Linear depolarization estimation

Calculation of volume and particle depolarization coefficient.

depolarization.calibration_constant_cross_parallel_profile(signal_cross_plus45, signal_cross_minus45, signal_parallel_plus45, signal_parallel_minus45, t_cross, t_parallel, r_cross, r_parallel)

Calculate the calibration constant in a lidar system that is able to detect the cross-to-parallel depolarization ratio.

Parameters

signal_cross_plus45: vector

The input vertical profile from the cross channel. Calibrator angle phi=45.

signal_parallel_plus45: vector

The input vertical profile from the total channel. Calibrator angle phi=45.

signal_cross_minus45: vector

The input vertical profile from the cross channel. Calibrator angle phi=-45.

signal_parallel_minus45: vector

The input vertical profile from the total channel. Calibrator angle phi=-45.

t_cross: float

Transmittance of cross component through transmitted path.

t_parallel: float

Transmittance of parallel component through transmitted path.

r_cross: float

Transmittance of cross component through reflected path.

r_parallel: float

Transmittance of parallel component through reflected path.

Returns

v_star_mean: float

Calibration constant’s mean value (vertical axis).

v_star_sem: float

Calibration constant’s standard error of the mean (vertical axis).

Notes

The calibration constant is calculated by the following formula:

\[V^* = \frac{[1 + \delta^V tan^2 (\phi)]T_p + [tan^2 (\phi) + \delta^V]T_s} {[1 + \delta^V tan^2 (\phi)]R_p + [tan^2 (\phi) + \delta^V]R_s} \cdot \delta^* (\phi)\]

References

Freudenthaler, V. et al. Depolarization ratio profiling at several wavelengths in pure Saharan dust during SAMUM 2006. Tellus, 61B, 165-179 (2008)

depolarization.calibration_constant_cross_total_profile(signal_cross_plus45, signal_cross_minus45, signal_total_plus45, signal_total_minus45, r_cross, r_total)

Calculate the calibration constant profile, in a lidar system that is able to detect the cross-to-total depolarization ratio.

Parameters

signal_cross_plus45: vector

The input vertical profile from the cross channel. Calibrator angle phi=45.

signal_total_plus45: array

The input vertical profile from the total channel. Calibrator angle phi=45.

signal_cross_minus45: vector

The input vertical profile from the cross channel. Calibrator angle phi=-45.

signal_total_minus45: vector

The input vertical profile from the total channel. Calibrator angle phi=-45.

r_cross: float

The transmission ratio of the cross channel (Rc).

r_total: float

The transmission ratio of the total channel (Rt).

Returns

c_profile: vector

The vertical profile of the calibration constant.

Notes

The calibration constant is calculated by the following formula:

\[C = \frac{1 + R_t}{1 + R_c} \cdot \sqrt{\delta'_{+45} \cdot \delta'_{-45}}\]

References

Engelmann, R. et al. The automated multiwavelength Raman polarization and water-vapor lidar Polly XT: the neXT generation. Atmos. Meas. Tech., 9, 1767-1784 (2016)

depolarization.calibration_constant_value(calibration_constant_profile, first_bin, bin_length, lower_limit, upper_limit)

Calculate the mean calibration constant and its standard error of the mean, from the calibration constant profile.

Parameters

c_profile: vector

The vertical profile of the calibration constant.

first_bin: integer

The first bin of the system.

bin_length: float

The length of each bin. (in meters)

lower_limit: float

The lower vertical limit for the calculation. (in meters)

upper_limit: float

The lower vertical limit for the calculation. (in meters)

Returns

c_mean: float

Calibration constant’s mean value (vertical axis).

c_sem: float

Calibration constant’s standard error of the mean (vertical axis).

depolarization.particle_depolarization(delta_m, delta_v, molecular_backscatter, particle_backscatter)

Calculate the linear particle depolarization ratio.

Parameters

delta_m: vector

The linear molecular depolarization ratio.

delta_v: vector

The linear volume depolarization ratio.

molecular_backscatter: vector

The molecular component of the total backscatter coefficient.

particle_backscatter: vector

The particle component of the total backscatter coefficient.

Returns

delta_p: vector

The linear particle depolarization ratio.

Notes

The linear particle depolarization ratio is calculated by the formula:

\[\delta^p = \frac{(1 + \delta^m)\delta^V \mathbf{R} - (1 + \delta^V)\delta^m} {(1 + \delta^m)\mathbf{R} - (1 + \delta^V)}\]

References

Freudenthaler, V. et al. Depolarization ratio profiling at several wavelengths in pure Saharan dust during SAMUM 2006. Tellus, 61B, 165-179 (2008)

depolarization.volume_depolarization_cross_parallel(signal_cross, signal_parallel, t_cross, t_parallel, r_cross, r_parallel, v_star)

Calculate the linear volume depolarization ratio in a lidar system that is able to detect the cross-to-parallel depolarization ratio. The calibration factor from the delta-90 calibration is being used.

Parameters

signal_cross: vector

The input vertical profile from the cross channel. Normal measurement (phi=0).

signal_parallel: vector

The input vertical profile from the parallel channel. Normal measurement (phi=0).

t_cross: float

Transmittance of cross component through transmitted path.

t_parallel: float

Transmittance of parallel component through transmitted path.

r_cross: float

Transmittance of cross component through reflected path.

r_parallel: float

Transmittance of parallel component through reflected path.

v_star: float

The calibration constant.

Returns

delta_v: vector

The linear volume depolarization.

Notes

The linear volume depolarization ratio is calculated by the formula:

\[\delta^V = \frac{\frac{\delta^*}{V^*}T_p - R_p}{R_s - \frac{\delta^*}{V^*}T_s}\]

References

Freudenthaler, V. et al. Depolarization ratio profiling at several wavelengths in pure Saharan dust during SAMUM 2006. Tellus, 61B, 165-179 (2008)

depolarization.volume_depolarization_cross_total(signal_cross, signal_total, r_cross, r_total, c)

Calculate the linear volume depolarization ratio in a lidar system that is able to detect the cross-to-total depolarization ratio. The calibration factor from the delta-90 calibration is being used.

Parameters

signal_cross: vector

The input vertical profile from the cross channel. Normal measurement (phi=0).

signal_total: vector

The input vertical profile from the total channel. Normal measurement (phi=0).

r_cross: float

The transmission ratio of the cross channel (Rc).

r_total: float

The transmission ratio of the total channel (Rt).

c: float

The calibration constant.

Returns

delta_v: vector

The linear volume depolarization.

Notes

The linear volume depolarization ratio is calculated by the formula:

\[\delta^V = \frac{1 - \frac{\delta'}{C}}{\frac{\delta'R_t}{C} - R_C}\]

References

Engelmann, R. et al. The automated multiwavelength Raman polarization and water-vapor lidar Polly XT: the neXT generation. Atmos. Meas. Tech., 9, 1767-1784 (2016)