Estimation of Avalanche Development and Frontal Velocities Based on the Spectrogram of the Seismic Signals Generated at the Vallée de la Sionne Test Site

Abstract

The changes in the seismic signals generated by avalanches recorded at three sites along a path at the Vallée de la Sionne (VdlS) experimental site are presented. We discuss and correlate the differences in the duration, signal amplitudes, and frequency content of the sections (Signal ONset (ON), Signal Body (SBO), and Signal TAil and Signal ENd STA-SEN) of the spectrograms with the evolution of the powder, transitional and wet snow avalanches along a path. The development of the avalanche front was quantified using the exponential function in time F (t) = K’ exp (β t) fitted to the shape of the signal ONset (SON section of the spectrogram. The speed of the avalanche front is contained in β. To this end, a new method was developed. The three seismic components were converted into one seismic component (FS), when expressing the vector in polar coordinates. We linked the theoretical function of the shape of the FS-SON section of the spectrogram to the numerical coefficients of its shape after considering the spectrogram as an image. This allowed us to obtain the coefficients K’ and β. For this purpose, the Hough Transform (HT) was applied to the image. The values of the resulting coefficients K’ and β are included in different ranges in accordance with the three types of avalanche. Curves created with these coefficients enable us to estimate the development of the different avalanche types along the path. Our results show the feasibility of classifying the type of avalanche through these coefficients. Average speeds of the avalanches approaching the recording sites were estimated. The speed values of wet and transitional avalanches are consistent with those derived from GEODAR (GEOphysical Doppler radAR) measurements, when available. The absence of agreement in the speed values obtained from seismic signals and GEODAR measurements for powder snow avalanches indicates, for this type of avalanche, a different source of the measured signal. Hence, the use of the two measuring systems proves to be complementary.

1. Introduction

It is well known that snow avalanches generate seismic signals and that they can be used for monitoring e.g., [1,2,3,4,5,6,7,8,9,10,11,12]. At present, seismic signals are also employed for avalanche location [13,14,15], and to infer avalanche dynamic properties [13,15,16,17,18,19,20,21,22]. The use of the spectrograms of the seismic signals was crucial to these studies [23]. New insights were provided by comparing the seismic signals with the results from the measurements of the FCMW radar [24] co-located with a seismic sensor at the Vallée de la Sionne (VdlS) test site [25,26,27]. The results of FCMW and of seismic sensors at two strategic sites of the avalanche path, the track zone and the run-out zone, were compared in detail in [19]. A division of the spectrogram into sections corresponding to avalanches approaching a sensor: SON (Signal ONset), SOV (Signal OVer), and SEN (Signal ENd) depending on whether the avalanche approaches the sensor, is over it, or is moving away from it, respectively, was made [19]. The SOV section can be subdivided into two sections, SBO (Signal Body) and STA (Signal Tail). These sections display different features as regards the frequency content and seismic patterns according to the avalanche type. More recently, a joint analysis of the spectrograms of seismic signals and of the images of the GEODAR (GEOphysical Doppler radAR) measurements [28,29,30] carried out with snow avalanches at the VdlS indicates the agreement of the observations and the complementarity of the two different measuring systems (seismic and GEODAR) [31]. This enables us to validate the seismic observations. The data obtained from GEODAR provide images of the evolution of the avalanche when projected into a longitudinal section [30]. To date, a detailed description of the evolution of the avalanche through the seismic signals (spectrogram and time series) along a path for the different types of avalanches has not existed. We have now been able to provide a detailed description of this owing to the seismic installation at VdlS.

With regard to the determination of the avalanche speed, a variety of approaches using equipment different from seismic sensors have been adopted for different avalanche types. Internal avalanche profile velocities have been estimated using impact pressure and optical sensors, e.g., [32,33,34,35,36,37]. Dense avalanche front velocities were initially estimated from FMCW X-band by [38,39]. Anemometer, e.g., [40] and Videogrametry, e.g., [41] have also been used to measure the velocity of the avalanches along the avalanche path. More recently, photo and video analyses have proved useful, e.g., [42,43,44]. Doppler radar measurements were employed in [45,46,47]. Recently, [30,31,48] used GEODAR images [49] to obtain the internal flow velocities as well as the evolution of the avalanche speed along the path.

As for the use of seismic signals to estimate avalanche internal velocity profiles, a successful approach has been attempted, making use of wireless accelerometers in small-scale avalanche experiments [50]. However, it has not been possible to transfer this system to full-scale experiments because of the unsuitable logistical conditions. Seismic methods have been used in avalanche speed determination. The mean velocity of a dry avalanche employing one geophone was calculated in [51]. The evolution of the front speed of avalanches at the experimental site of Ryggfonn using the arrival times of the SOV section at geophones along a path was estimated [16]. These authors found a different speed behavior along the path for wet and powder avalanches. In this case, the avalanches passed over the sensors. Employing beam-forming techniques, [13] estimated front speeds of avalanches using a seismic array for avalanches at distances of about 3 km. More recently [12], inferred apparent velocities of avalanches for events near Davos (Switzerland), employing geophone array data and the processing methods of localization MUSIC (multiple signal classification [52]) and beam-forming. Using the ASL method (amplitude source location) with the data of a dense permanent short-period seismic network installed around Mt. Fuji volcano [15], localized avalanches and calculated their average front speeds after a control by numerical simulations.

In all these cases, more than one seismogram (or its equivalent, more than one seismic station) was necessary to estimate the speed of one avalanche. In contrast, the method we propose uses only one spectrogram to estimate the speed as the avalanche approaches the sensor.

Given that a spectrogram can be analyzed as an image, we use the linear Hough Transform (HT) [53] to estimate the parameters of its geometrical shape. HT is used for digital image processing. It consists of the detection of regular objects determined by known equations. The original HT was designed to detect straight lines and circles. It is considered as a discretization of the Radon Transform [53] defined as an integral along a line. This method is currently used when the analytical equation of the borderline of the object is known. The linear HT has been applied in different fieldworks to a) detect geological faults from seismic sections [54], b) enhance geological structures from two-dimensional image-profiles constructed from ground-penetrating radar [55], and c) develop face recognition software [56].

In this paper, we first introduce the characteristics of the VdlS test site, the instrumentation, and the avalanches used in the study. We then present the data processing methods to obtain the spectrograms from the raw data. Next, the evolution of the seismic signals (time series and spectrograms) of the three types of avalanche (Powder-snow, transitional and wet-snow) recorded at three sites along the path are described. We seek to show how the evolution of the avalanches descending the slope is replicated in the spectrogram seismic signals. After this, we devote a section, with two subsections, to the quantification methodology and results. In the first subsection we present the new method developed for the quantification of the SON section of the spectrogram using HT. To facilitate comprehension, the results obtained are represented in curves. A second subsection is dedicated to the estimation of the avalanche front speed. The discussion constitutes the next section and is followed by the conclusions. Two annexes and supplementary material accompany the main text.

2. Vallée de la Sionne (VdlS) Test Site, Instrumentation, and Avalanche Events

Figure 1 shows an image of the landscape of the VdlS test site (Valais, Switzerland). The characteristics of the site and the instrumentation installed has been described in detail in earlier contributions [25,26,27,57]. The UB group installed seismic sensors of three components along the avalanche path at the VdlS experimental site in 1997 [22]. Over the years we have increased the number of sensors from two to four (Figure 1 and Figure 2). At present, the seismic sensors are installed in a linear array at three strategic sites of the avalanche path: the release zone (cavern A), the track zone (cavern B), and the run-out zone (cavern C) with a common base of time (Figure 1 and Figure 2). Moreover, an infrasound sensor installed a few meters from the fourth seismic sensor (cavern D) at the opposite slope has been in operation since 2008. The seismic data were obtained at stations equipped with a three-component seismometer Mark L-4C-3D (Mark Products) of eigenfrequency 1 Hz and a data acquisition system Reftek–130 (Trimble). One of the major advantages of our installation is that the seismic data recording system does not depend on a triggering system and obtains data continuously, allowing us to record the signals generated by the entire avalanche and other seismic events (e.g., earthquakes, [58]).

Avalanches of a different type of flow descending the same path from a similar release location were chosen.

Table 1. Main characteristics of the selected avalanches. Aval. N.: Avalanche name used in this paper. Type: Avalanche type: POW—powder snow, TRANS—transitional, and WET—wet snow. Size: L—large, M—medium, S—small. Classification after SLF and [19]. (+) from [59]. (A): artificially triggered avalanche. Date: Date of occurrence. N°: SLF internal code. Release location (See Figure 1). Av12 WET–L descended Gullies 1 and 2 (After [19]). GEODAR (GEOphysical Doppler radAR): Information from images, B&C: for avalanches reaching sites B and C. In the GEODAR column, the letters have been replaced by numbers that correspond to the ranges of the start and/or end of the GEODAR signal whenever the avalanche does not reach sites B or C. Snow cover information after [19].

Aval. N.- Type- Size	Date	Nº	Release Location	Runout [m]	GEODAR	Snow Cover at B [m]	Snow Cover at C [m]
Av1 POW-L (A)	31.01.03	504	Pra–CB1	2200	-	[3–1.5]	-
Av2 TRANS (+)	30.12.09	210_0003	CB1	2000	-
Av3 WET-M	01.02.13	3018	CB1	1000	B&900	[5.5–4.6]	-
Av4 TRANS- L	01.02.13	3019	CB1	1900	B&C	[5.5–4]	[1.75–0.5]
Av5 POW- L	05.02.13	3024	CB1	1800	B&C	4.2	-
Av6 WET-M	13.04.13	3050	CB1	1200	B&1000	1.5	-
Av7 WET-M	14.04.13	3053	CB1	1300	B&950	2	-
Av8 POW-L (A)	03.02.15	17	CB1	2000	B&C	[2–0]	[3.5–2.5]
Av9 POW-M	17.12.11	3012	-	1300	-	[3.5–1.5]	[3.5–2.5]
Av10 POW-S	30.12.11	3023	Pra-CB1	1000	1200&650	[x–4.25]	-
Av11 TRANS-L	02.02.13	3021	CB1-CB2	1800	B&C	[x–4]	[1.5–0.5]
Av12 WET-L	01.02.13	3020	CB1-CB2	2000	1300&C	[4.5–4]

shows the twelve selected avalanches with their main characteristics described in different publications (Table 1 in [19], [30,48,57,59]). We analyzed five powder-snow avalanches, three transitional avalanches (avalanches that start moving as a powder avalanche and transform into a wet flow at lower elevations of the path), and four wet-snow avalanches. Most of the avalanches satisfy the following requirements: they were released in the same area (CB1) and descended along the same track zone where Gully 1 connects cavern B to C (Figure 1). Only three avalanches did not meet these requirements: they were released in the nearby area of CB2 and Pra (Pra Rua). We considered these avalanches in our study bearing in mind that they do not fulfill our requirements. With this selection, we ensure the validation of the avalanche comparison. All the avalanches were recorded at the B, C, and D sites (Figure 2). All the powder-snow and transitional avalanches flowed over B and C. However, wet-snow avalanches were smaller and they did not reach sensor C. All this information was obtained from the GEODAR images [49] and from earlier analyses [19].

3. Data Processing

The raw data obtained at 100 s.p.s. in the field were converted to ground motion (m/s) and subjected to a process of homogenization which has been reported in earlier publications, e.g., [16,19,23]. The signals were filtered (1 Hz to 40 Hz) with a 4th order Butterworth band-pass filter. Spectrograms were obtained using the short-time FFT (e.g., [60]) with a Hanning window (length 0.64 s) and 50% (0.32 s) of overlapping. Spectrograms indicate the amplitude of the signal (PSD) according to the frequencies in intervals of 1.56 Hz. The spectrogram amplitudes are represented in 10⁻¹ dB and in a color scale. For a correct reading of the spectrograms, see ([57] and references therein).

Given our interest in the estimation of the evolution of the avalanche front, we studied the seismic amplitudes of the waves travelling towards the sensors down the slope corresponding to the SON section of the spectrogram. The seismic amplitudes obtained in the seismic components oriented (N-S, E-W, Z) were first reorganized to obtain the amplitudes distributed in the directions (S, TS, No), (No) being orthogonal to the slope, (S) the direction of the flux along the slope, and (TS) being orthogonal to these two. This process is described in Appendix A.1. The amplitudes after this process were equally distributed in the three directions (S, TS, No), with no significant amplitude in the direction of the slope (S). Figure A1 in the Appendix A.2 shows the seismic amplitudes distributed in the new directions for the three types of avalanche. The simple explanation for this behavior is that the propagation of the wave field is in all directions. Owing to the ground heterogeneities and to the impacts of snow blocks on the ground, no pure superficial waves were generated, at least in the SON section. Thus, we opted to use all the information obtained in the three seismic components. To this end, the seismic vector constituted by the three seismic components was represented in polar coordinates. The advantage of this representation is that the whole amplitude of the vector is contained in only one coordinate (the radial coordinate) in contrast to the representation in Cartesian coordinates. In this new representation, we have only one seismogram termed final seismogram (FS), which contains all the amplitude information. The properties of the vector are maintained under coordinate transformation. Further details can be found in [61,62] and in Appendix A.2. As a result, the amplitudes of the three seismic components (Cartesian coordinates) were contained in one Final Seismic signal (FS) that helped us to study the evolution of the signals recorded along the avalanche path at caverns B, C, and D.

4. Avalanche Seismic Signal Evolution

One of the characteristics observed in the seismic signals of a single avalanche is their evolution in time and frequency content (spectrograms) along the avalanche path, reflecting the evolution of the flow at different elevations. Each avalanche type displays different patterns in the evolution of the seismic signal.

To illustrate the evolution of the flows along the path three representative avalanches each of a different type (Av4 TRANS-L (TRANS), Av5 POW-L (POW), and Av7 WET-M (WET)) were selected from the avalanches in Table 1. The time series of the final seismogram (FS) of these avalanches recorded at sites B, C, and D (Figure 1 and Figure 2) were displayed using a common time (top Figure 3, Figure 4 and Figure 5). The spectrograms were calculated and displayed in the same manner at the bottom of Figure 3, Figure 4 and Figure 5. Appendix B and

Table A1. Arrival times and duration of the outstanding features (OF) in Figure A3. In bold sections mentioned in the text.

			TRANS- L					POW- M					WET-M
	OF	Time (s)	DURATION (s)	Av. Speed (m/s)		OF	Time (s)	DURATION (s)	Av. Speed (m/s)		OF	Time (s)	DURATION (s)	Av. Speed (m/s)
SON_B	o	32	o-a	SON (o-a)	SON_B	o	36	o-a	SON (o-a)	SON_B	o	19	o-d	SON (o-a)
			38	26		a’	66	34	29		c	34	52	17
			a-c	B-C				a-c	B-C		c’	57	o-a
			105	6.6				23	30		d	71	57
SOB_B	a	70	a-b		SOB_B	a	70	a-b		SOB_B	a	76	a-b
			35		STA_B	b	92	22					28
			a-d					b-f		STA_B	b	104	a-e
			208		SEN_B	f	160	68			e	127	51
STA_B	b	105	o-d					a-h			f	159
			246			g	171	114					o-i
			d-b					o-h					206
SEN_B	d	278	173			h	184	148		END_B	i	225	b-i
						i	191	o-i		SEN_B			121
SON_C	o	32	o-c					155
			143							SON_C	o	19	o-h
			a-c		SON_C	o	36	o-c					167
			105					57		STA_C	f’	163
SOB_C	c	175	c-e		SOB_C	c	93	c-d			g	181	a-i
			153			d	103	10			h	186	149
STA_C	e	328	c-f		STA_C	e	110	d-e
	f	348	173					7		SON_D	c	34
	g	358	c-g			g	171			STA_D	f’	163
			183		SEN_C	h	184	e-h			g	181
	h	410	c-i			i	191	74
SEN_C	i	441	266					o-i
			o-i					155
			409
SON_D	o	32	o-i		SON_D	o	36	o-e
	h	410	409			e	110	74
SEN_D	i	441				h	184

give a detailed explanation of the arrival times and outstanding features (OF) of the sections (SON, SBO, STA, and SEN) of the spectrograms, which are consistent with the images of the GEODAR [49]. Here, we highlight the most distinctive features of the signals of the three avalanche types. Note the differences in the duration of the signals as well as in the shape of each recording site and flow type. In the time series, note the shorter duration (150 s) of the powder-snow avalanche signals (POW, Figure 3) with respect the durations of the transitional avalanche (425 s) (TRANS, Figure 4) and of the WET avalanche (220 s) (Figure 5). The spindle-cigar shape in the time series referred to in the literature as a typical pattern of seismic moving sources, i.e., snow avalanches, debris flows, lahars, etc., [8,13,15,40,63,64] is more prominent in seismograms at sites C and D, located at some distance from the start of the avalanche, whereas wave packets of higher amplitudes are observed in the seismograms at B. Note that the sections of the spectrograms are different depending on the avalanche type and site. POW (Figure 3 and Figure A3e) and TRANS (Figure 4 and Figure A3d) avalanches show a clear SON section at B and C in contrast to the WET avalanche (Figure 5 and Figure A3f) in which the SON section is not clear since it shows peaks of energy associated with impacts. No specific SON section is observed at any of the three avalanches at D.

The SBO section in the POW avalanche is short at B (22 s) and at C (10 s), and no section is observed at D. The duration of this section in the TRANS avalanche at B is longer (35 s) than the section in the POW and the WET (28 s) avalanches. At C, in contrast to the POW avalanche, the SBO section in the TRANS avalanche is very long (173 s) and energetic. The SBO section in the WET avalanche is not very well observed at C and it is similar to the SBO section in the TRANS avalanche at D. No SBO section is observed at any of the three avalanches at D. The STA and SEN sections are longer in the WET (121 s) and TRANS avalanches (123 s) than in the POW avalanche (68 s) at B. This section is almost absent in the POW and the WET avalanches at C. In the TRANS avalanche, it disappears at the expense of the SBO section at C. The spectrogram in the TRANS avalanche at D is similar to that of the WET avalanche at C, showing mainly energy of low frequency content. Note, also, the energetic phase at the end of the signals observed in the three avalanches that corresponds to the stopping phase [22].

We obtained an average of the avalanche front speed between the two recording sites (B and C) by considering the time taken by the avalanche to travel from one site to the other (difference in the arrival time of the SBO sections) and the distance between the two sites (Table A1). This is possible if the avalanches reach the sensors. However, this condition is not always fulfilled.

To resolve this problem, we developed a method to quantify the SON section of the FS spectrograms in order to derive the speed of the avalanche front. We took advantage of the close link between the shape displayed by the SON sections of the spectrograms and the characteristics of the avalanche. Our analysis of hundreds of seismograms of diverse snow avalanches at the different sites provides robust evidence of this behavior.

5. Methodology and Quantification

5.1. Quantification of the Spectrogram SON Section

The increase in amplitudes of the frequencies in time in a quasi-exponential shape of the SON section of the spectrograms can be expressed as Equation (1)

F (t, f) = K (t, f) e^{β (f, t) t}

(1)

where F (with units of Hz) is a function that limits the amplitude of the spectrogram, which is a function of time, t(s), and frequency, f(s⁻¹). K(s⁻¹) and β(s⁻¹) are functions that describe the shape of the spectrogram. To obtain Equation (1), following [65] we first deduced the equation of the decrease in the signal amplitudes A (t’, f) (m/s) at the sensor at a distance r(t’), in m, from the source (avalanche)

A (t’, f) = A_s (t’, f)/ (√ (2π h r(t’))) e ^{− α(f) r (t’)}

(2)

where

α (f) = π f/(Q(f)c(f))

(3)

Equation (2) considers the energy flux of the surface seismic waves through a cylindrical surface of radius r(t’), varying in time (t’) and height h (m) [16]. A_s (t’, f) (m²/s) is the amplitude at the source. t’(s) is time from the source. α (f) (m⁻¹) is the intrinsic attenuation factor and the other term corresponds to the geometrical spreading. The ground seismic characteristics are involved in the ground quality factor, Q (f,), and in c(f) (m/s), the phase velocity of the surface waves. Equation (2) can be expressed as an exponential decreasing function

A (t’, f) = K (t’, f) e^{−β(f) t’}

(4)

where r(t’) = S (t’) t’, S(t’) being the speed of the avalanche, in m/s,

β (f, t’) = α(f) S (t’)

(5)

and

K(t’, f) = As (t’, f))/√(2π h r(t’))

(6)

When a change of the time/distance reference is applied, Equation (4) becomes

A(t, f) = K(t, f) e^{β(f, t) t}

(7)

which is the same as Equation (1). t accounts for the change of reference in time (t = −t’). In this case, the origin of time and distances is the sensor instead of the source. Distances decrease as the avalanche approaches the sensor. In Equation (7) the amplitudes increase in time as in the spectrograms. Figure S3 in the supplementary material shows, as an example, a synthetic spectrogram using Equation (7). This figure reproduces the shape of the spectrograms.

To obtain parameters K and β of Equation (7), the Hough Transform (HT) [53] was applied to the spectrograms considered as an image. To this end, the amplitudes of the spectrogram (Figure 6a) were band-pass filtered in amplitudes (for all the times and the complete set of frequencies), selecting the range that was interesting to highlight (Figure 6b). The filtered image of the spectrogram was binarized to distinguish the arrival of the more energetic frequencies (Figure 6c). The binarized image was convolved with an edge detector mask [66] to highlight the lines of interest after a change of coordinates between the originals of the spectrogram (f, t) and those of the binary image (y, x). The HT was then applied to the final binary image (black and white) (Figure 6d), where the straight lines are defined by the limit of the white, the black corresponding to the background. Different fits defined by parameters Ai (slope) and Bi (y-intercept) were obtained through the application of the HT (Figure 6e) to each spectrogram. The averages of all the possible fits of Ai and Bi yield the slope, a, and the y-intercept, b, of

y(x) = ax + b

(8)

which allows us, after a change of variable, to obtain the functional relationship between time and frequency that limits threshold of amplitudes selected in the spectrogram

F(t) = a t + b

(9)

A relation between Equations (7) and (9) is necessary, i.e., between the coefficients K and β and a and b. However, an analysis of the dependence of the parameters in frequency is previously required. Note that the HT method uses a binarization process and that for each time HT discriminates the amplitudes to determine the shape of the SON section of the spectrogram by a threshold value. The dependence on frequency of β (f, t) (Equation (5)) is due to α(f), and not to r(t), which is not frequency dependent. The effect of the variation in distance r (t) leads to a variation in the amplitude values of the spectrogram in the same ratio for each time. As regards K (t, f) (Equation (6)), the same effect causes the dependence in r(t). Note that the dependence on f is through the source (As (t, f)). As a result, the amplitude ratio of the values for the threshold for each time (distance) does not change and does not affect the shape of the spectrogram detected by the HT method. Moreover, in Equation (2), an increase in amplitudes owing to the incorporation of snow as the avalanche approaches the sensor was neglected. The reason for this is that the increase in mass would affect the amplitudes at the same ratio for all frequencies at each time.

Following the above analysis, we may assume that K in Equation (7) is constant termed K’ in order to compare Equation (7) with Equation (9) yielding

A (t, f) = K’ e^{β(f, t)t}

(10)

To obtain the link between Equations (9) and (10), we apply the function ln to Equation (10) to its linearization

ln (A (t, f)) = ln K’ + β (f, t) t

(11)

Given that we are considering a threshold in the spectrogram amplitudes (upper limit) we impose Equation (12) on the implicit function Equation (11)

ln (A (t, f)) = Const.

(12)

As a result, Equation (13) is obtained which is a function that fulfills a fixed amplitude in the spectrogram

F’ (t) = ln K’ + β t

(13)

yielding to

F (t) =K’ e^βt

(14)

The function of the frequency in time F(t) corresponds to the upper limit of the amplitude (Equation (14)) considered in the spectrogram.

The parallelism of the linear Equations (9) and (13) shows that the y-intercept term of the linear equation, b, has the role of ln K’ and the slope, a, has the role of β. Note that the spectrogram image is represented as a logarithmic function of the amplitudes. The identification of the left sides of Equations (9) and (13) is therefore correct. This comparison leads to β being frequency independent. This result is not inconsistent since the value of the slope, a, is obtained from the average of the different slopes of the fits to the spectrogram image.

The above process was applied to the SON section of the spectrogram of the final seismogram (FS) of the different avalanches recorded at sites B and C (Table 1). The extracted parameters ${\mathit{\kappa }}_{\mathit{i}}^{\prime }$ and βi (Equation (14)) for all the avalanches and sites are shown in

Table 2. Values for the coefficients κ’ and β with the standard error of β, σ_β, for the sites B and C obtained after the application our method. Gray areas correspond to the avalanches that do not fulfill the required conditions. e.g., Release area in CB1.

		Site B			Site C
Type	K’B(s−1)	β B(s−1)	σβB (s−1)	K’C(s−1)	β C(s−1)	σβC(s−1)
Av1 POW-L	1.04	0.19	0.68
Av2 TRANS-WET	1.33	0.11	0.48
Av3 WET-M	14.44	0.07	0.11	411.58	0.05	0.13
Av4 TRANS- L	3.10	0.10	0.07	12.18	0.05	0.03
Av5 POW-L	1.09	0.14	0.12	1.26	0.09	0.18
Av6 WET-M	3.32	0.05	0.02	2.72	0.06	0.02
Av7 WET-M	46.53	0.06	0.09	4.48	0.03	0.01
Av8 POW-L	2.01	0.31	0.10	3.32	0.09	0.07
Av9 POW-M	1.20	0.22	0.26	24.78	0.10	0.03
Av10 POW-S	66171.16	0.08	0.08	1.02	0.13	0.05
Av11 TRANS-L	403.43	0.06	0.03	1.01	0.12	0.90
Av12 WET-L	1.22	0.11	0.08	1211.97	0.07	0.09

.

Figure 7 and Figure 8 show the curves for Equation (14) of the SON section of the spectrogram of the avalanches for sites B and C, with the values in Table 2 to facilitate comprehension of these results. The curves are identified as in Table 1. Capital letters B or C are added to the avalanche name in accordance with the site. In these graphs, the errors are not considered to facilitate visualization. The curves were shifted in time to compare their curvatures. This shift corrects the different origins of time of the spectrograms in the image processing and does not affect the curvature.

5.1.1. Curves at B

Note that the curves are bunched, according to their avalanche type (Figure 7). The curves of the POW avalanches (in light blue) are steeper, showing a sharp increase in frequency greater than that of the TRANS (in light purple) and especially the WET (in light brown) avalanches. Note that the behavior of Av10B POW-M differs from that of the other POW avalanches, being more similar to that of WET avalanches. This behavior will be considered in the discussion. Curves for WET avalanches present a gentler increase in frequency with time. A comment regarding avalanches Av11B TRANS-L and Av12B WET-L defined as transitional and wet, respectively, in the classification in [19] is opportune. The behavior of Av11B TRANS-L is closer to that of the wet avalanches, whereas Av12B WET-L has a shape similar to that of the transitional avalanche Av4B TRANS-L. In the discussion this discrepancy in behavior with respect to their classification will be considered.

5.1.2. Curves at C

At this location, the POW curves (in dark blue) are bunched including Av10C POW-M (Figure 8). The curvatures of Av11C TRANS-L and Av12C WET-L are similar to those of the POW avalanches, differing from that of the transitional avalanche (Av4C TRANS-L, in dark purple), which has a gentler curvature. The curves of the WET avalanches (in dark brown) are gentler than those of the other avalanche types. As for Avalanche Av10 POW-M, while the behavior at B differs from that of the POW avalanches, its behavior coincides at C. This behavior will be considered in the discussion. Although the WET avalanches do not reach sensor C (Table 1), we regard the values obtained as correct because the coefficients correspond to the SON section when the avalanche does not reach the sensor.

5.1.3. Evolution of Curves from B to C

We now focus our attention on the evolution of the avalanches from sites B to C. Since the figure of all the curves at B and C plotted jointly is confusing, the curves are presented according to the avalanche type. Figure 9 corresponds to powder avalanches. Note that the curves are gentler at C than at B, maintaining the two groups apart. Figure 10 shows the curves of the transitional avalanches recorded at B and C. Curves of avalanche Av12 WET-L are included in the plot because the curvatures at sites B and C are similar to those of the TRANS avalanches. Figure 11 corresponds to the curves of wet avalanches recorded at B and C. In general, the curvature of the curves at B and C are fairly similar.

5.2. Avalanche Speed Estimation

Equation (5) allows us to estimate the speed of the front of the avalanche (S) using the β(s⁻¹) values obtained for each spectrogram. However, the parameter α(m−1) depends on the seismic characteristics of the ground (Equation (3)). In the absence of measured values of α, there is an extensive debate about how to fix this value. Different authors have assumed that α (m⁻¹) is either constant or dependent on the frequency, e.g., [16,67,68,69]. This selection is crucial for the calculation of the energy released by a mass in movement. The choice of α has also repercussions in the calculation of the speed of the avalanche (Equation (5)). In the specific case of VdlS, no values of α (m⁻¹) are known, although attenuation parameters κ (s⁻¹) (spectral decay factor) in the Swiss territory have been calculated [70], even if they are not the same. To resolve the absence of α values, a procedure was adopted to determine them in order to estimate the avalanche speed (S). Figure 12 shows a diagram of the procedure. To fix α from the calculated βi (from HT), we took advantage of the estimates of the avalanche velocity (V_GDRi), obtained from the GEODAR system [30,48,49]. This velocity is defined as the ground-parallel velocity down the talweg in [30] and it is the projection of the velocity along the path similar to our speeds (S). We maintain below this difference (speed S (obtained from seismic) and velocity V_GDR (derived from GEODAR)) in order to facilitate comprehension. Note that both values are not derived from a direct measurement. We used, when available, the avalanche velocity values (V_GDRi). In fact, we focused our attention on the values corresponding to the few seconds before the avalanche reached the sensor. In other words, we considered the velocities (V_GDR) corresponding to the SON section of the avalanche. We assumed an error of 10% in our visual determination from the GEODAR curves. The αi values were calculated from the βi values obtained from HT for each avalanche and for each site (B and C) using Equation (5) (Figure 12). To estimate the speed of the avalanches, we took the average, α_a, obtained from the different αi for the two sites (

Table 3. VB_GDR and VC_GDR are the velocities derived from the GEODAR images for the different avalanches at sites B and C. An error estimation of 10% is assumed. Values are absent because of the lack of information in the GEODAR. αa is the averaged coefficient for sites B and C considering the αi values of all the available avalanches, and αe is the averaged coefficient for sites B and C excluding the powder avalanches. SBap and SBa and SCap and SCa are the avalanche speeds calculated from Equation (5) using αa and αe, respectively, for sites B and C and βi in Table 2.

AVALANCHE TYPE	VBGDR (m/s)	SBap (m/s)	SBa (m/s)	VCGDR (m/s)	SCap (m/s)	SCa (m/s)	αi (B) (m−1)	αi (C) (m−1)
Av1 POW-L		54 ± 194	75 ± 273
Av2 TRANS-		31 ± 136	44 ± 192
Av3 WET-M	20 ± 2	20 ± 32	28 ± 47		5 ± 17	3 ± 12	0.004 ± 0.006
Av4 TRANS- L	38 ± 4	28 ± 21	40 ± 35	4 ± 0.4	5 ± 11	3 ± 9	0.003 ± 0.002	0.015 ± 0.009
Av5 POW- M	31 ± 3	40 ± 37	55 ± 58	26 ± 3	9 ± 26	6 ± 19	0.005 ± 0.004	0.003 ± 0.007
Av6 WET-M	26 ± 3	13 ± 7	18 ± 13		6 ± 13	4 ± 10	0.002 ± 0.001
Av7 WET-M	20 ± 2	16 ± 27	22 ± 39		3 ± 7	2 ± 5	0.003 ± 0.005
Av8 POW-L	41 ± 4	88 ± 42	123 ± 84	26 ± 3	9 ± 20	6 ± 16	0.008 ± 0.003	0.003 ± 0.003
Av9 POW-M		62 ± 77	87 ± 115		10 ± 22	7 ± 17
Av10 POW-S		23 ± 24	32 ± 36	13 ± 1	13 ± 28	9 ± 22		0.01 ± 0.004
Av11 TRANS-L	31 ± 3	17 ± 11	24 ± 20	8 ± 1	12 ± 91	8 ± 63	0.002 ± 0.001	0.01 ± 0.113
Av12 W/TRANS-L		31 ± 25	44 ± 41	5 ± 1	7 ± 17	5 ± 13		0.016 ± 0.019
αa							0.004 ± 0.001	0.010 ± 0.023
αe							0.0025 ± 0.0015	0.015 ± 0.038

). We obtained different values α_a for sites B (0.004 ± 0.001 m⁻¹) and C (0.010 ± 0.023 m⁻¹), the values being higher at C than at B (Table 3). In this calculation, avalanches.

Av11C TRANS-L and Av12C WET-L were ignored because they do not meet our requirements. We also calculated the median, αm. Given that the difference was not significant when considering the errors, we maintained α_a for the calculations. The avalanche speeds S were obtained using α_a and Equation (5) for the βi values (Figure 12). The speed values for both sites (SBa and SCa) and the values derived from V_GDR, when available, are given in Table 3. Some velocity (V_GDR) values at C are absent because GEODAR signal stops before reaching the range where the seismic sensor is located.

Likewise, some values are absent at B since the GEODAR signal does not provide images of the start of the avalanche [53]. This is a limitation of the triggering system. As mentioned above, our seismic recording system does not depend on the VdlS triggering system, thereby enabling us to obtain the entire avalanche signal.

6. Discussion

6.1. Avalanche Evolution

The spectrograms of the seismic signals (FS) of the three types of avalanche present differences along the path. In the POW avalanche (Figure 3), the arrival times of the wave packets in the signals at the three sites are closer together, whereas in the TRANS (Figure 4) and WET avalanches (Figure 5) the arrival times are spaced out. The SBO section of the TRANS avalanche is longer than that of the POW avalanche at C, although at B it presents a similar duration. This is not observed in the WET avalanche (Figure 5) in which the spectrogram at C is similar to that of the TRANS avalanche at D.

Moreover, each signal has several wave packets that arrive at different times (Figure 3, Figure 4 and Figure 5) depending on the velocity of the avalanche and on the internal surges or parts of the avalanche moving behind the avalanche front [19]. As mentioned earlier, a TRANS avalanche is a POW avalanche that evolves into a WET avalanche along the path and, in principle, TRANS avalanches have no specific rheology. The specific characteristics of the spectrograms allow us to distinguish between the different avalanche types given that the sections appear at different locations (and times) along the path (note the difference in the SBO section at C in both avalanches (TRANS and WET) and the similarity of the spectrograms at D (for TRANS) and C (for WET)). In contrast to the TRANS avalanche, the POW avalanche does not incorporate snow that produces energetic seismic signals from B to C. The effect of the incorporation of snow from B to C in the WET avalanche is insignificant in the spectrogram, and the flux (reflected by the spectrogram) at C is similar to that at the end of the TRANS avalanche at D. The examples presented show that the entire spectrogram of the seismic signals reflects the avalanche type and its evolution characteristics along the path, always in accordance with the perspective of the seismic signal analysis.

6.2. Averaged Curves

Although the majority of the curves constructed with the κ’ and β values obtained when applying our method agree with their avalanche type, the curves of Av10B POW-M, Av11B TRANS-L, and Av12B WET-L do not follow the general trend. A possible explanation for the behavior of Av10B at B could be that this avalanche, as reported in [19], was not released exactly in CB1 (in Pra-CB1, Figure 1). Given that it descended Gully 1 (Figure 1) as did the other avalanches reaching C, it presents the same behavior as the other POW avalanches at C. The Av11 TRANS-L and Av12 WET-L avalanches at C display a behavior akin to that of the POW avalanches (Figure 8), whereas at B the former acts as a WET avalanche while the latter behaves as a TRANS avalanche (Figure 7). The different starting zone of these avalanches (the release area was CB1 + CB2, Table 1) could give rise to a different state of the avalanche flow regime, resulting in a discrepancy in the behavior of the curves with respect to the other curves.

To deepen our understanding of the avalanche evolution, we calculated the mean values β_m and K’_m of the parameters according to the avalanche type and site in Table 2 (

Table 4. Mean values of the κ’ and β parameters and standard errors according to the avalanche type and site, and ratio between the β values (Table 2). E= ϐmC/ ϐmB. Gray: the values correspond to only one TRANS avalanche.

Type	K’mB (s−1)	σK’mB (s−1)	ϐmB (s−1)	σϐmB (s−1)	K’mC (s−1)	σK’Mc (s−1)	ϐmC (s−1)	σϐmC (s−1)	E	σ
POW	1.34	0.79	0.22	0.47	9.79	18.42	0.09	0.11	0.43	1.45
TRANS	2.21	1.25	0.11	0.29	12.18		0.05	0.03	0.49	1.60
WET	21.40	31.73	0.06	0.07	139.59	333.12	0.05	0.09	0.83	2.61

). We did not consider Avalanches Av10 POW-M and Av11 TRANS-L and Av12 WET-L so as not to alter the results because they do not meet our requirements. Although the errors are considerable with respect to these values, information on the evolution can be extracted from them. The average value of the exponent β_m and K’_m differ according to the avalanche type. For cavern B, situated upslope, the value of β_mB for the POW avalanches is higher than that for the TRANS avalanches, which is in turn higher than that for the WET avalanches. This order is inverse for K’_mB. As for site C, situated 920 m downslope, the β_mC value for the POW avalanches exceeds those of the TRANS avalanches and WET avalanches.

The behavior of K’_mC values is similar to that of K’_mB values. Given that there is only one valid transitional avalanche at C, it is meaningless to draw conclusions on transitional avalanches from Table 4. Conclusions similar to those of the individual avalanches may be drawn for the three avalanche types and their evolution from B to C. The ratio between the exponents defined as index E= β_mC/β_mB (Table 4) provide us with an index of the evolution of the avalanches between the two sites. A value of E close to 1 indicates an absence of evolution of the avalanches from B to C. Thus, index E indicates a higher evolution from B to C for the SON section of the powder avalanches than for the wet avalanches. Figure 13 displays the curves for the averaged values at B (Figure 13a) and C (Figure 13b) and the evolution from B to C for POW (Figure 13c), TRANS and WET (Figure 13d) avalanches in order to facilitate visualization. Figure 13d shows that the evolution from B to C is more marked in TRANS avalanches than in WET avalanches. The curve of the TRANS avalanche is close to that of the POW avalanches (Figure 13a) at B, whereas at C the curve is close to that of the WET avalanches (Figure 13b). Note that although TRANS and WET avalanches at C show a similar behavior in the SON section, the differences in the spectrograms (mainly in the SBO section (Figure 4 and Figure 5)) indicate differences in the flow regime.

6.3. Front Avalanche Speed Estimation

Table 3 shows the front avalanche speeds for both sites (SB_a and SC_a) using α_a and the values deduced from V_GDR. Note that although the errors are large, the speed values deduced from the HT are, in general, close to the values of the avalanche velocities (V_GDR) derived from GEODAR data. There is, however, a discrepancy between the values of the velocity deduced from our calculations and the speeds derived from GEODAR measurements for POW avalanches. At B, the speeds are generally higher than the velocities deduced from GEODAR, whereas at C this is the inverse. A comparative study of the spectrograms and GEODAR images concerning the difference of the arrival times of the energy reveals that the GEODAR images and the seismic signals do not reflect the same phenomenon [31]. This agrees with the assertion that GEODAR only detects the denser flow structures below the diluted powder cloud [30], and also with the fact that the entrainment of the powder part of the avalanche generates seismic signals [19,57]. Therefore, the use of the two complementary measurement systems is useful. However, this needs to be corroborated in future studies. Bearing this in mind, a second more realistic attempt was made to calculate the avalanche speeds. The elimination of the αi values of the POW avalanches to calculate an averaged value, α_e, of this seismic ground property is reasonable if GEODAR and seismic signals do not reflect the same phenomenon in POW avalanches. Thereafter, we obtained the averaged value α_e using the procedure indicated in Figure 12, excluding POW avalanches, and Av11C TRANS-L and Av12C WET-L avalanches. The values obtained were 0.0025 ± 0.0015 m⁻¹ for site B and 0.015 ± 0.038 m⁻¹ for site C (Table 3). Although it seems that the procedure of obtaining α (Figure 12) is circular, this is the only one we are able to follow in the absence of real seismic characteristics of the ground. The α_e values for sites B and C (Table 3) are of the order of the values derived from other studies, e.g., [16,66,69]. These values yield a quality factor Q (f) (Equation (3)) for the same frequency and phase velocity that is higher at B than, downslope, at C. This result is plausible since the seismic characteristics of the two sites are different (geological effects and snow-cover e.g., [25,57]) with the result that this difference is reflected in the calculated α_a.

In Figure 14 and Figure 15, the speed values SB_ap and SC_ap (Table 3) for the different avalanche types are plotted jointly with the velocities derived from GEODAR (V_GDR) when available for sites B and C. Using this α_e, the coincidence of the speed values with the velocities derived from GEODAR is higher for TRANS and WET avalanches. The agreement of the values of the TRANS avalanches for the two types of measurements is satisfactory at C. The speed values obtained for POW avalanches are plotted in the figures, but are not compared with the V_GDR values for the reason explained above. In Figure 16, the speed values at B and C are plotted for TRANS and WET avalanches, with their averaged values. Although the standard deviations calculated with the errors in Table 3 are very high, the averaged values yield information that merits a comment. The averaged value of the front speed for TRANS avalanches is 42 ± 3 m/s (deviation corresponds to the averaged value) at B. Note that the front speed value of the Av12 WET-L at B is similar to that of the TRANS avalanches. The averaged value of the front speed for WET avalanches is 23 ± 7 m/s at B. This value is similar to that obtained for the Av11 TRANS-L. The speed values of the aforementioned avalanches were calculated using the α_e value (equation 5) with the result that the speed values of these avalanches are consistent with the behavior of the corresponding curves. At C, the speed values for the two types of avalanche are similar within the range of (3–8) m/s with an average speed value of 4 ± 4 m/s. The speed values at B are higher and more dispersed than those at C. In both types of avalanche, a decrease in the speed is observed, the decrease being greater for TRANS avalanches than for WET avalanches. At site C, no values of GEODAR velocities for WET avalanches exist because the avalanches stopped before reaching the site as indicated by the images [49]. Although WET avalanches 6 and 7 (Figure 15) did not reach sensor C (Table 1), we maintained the speed values because, as mentioned above, the SON section corresponds to waves travelling before the avalanche reaches the sensor.

7. Conclusions

The division into sections (SON, SOB, and STA-SEN) of the spectrograms of the seismic signals generated by snow avalanches is useful to obtain information about avalanche behavior. These sections differ in duration, amplitude, and frequency content according to the avalanche type, and provide detailed information of their evolution along a path.

The application of the Hough Transform to the spectrograms when considered as an image allows us to parameterize the shape of the SON section and deduce the avalanche front speeds. Anaelastic attenuation coefficients, α(m⁻¹), were obtained for two sites at VdlS. These coefficients characterize these sites seismically and can be used in further studies in the absence of values obtained from specific experiments.