D, \Psi) = g(x)\Psi^{\alpha}, \quad x = D\Psi^{-\beta}, \qquad [\text{A1a, A1b}]$$

648 where $g(x)$ is a raindrop size distribution that is scaled to the unit value of $\Psi$ (e.g., for $I$ this is 1 mm/h)
649 and $\alpha$ and $\beta$ are dimensionless exponents. Substituting the scaling law (equation A1a) into equation 6 for
650 impulse power leads to the power-law scaling relation:

$$J_p \propto \Psi^{\alpha+8.34\beta} \qquad [\text{A2}]$$

651 If we consider $I$ as reference variable $\Psi$, we can also substitute the scaling law (equation A1a) into
652 equation 1a for rainfall intensity, which leads to the self-consistency constraint of $\alpha + 4\beta = 1$ that is
653 required to ensure $I = I$. Note that the constraint differs from Uijlenhoet et al. (2003) because we
654 consider $N_{i,j}(D_i)$ as the drop density as a flux-based density and not as a volume-based density.
655 Similarly, if we use $E$ as the reference variable, we obtain a self-consistency constraint of $\alpha + 5.34\beta = 1$
656 using equation 1b.

657 With the exponents $\alpha$ and $\beta$, a range of dependencies between the drop size distribution and the
658 reference variable, $I$ or $E$ in our case, can be quantified. The endmember cases of this range are
659 provided by ‘number-controlled’ rainfall on one hand, where $\beta = 0$, and ‘size-controlled’ rainfall on the
660 other hand, where $\alpha = -\beta$. For ‘intermediate’ rainfall, with equal proportions of size- and number-
661 controlled rainfall, $\alpha = 0$ (Uijlenhoet et al., 2003). If we apply equation A2 for these 3 cases, taking into
662 account the self-consistency constraints for $I$ and $E$, we obtain the scaling laws derived in section 2.3 and
663 summarized in Table A1.

<table><caption>Table A1: Scaling relations for idealized rainfall cases.</caption><thead><tr><th>'Number-controlled' rainfall</th><th>'Intermediate' rainfall</th><th>'Size-controlled' rainfall</th></tr></thead><tbody><tr><td>P ∝ I</td><td>P ∝ I<sup>2.1</sup></td><td>P ∝ I<sup>2.4</sup></td></tr><tr><td>P ∝ E</td><td>P ∝ E<sup>1.6</sup></td><td>P ∝ E<sup>1.7</sup></td></tr><tr><td>P ∝ J<sub>P</sub></td><td>P ∝ J<sub>P</sub></td><td>P ∝ J<sub>P</sub></td></tr></tbody></table>

665
667 Following Tsai et al. (2012), we quantify $w_p$, $w_g$ and $K$ as follows:

$$w_p = w_{p0}(f/f_0)^\xi, \qquad [B1]$$

$$w_g = \frac{w_p}{(1 + \xi)}, \qquad [B2]$$

$$K = K_0(f/f_0)^\eta, \qquad [B3]$$

668 where $f_0 = 1$ Hz and $\xi$ and $\eta$ are dimensionless exponents. Table B1 provides an overview of the value 669 used to quantify $w_p$, $w_g$ and $K$ in the studies by Tsai et al. (2012), Bakker et al. (2020) and Lagarde et al. 670 (2021).

671 The values for $w_{p0}$ and $\xi$ in Tsai et al. (2012) are based on depth-velocity profiles for a generic bedrock 672 subsurface (Boore & Joyner, 1997). The value for $K$ was assumed constant, $K=20$, for high frequencies 673 based on Anderson and Hough (1984).

674 The values for $w_{p0}$ and $\xi$ in Bakker et al. (2020) are based on arrival times of a series of large-rock 675 impacts along a transect between geophones on opposite sides of a braided river. The values for $K_0$ and 676 $\eta$ were derived from the decay in seismic power from the same active experiment.

677 The parameters in Lagarde et al. (2021) were derived from sledgehammer blows that were recorded by a 678 network of geophones on both sides of the river. They first derived shear wave velocity as a function of 679 depth from the inversion of Rayleigh wave dispersion curves and subsequently modelled $w_p$, $w_g$ and $K$. 680

681 Table B1: Wave velocity and quality factors used to explore the sensitivity of seismic power attenuation in Figure 11.

<table><thead><tr><th>Tsai et al. (2012)<br>generic bedrock subsurface</th><th>Bakker et al. (2020)<br>river gravel bed</th><th>Lagarde et al. (2021)<br>river overbank deposits</th></tr></thead><tbody><tr><td>$w_{p0}$ = 1295<br>$\xi$ = 0.374</td><td>$w_{p0}$ = 680<br>$\xi$ = 0.31<br>$w_p \ge 180$</td><td>$w_{p0}$ = 3190<br>$\xi$ = 0.59</td></tr><tr><td>$K_0$ = 20<br>$\eta$ = 0</td><td>$K_0$ = 11.47<br>$\eta$ = 0.15</td><td>$K_0$ = 10<br>$\eta$ = 0</td></tr></tbody></table>
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Figure 1: (a) Overview aerial photograph (courtesy of Esri-World Imagery) of the Galabre catchment (1:100'000), France; (b) Parsivel² disdrometer at Ainac with an indication of the monitored surface area A; (c) Sensor geophone installed at Ainac. Zoomed-in overview photos (1:5'000) of: (d) meteorological station Robine-sur-Galabre (R-s-G); (e) meteorological station and geophone Ainac; (f) geophone and R-s-G upstream river monitoring station; and (g) geophone and R-s-G downstream river monitoring station. The maximum area for which the geophone is expected to be sensitive to rainfall is indicated in red, with a 25-m radius around the geophone (see section 5.2).

Figure 2: Number of drops observed per category drop diameter and drop velocity at R-s-G meteorological station from February 2019 to December 2020. The dashed line shows terminal velocity as a function of drop diameter as predicted by Atlas and Ulbrich (1977), see equation 2. A maximum drop velocity of 9.5 m/s is indicated following Carollo and Ferro (2015).

Figure 3 (a) Mean drop size distribution for different rainfall intensities at R-s-G meteorological station from February 2019 to December 2020; (b) Impulse power modelled as a function of D for the drop size distribution per intensity in (a) and using $v=v_t$. The insert shows cumulative volume (V) and cumulative impulse power with drop size.

Figure 4: (a) Correlation of disdrometer-derived impulse power with (a) rainfall intensity and (b) rainfall kinetic energy. Scaling relations for ‘number-controlled’, ‘size-controlled’ and ‘intermediate’ rainfall cases (section 2.3) are shown for reference.

Figure 5: (a) Spectrogram and (b) rainfall intensity (log-scale) at Ainac for the period April-May 2021. (c, d) provide a zoomed-in detail of a rainfall event on 11 April 2021, where rainfall intensity in (d) is shown on a linear scale. At ~165 Hz there is a spurious frequency (instrument artefact; e.g. Faber and Maxwell, 1997).

Figure 6: Seismic power (80–120 Hz) versus rainfall intensity (a), kinetic energy (b) and impulse power (c) at Ainac for the period April-May 2021. Regression lines are power-law functions fitted to the data in the boxed areas.
Figure 7: Correlation between seismic power and rainfall physical measures as a function of frequency (bins of 5 Hz), with (a) the scaling exponent, i.e. the slope of the regression line in Figure 6, and (b) the corresponding Spearman's r-value, at Ainac for the period April-May 2021.

Figure 8: Spectrograms at R-s-G (a) upstream site and (b) downstream site, and (c) rainfall intensity (log-scale) at R-s-G meteorological station for the period October-December 2019. (e, f, g) provide a zoomed-in detail of a rainfall-runoff event on the 15th of October 2019. (g) In addition to the rainfall intensity (linear scale), flow discharge at R-s-G upstream station is shown. The dashed boxed area in (e) indicates site-specific river-generated seismic power (see main text for details).

Figure 9: Correlation and scaling relations of seismic power (80–120 Hz) at R-s-G upstream and downstream sites with (a, d) rainfall intensity, (b, e) kinetic energy, and (c, f) impulse power, for February 2019 to December 2020. Flow discharge, Q, at the upstream site is indicated with the color scale; we account for a mean delay of 40-min at the downstream site. Regression lines are fitted to the data in the boxed areas for Q<0.2 m³/s; regression lines for Ainac, fitted in Figure 6, are provided as reference (dashed lines).

Figure 10: Correlation between seismic power and rainfall physical measures as a function of frequency (frequency bins of 5 Hz) for (a, b) R-s-G upstream site and (c, d) R-s-G downstream site, for February 2019 to December 2020. The scaling exponent (a and c), i.e. the slope of the regression lines in Figure 9, and the corresponding Spearman's r-value (b and d) are shown; the corresponding plots for Ainac (Figure 7) are provided as reference.

Figure 11: (a) Modelled relative seismic power spectral density contributions with radial distance from a geophone and (b) cumulative seismic power with distance. Seismic power is given relative to $P(f=100 \text{ Hz}, r=1 \text{ m})$ and is determined through discretizing the integral in equation 7 with $dr=1 \text{ cm}$ up to a maximum of $r=1 \text{ km}$. Radial distances are indicated that correspond to the maximum seismic power contribution (a) and 50% and 90% of the total estimated seismic power contributions (b). We compare (c) the seismic power and (d) the radial distance within which 90% of the total seismic power is expected, $r(P_{90}\%)$, based on Bakker et al. (2020) and Lagarde et al. (2021), relative to the corresponding values that we obtained for Tsai et al. (2012) shown in (a,b). For the relevant seismic wave propagation values from these studies, see Appendix B.
<caption>(b)</caption>

A

<figure><img src="image_4.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

6°10.800'

6°12.000'

6°13.200'

6°14.400'

(a)

44°14.400'

<figure><img src="image_9.png" /></figure>

44°13.200'

44°12.000'

<figure><img src="image_10.png" /></figure>

44°10.800'

44°9.600'

<figure><img src="image_11.png" /></figure>

44°8.400'

44°14.000'

44°13.200'

44°12.000'

44°10.800'

44°9.600'

<figure><img src="image_12.png" /></figure>

6°10.800'

6°12.000'

6°13.200'

6°14.400'
<figure><img src="image_13.png" /></figure>

$v_t \propto D_i^{0.67}$
(Atlas & Ulbrich, 1977)

$\max(v_t) \approx 9.5$
<figure><img src="image_14.png" /></figure>

p(D) total

<figure><img src="image_15.png" /></figure>

D(l) with max.
contrib. to J<sub>p</sub>

<figure><img src="image_16.png" /></figure>

Drop diameter D [mm]
<figure><img src="image_17.png" /></figure>
<figure><img src="image_18.png" /></figure>
Seismic power $P(f = 80-120 \text{ Hz})$ [dB rel. (m/s)$^2$]

<figure><img src="image_19.png" /></figure>

Rainfall Intensity
$I$ [mm/h]

0.1

0.7

10

<figure><img src="image_20.png" /></figure>

Rainfall Kinetic Energy
$E$ [J/m$^2$/h]

0.5

100

<figure><img src="image_21.png" /></figure>

Rainfall Impulse power
$J_p$ [N$^2$s/m$^2$]

10

10$^6$

10$^4$

Seismic power $P(f = 80-120 \text{ Hz})$ [dB rel. (m/s)$^2$]
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>
<figure><img src="image_24.png" /></figure>
<figure><img src="image_25.png" /></figure>
<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>
<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>