# weatherTimeSeries

Simulate I/Q signals for weather returns using a Monte Carlo approach

*Since R2024a*

## Syntax

## Description

This function generates I/Q signals for monostatic polarimetric weather radar
systems. Each `weatherTimeSeries`

simulation produces an independent radar return
from a particular resolution volume, or cell, that is modeled as a complex stationary Gaussian
random process using Monte Carlo method (see Algorithms).

## Examples

## Input Arguments

## Output Arguments

## Algorithms

`weatherTimeSeries`

simulates I/Q signals using a numerical Monte Carlo
approach combined with a statistical model that is based on the expected behavior of radar
returns from weather phenomena [2]
[3].
`weatherTimeSeries`

calculates scattering amplitudes for every time step using
relevant scattering parameters specified as input arguments to generate a time series.

Statistical model assumptions for radar returns from weather phenomena:

I/Q signals follow Gaussian distribution

Signal amplitude follows Rayleigh distribution

Signal phase follows uniform distribution

The first step in the Monte Carlo approach is to specify scattering amplitudes and generate random numbers for particle position and motion. Scattering amplitudes for a given resolution volume are modeled as combinations from random multiple scattering centers. Assume that scatterers within the resolution volume have the same size and a canting angle of zero (which implies that off-diagonal elements of the backscattering matrices have a value of zero). Then we have

$${P}_{r}=N{\left|{V}_{hh}\right|}^{2},$$

where *P _{r}* is the received power for horizontal
polarization,

*N*is the number of scatterers in the resolution volume, and

*V*is the complex voltage for an individual scatterer for horizontal polarization. At least 20 scatterers (

_{hh}*N*≥

`20`

) are necessary to adequately model Gaussian random signals.The amplitude of *V _{hh}* can be obtained as

$$\left|{V}_{hh}\right|=\sqrt{{P}_{r}/N}\text{\hspace{0.17em}}.$$

Accordingly, the amplitude of the complex voltage for an individual scatterer for vertical polarization can be calculated as

$$\left|{V}_{vv}\right|=\left|{V}_{hh}\right|\text{\hspace{0.17em}}/\sqrt{{Z}_{dr}}\text{\hspace{0.17em}},$$

where *Z _{dr}* is the differential reflectivity on a
linear scale.

In addition, the correlation coefficient *ρ _{hv}* is
assumed to be reduced by a factor of $${\text{e}}^{{}^{-{\sigma}_{\delta}^{2}/2}}$$ due to the random scattering phase difference, where

*σ*is the standard deviation of random scattering phase difference, that is, $${\rho}_{hv}={\text{e}}^{{}^{-{\sigma}_{\delta}^{2}/2}}$$. Then we have

_{δ}$${\sigma}_{\delta}=\sqrt{-2\mathrm{ln}\left({\rho}_{hv}\right)}=\sqrt{{\sigma}^{2}{}_{\delta h}+{\sigma}^{2}{}_{\delta v}}\text{\hspace{0.17em}}.$$

For simplicity, it is assumed that $${\sigma}_{\delta h}={\sigma}_{\delta v}={\sigma}_{\delta}\text{\hspace{0.17em}}/\sqrt{2}$$, where *σ _{δh}* and

*σ*are standard deviations of the backscattering phase

_{δv}*δ*for horizontal polarization and the backscattering phase

_{h}*δ*for vertical polarization, respectively.

_{v}Next, the scattered wave field for each particle is calculated. The amplitudes of complex voltage for an individual scatterer can be expressed as

$${V}_{hh}=\left|{V}_{hh}\right|\cdot {\text{e}}^{-j{\delta}_{h}}\cdot {\text{e}}^{{}^{-j{\varphi}_{DP}}}$$

$${V}_{vv}=\left|{V}_{vv}\right|\cdot {\text{e}}^{-j{\delta}_{v}},$$

where $${\varphi}_{DP}$$ is the differential phase.

The final step is to calculate the total scattered wave fields. The total complex voltage of the resolution volume is

$${V}_{h}={\displaystyle {\sum}_{l=1}^{N}{V}_{hh}}\cdot {\text{e}}^{-j2{k}_{i}\cdot {r}_{l}}$$

$${V}_{v}={\displaystyle {\sum}_{l=1}^{N}{V}_{vv}}\cdot {\text{e}}^{-j2{k}_{i}\cdot {r}_{l}},$$

where $${k}_{i}$$ is the incident wave vector and $${r}_{l}$$ is the random position of each scatterer.

## References

[1] Doviak, R. and D. S. Zrnic.
*Doppler Radar and Weather Observations*, 2nd Ed. New York: Dover
Publications, 2006.

[2] Zhang, G. *Weather Radar
Polarimetry*. Boca Raton: CRC Press, 2016.

[3] Li, Z, et al. Polarimetric phased
array weather radar data quality evaluation through combined analysis, simulation, and
measurements. *IEEE Geosci. Remote Sens. Lett*., vol. 18, no. 6, pp.
1029–1033, Jun. 2021.

## Version History

**Introduced in R2024a**