# Ensemble simulation for Ubinas (2019)

## Fix column height

### General configuration
- Start time: 19 July 2019, 07:00 UTC
- End time: 21 July 2019, 00:00 UTC
- Meteo data: ERA-5 (model levels)
- Domain size: 600x400x60
- Species: Tephra

### Source:
- Column height: 4 km avl (fixed)
- Type: TOP-HAT
- Start time: 19 July 2019, 07:00 UTC
- End time: 19 July 2019, 18:00 UTC
- Start time: 19 July 2019, 07:00 UTC
- MFR: ESTIMATE-DEGRUYTER
- Thickness: 1 km
- Maximum particle size: 12 $\mu m$
- Granulometry file:

  0.011049   2300.0   0.900  0.296800505E-01   1   1      tephra     fine_ash-06  T
  0.007812   2300.0   0.900  0.163989677E-01   1   1      tephra     fine_ash-07  T
  0.005524   2300.0   0.900  0.806730489E-02   1   1      tephra     fine_ash-08  T
  0.003906   2300.0   0.900  0.353518749E-02   1   1      tephra     fine_ash-09  T
  0.002762   2300.0   0.900  0.137613813E-02   1   1      tephra     fine_ash-10  T
  0.001953   2300.0   0.900  0.679610383E-03   1   1      tephra     fine_ash-11  T

### Ensemble:
- Column height perturbation:
- Perturbation type: Relative (35%)
- PDF: Uniform
- Wind perturbation:
- Perturbation type: Relative (20%)
- PDF: Uniform

### Results: Probability

*Snapshot:*

*Video:*
<iframe src="https://drive.google.com/file/d/1InvdAw0dTiRrbbkyQCHUXk4EfAECehKp/preview" width="640" height="480"></iframe>

### Results: Statistical estimators for column mass

Possible estimators:
1. Ensemble median ($x_1$)
1. Ensemble mean ($x_2$)
1. Minimal distance ($x_3$)

Typically, we have a very positively skewed distribution. In the next figure, very high values of the skewness are found in most of cases, except when the ensemble mean-to-spread ratio is greater than 1. In consequence, $x_1 \ll x_2$ for most of grid elements.


For the minimal distance estimator ($x_3$), we use the solution corresponding to the ensemble member which minimizes the distance:
$$
\min_i \sqrt{(x_i - \overline x)^2 + (y_i - \overline y)^2}
$$
where $x_i,y_i$ are the coordinates of the center of mass for the ensemble member $i$, and $\overline x,\overline y$ are the averaged coordinates of the center of mass.

Conclusions:
- Ensemble median: significantly underestimates column mass
- Ensemble mean: too diffusive
- Minimal distance: It has the advantage of being a physical solution. Members with large MFR/Column height do not have a greater weight (as $x_1$ and $x_2$).

*Snapshot:*
The paths of the center of mass are represented by red lines.


*Video:*
<iframe src="https://drive.google.com/file/d/1gp2aWpbYKYYz-OJAnjaJzI3ZSvSyhMn8/preview" width="640" height="480"></iframe>

## Variable column height

Column height timeseries:

### Ensemble:
- Column height perturbation:
- Perturbation type: Relative (35%)
- PDF: Uniform
- Wind perturbation:
- Perturbation type: Relative (20%)
- PDF: Uniform

### Results: Probability

*Snapshot:*

*Video:*
<iframe src="https://drive.google.com/file/d/11OsA6h8-GnoHPxTVRflm7g7uvJqkCJ-c/preview" width="640" height="480"></iframe>

### Results: Statistical estimators for column mass

*Snapshot:*

*Video:*
<iframe src="https://drive.google.com/file/d/1RBwPETYkkIvZCa0a9OeyNLvfjezlcBSa/preview" width="640" height="480"></iframe>