# Attenuation¶

## Introduction¶

• Q is measuring the decay of wave amplitude during its propagation in the medium and is physically defined as the ratio of wave energy to the energy dissipated per cycle of oscillation.
• The actual dissipation mechanisms, as scattering, heat generation, frictional losses in the vibrating crystal lattice, etc., “causing Q” can be manifold and do not have to follow similar physics.
• Q is not very frequency-dependent over a large range of (lower) frequencies but observations are ambiguous at higher frequencies.
• Common choices are $$Q_s \sim 0.5 Q_p$$ and $$Q_s \sim 40-50V_s$$ (in km/s) (e.g. Olsen et al., 2009). $$Q_s = 100-200$$ for shallow sediments.
• In damaged fault zones $$Q_s$$ would be expected to be as low as on the order of 10 and potentially smear out slip rate on the fault without affecting rupture speeds.

## Implementation¶

• Implementing attenuation in SeisSol follows the ideas of a general Maxwell or Zener body.
• Each damping mechanism can be parametrized by its own relaxation frequency.
• It aims at resolving a frequency-independent Q with an adequate number of anelastic functions and relaxation frequencies to cover the frequency range under interest. Usually, 3 Maxwell bodies are enough for 5% error.

## Stability with Local time stepping¶

To ensure the stability of SeisSol using attenuation and local time-stepping (LTS), it seems necessary to limit the number of clusters of the local time stepping. Else, very large elements, rarely updated by the LTS, get unstable. This can be achieved by using the following patch, which in most cases should not affect the LTS speed-up.

--- a/src/Initializer/time_stepping/MultiRate.hpp
+++ b/src/Initializer/time_stepping/MultiRate.hpp
@@ -77,8 +77,12 @@ class seissol::initializers::time_stepping::MultiRate {
// first multi-rate interval
double l_lower = i_minimumTimeStepWidth;
double l_upper = i_multiRate*l_lower;
-
-      for( unsigned int l_id = 0; ; l_id++ ) {
+#if NUMBER_OF_QUANTITIES > 9
+      unsigned int l_id_max=6;
+#else
+      unsigned int l_id_max=std::numeric_limits<unsigned int>::max();
+#endif
+      for( unsigned int l_id = 0; l_id_max; l_id++ ) {
// the first cluster with an upper bound above the time step width is our
if( l_upper > i_timeStepWidth ) {
o_clusterTimeStepWidth = l_lower;
@@ -89,6 +93,11 @@ class seissol::initializers::time_stepping::MultiRate {
// update interval and continue searching
l_lower = l_upper;
l_upper = i_multiRate * l_lower;
+#if NUMBER_OF_QUANTITIES > 9
+        if(l_id==l_id_max-1) {
+          l_upper = std::numeric_limits<double>::max();
+          }
+#endif
}
}


l_id_max, the maximum cluster id, is here hardcoded to 6. This probably depends on the minimum mesh size and the order of accuracy used. The higher the order and the smaller the minimum mesh size, the larger l_id_max. As we did not investigate in detail the dependence of l_id_max with the simulation order and the minimum mesh size, we did not include the patch in the master branch yet.

## Compiling¶

In SeisSol build configuration script replace

equations = 'elastic'


by

equations = 'viscoelastic2'
numberOfMechanisms = 3


Note that the equations=’viscoelastic’ is operational but deprecated.

## Dispersion¶

The attenuation implementation implies dispersion in P and S wave velocities: that why we need to define a central frequency at which $$V_p/V_s$$ are exact. In addition, the effective Q values are not exactly equal to the desired Q, but are oscillating around those values. The variation of $$V_p$$, $$V_s$$, $$Q_p$$ and $$Q_s$$ with frequency can be visualized using ViscoelasticModComp.m.

## Parametrisation¶

Add Inside the parameter file of SeisSol, in the ‘&equations’ section (frequencies values to be adapted to the source frequency content):

FreqCentral=2.5
FreqRatio=100


The spatial variation of $$Q_s$$ and $$Q_p$$ are defined with easi in the MaterialFileName. Here is an example of easi file, in which $$Q_s$$ and $$Q_p$$ are directly related to the shear wave speed $$V_s$$:

!ASAGI
file: ../material/vmodel_500.nc
parameters: [rho, mu, lambda]
var: data
components: !FunctionMap
map:
rho:    return rho;
mu:     return mu;
lambda: return lambda;
Qs:     return 0.1 * sqrt(mu/rho);
Qp:     return 0.2 * sqrt(mu/rho);


## FreqCentral and FreqRatio¶

The relaxation frequencies are logarithmically equispaced, i.e.
$$log(w_{i+1})-log(w_i) =$$ constant.

In the parameter file, one has to give a frequency ratio of maximum to minimum frequency and a central frequency. For example, in the case of 3 mechanisms the following relations define the relaxation frequencies:

$$w_2 = FreqCentral$$
$$log(w_3)-log(w_2) = log(w_2) - log(w_1)$$
$$w_3 / w_1 = FreqRatio$$

Outside of the frequency band $$w_1 - w_3$$, Q goes to infinity, yielding elastic behavior.