Model eruptionTo test models of eruptions against measured data, a Doppler radar simula-tor was developed. It uses the reflection coefficients from Mie scattering cal-culations to generate synthetic measurements. For simplicity, a basic plume model consisting of a vertical upwind field and a horizontal sidewind field is assumed. The upwind velocity decreases exponentially with the square of the distance from the plume center. The horizontal windfield is held constant.

For the simulation runs, the ash is divided into 10 000 packages, each of them having a fixed particle radius. The number of particles in each package is determined by the Weibull distribution. The volume of a package then fol-lows as the product of particle number and volume. The starting velocities are assigned randomly within prescribed limits.

Conclusions• Doppler radar measurements have to be analyzed with great care

and – if possible – using wind field data.• Radius estimates from particle settling velocities are prone to er-

rors because of sidewind.• The mean radius of the ash particles can be estimated using m ea-

surements at multiple frequencies.

To Do• Repeat the scattering calculations for different particle shapes• Combine the radar model with a realistic plume model, e.g. ATHAM

(PhD project of Lea Scharff)• Test the Idea in reality (ouch!)

Radius from Mie theoryThe Mie theory describes the electromagnetic field caused by a plane wave interacting with a homogeneous, spherical particle at arbitrary ratios of wave-length and particle radius.

The field is decomposed into an incoming plane wave, a field inside the particle and a scattered field outside. Each of the three fields is expanded in spherical vector functions. These functions are derived from the spherical har-monics and form a complete set of orthogonal functions. At the surface of the sphere, continuity of the field components parallel to the surface is required. This property is used in conjuncition with the orthogonality to determine the expansion coefficients.

The reflected fraction of the incoming wave strongly depends on the ratio between particle radius and wavelength. Thus, the same particle reflects dif-ferent fractions of the incoming energy at different wavelengths. This effect can be used to determine the radius of the particle by measuring the reflected energy at different wavelengths and fitting the theoretical curve to the mea-surements.

Radius from terminal velocityThe classical approach to determine the radius of falling particles is to assume the particles are falling out vertically at their terminal settling velocity. Using this assumption one transforms the measured in-beam-velocity into a vertical velocity. From this vertical velocity, one determines the particle radius from the drag laws.

Volcanic mass flux rates from Doppler radar observationsFlorian Ziemen1 (florian.ziemen@zmaw.de), Lea Scharff 2 (lea.scharff@zmaw.de), Matthias Hort 2 (matthias.hort@zmaw.de)

1 now at Max Planck Institute for Meteorology, Hamburg; 2 University of Hamburg, Institute of Geophysics

The whole storyVolcanic eruptions pose a major thread to the local infrastructure and population. Sometimes volcanic ash clouds require re-routing of air-traffic; on the ground, the ash threatens lives, and can destroy houses as well as crops and infrastructure. Thus, the decision makers need es-timates of the amount of ash released during an eruption as fast as pos-sible. However, estimating this mass flux is difficult. In recent years it has been proposed that the mass flux could be derived from Doppler radar observations of volcanic ash clouds. Here we present a study ex-ploring the potential of this method.

To obtain the mass flux from radar measurements, one needs to know the particle size distribution (PSD) of the erupted ash. Studies of vari-ous historic eruptions suggest that the PSD can be described using a Weibull distribution. The Weibull distribution is characterized by three parameters, a shape factor (steepness) and its mean value (particle ra-dius) and a linear scaling of the amplitude. For the shape factor, val-ues between one and two yield good fits of measured data. The mean particle size and amplitude have to be determined for each individual eruption. The traditional approach to estimate the mean particle size using the particle’s terminal settling velocities is biased by the wind and thus is unreliable. We present an approach to determine the mean particle size by using measurements at multiple carrier frequencies and backscatter efficiencies calculated using Mie theory. This method is independent of sidewind effects and thus could yield more reliable estimates of the amount of mass released in an eruption than the ap-proach using terminal velocities. We demonstrate the feasibility using synthetic measurements at five wavelengths between 6 and 96 mm.

f (r) = Akrk−1

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/m

The Weibull DistributionFor ash particles the Weibull distribution can be used as a PSD by defining the probability f of a particle having the radius r by

Here, A is the amplitude, k is the shape factor of the distribution that deter-mines the steepness and θ determines the mean radius of the distribution.

The MVR-4 Doppler radar system with an 1.2 m satellite dish.

A dataset from an eruption at Colima volcano.

The amount of ash required to generate the same echo intensity at different mean radii of a Weibull PSD varies strongly. Thus, the PSD needs to be known to infer the amount of ash from radar measurements. Calculated for a wavelength of λ = 12 mm and a Weibull shape factor of k=1.5.

An Eruption at Colima volcano. One can clearly see the ash falling out non-vertically.

The relationship between measured velocity and assumed particle settling velocity. Since the horizontal velocity component usually is unknown, it is assumed to be zero. This may lead to wrong estimates of the particle radius.

The relationship between radius and terminal settling velocity. At the end of the synthetic measurements displayed to the far right, the particles reach an in-beam velocity of 18 m/s. Because of the assumed sidewind of 10 m/s this corresponds to a real settling velocity of also 18 m/s (pure chance the numbers are the same). If one would instead compensate for the beam angle and assume a vertical velocity of 31 m/s, this would correspond to a mean radius of 18 mm instead of 6 mm.

In Rayleigh scattering this would lead to a underestimation of the mass flux by a factor of 27. In the case of the MVR4 with Mie scattering at r ~ 1 mm the error would be an overestimation of the mass flux by a factor of about eight.

The first four fundamental modes of the sphere displayed as pairs of electric modes (left) and magnetic modes (right).

Figure copied from Mie: Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der Physik, 1908.

The dependency of the backscattered energy on the relationship between the mean radius r of spherical particles and the wavelength λ.

Increasing or decreasing the number of particles shifts the curve up or down; increasing or decreasing the mean radius shifts the curve right or left. Since this is a log-log-plot the shape of the curve remains constant.

By measuring at different frequencies, the curve can be fitted and the mean particle diameter can be determined.

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The volume distribution and the backscattering cross section (color coded) of the ash packages used in the simulation. Because of the strong dependence of the scattering properties on the radius, the largest volume does not correspond to the largest backscattering cross section. Calculated for a wavelength of λ=12mm.

In the model run displayed here, the wind is blowing at 10 m/s towards the radar. The radar is installed at 2 km horizontal distance from the vent and 1.3 km below the vent (radar at (x = -2000 m, z = 0), vent at (x = 0, z = 1300 m)). The radar beam is aimed 100 m above the vent; the beam angle with the horizontal plane is 35 degrees.

In the measurements one can see the strong, impulsive start of the eruption (t ≤ 5s) when the particles cross the beam flying upwards. The particles are then sorted by grain size. The large particles fall out quickly while the small particles, which generate most of the echo early during the eruption, slowly cross the radar beam again towards the end of the measurement and form a long tail visible in the pictures on the far right. If the wind is not blowing directly towards the radar, the tail is weaker or non-existent.

Synthetic measurements at different wavelengths.

The evolution of a synthetic eruption at different timesteps.

vent

beam

λ = 96 mm

λ = 6 mm

λ = 12 mm

λ = 24 mm

λ = 48 mm

t = 1s

t = 5s

t = 9s

t = 13s

t = 17s

Literature• Barrick, D. E., 1973. FM/CW radar signals and digital processing,

Technical Report, NOAA Environment Research Laboratories• Gouhier, Mathieu and Donnadieu, Franck, 2008. Mass estimations

of ejecta from Strombolian explosions by inversion of Doppler radar measurements, Journal of Geophysical Research

AcknowledgementsWe would like to express our gratitute to the following people who made this work possible• Stefan Kinne for providing the Mie scattering code.• Gerhard Peters and Bernd Fischer for providing invaluable insight into the

deep interior of the radar system

• Mie, Gustav, 1908. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der Physik

• Stratton, J. A., 1941. Electromagnetic Theory• Weibull, Waloddi, 1951. A statistical distribution function of wide applica-

bility, Journal of Applied Mechanics

We furthermore greatfully aknowledge the stipened of the Foundation of German Business (Stiftung der Deutschen Wirtschaft) to Florian Ziemen during his time as a student.

• Ziemen, Florian, 2008. Theoretische Betrachtungen zu Dopplerradarmes-sungen an Vulkanen, Diploma thesis (available online)