Estimate optimization parameters for incoherent backscatter heterodyne lidar including unknown return signal bandwidth

Barry J. Rye

Barry J. Rye

^{}The author is with the Environmental Technology Laboratory, R/E/ET2, National Oceanic and Atmospheric Administration, Cooperative Institute for Research in Environmental Sciences, University of Colorado, 325 Broadway, Boulder, Colorado 80303.

Barry J. Rye, "Estimate optimization parameters for incoherent backscatter heterodyne lidar including unknown return signal bandwidth," Appl. Opt. 39, 6086-6096 (2000)

The conditions for optimizing the precision of heterodyne
atmospheric lidar measurements using extended (deep) targets are
investigated. The minimum standard deviation of each unknown
(return power, Doppler shift, and signal bandwidth) is
approximately twice the optical limit at best and is only weakly
dependent on knowledge of the other parameters at optimal power
levels. Somewhat stronger signal power levels are needed for
bandwidth estimation. Results are displayed as a function of a
time–bandwidth product to clarify the trade-off between estimate
precision and range weighting. Realization under ideal conditions
is confirmed by use of simulations.

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The values were obtained from CRLB’s
calculated for Levin’s spectral model with a Gaussian signal spectrum,
given a priori knowledge of the parameters not
estimated. Shown are values of the function
f^{(i)}(x) [for use in Eq.
(3)], the tuning parameter α =
δ/[√(2π)f_{2}], and the estimator noise
figure F_{
E
}, and √F_{
E
}
[the last being the factor by which the standard deviation of the
estimate exceeds the ideal value of Eqs. (2)] at the optimal
operating points for heterodyne systems, according to this
model. The values for δ and f_{1} were given
previously,1 except that the incorrect value previously
given for
(F_{
E
})_{min}(f_{1})
is corrected here. The formula for
f^{(2)}(x) is that of
Zrnic16 and also corrects a minor error in one given
previously.15

Table 2

Optimal Parameters and Precision for Pulse-Broadened
Return from Lidar with Short Time–Bandwidth
Producta

θ_{
i
}

α_{min}

SNR (dB)

√ (F_{
E
})_{min}

n

σ_{BZ}(θ_{
i
})/θ_{
i
} (%)

σ_{BZ}(θ_{
i
}) (m/s)

δ|f_{
1
}, f_{
2
}

2.5

-9.3

2.1

85

21

—

δ

3.4

-8.0

2.2

64

22

—

f_{
1
}, f_{
1
}|δ, f_{
2
}

8.2

-4.2

3.3

27

—

0.31

f_{
2
}

17.8

-0.8

5.3

12

37

0.35

f_{
2
}|δ, f_{
1
}

23.9

0.5

5.1

9

36

0.34

Optimal operating parameters for
estimation of the unknown given as θ_{
i
},
by use of heterodyne lidar with wavelength 10 µm (or 2
µm), pulse duration T_{FWHM} =
1 µs (200 ns), range-gate duration T =
1 µs (200 ns), receiver bandwidth
F_{
S
} = 10 MHz (50 MHz) (both equivalent
to ±25 m/s), f_{2} = 0.019,
f_{2}M = 0.19. Also given are the
wideband SNR (obtained from α_{min}) and the number of
pulses n needed to generate on average 100 effective
photocounts. The value of the CRLB is quoted as the relative error
for estimates of SNR (δ) and bandwidth
(f_{2}) and as the standard deviation of the
equivalent velocity estimates in meters per second for Doppler shift
(f_{1}) and bandwidth.

Table 3

Optimal Parameters and Precision for Broadband Return from
Lidara

θ_{
i
}

α_{min}

SNR (dB)

√(F_{
E
})_{min}

n

σ_{BZ}(θ_{
i
})/θ_{
i
} (%)

σ_{BZ}(θ_{
i
}) (m/s)

δ

1.9

-3.4

2.2

22

22

—

f_{
1
}

3.5

-0.9

2.4

13

—

1.11

f_{
2
}

6.4

-1.8

2.9

7

20

0.95

As Table 2, except that return bandwidth
is five times greater than the pulse (f_{2} =
0.094, f_{2}M = 0.94).

The values were obtained from CRLB’s
calculated for Levin’s spectral model with a Gaussian signal spectrum,
given a priori knowledge of the parameters not
estimated. Shown are values of the function
f^{(i)}(x) [for use in Eq.
(3)], the tuning parameter α =
δ/[√(2π)f_{2}], and the estimator noise
figure F_{
E
}, and √F_{
E
}
[the last being the factor by which the standard deviation of the
estimate exceeds the ideal value of Eqs. (2)] at the optimal
operating points for heterodyne systems, according to this
model. The values for δ and f_{1} were given
previously,1 except that the incorrect value previously
given for
(F_{
E
})_{min}(f_{1})
is corrected here. The formula for
f^{(2)}(x) is that of
Zrnic16 and also corrects a minor error in one given
previously.15

Table 2

Optimal Parameters and Precision for Pulse-Broadened
Return from Lidar with Short Time–Bandwidth
Producta

θ_{
i
}

α_{min}

SNR (dB)

√ (F_{
E
})_{min}

n

σ_{BZ}(θ_{
i
})/θ_{
i
} (%)

σ_{BZ}(θ_{
i
}) (m/s)

δ|f_{
1
}, f_{
2
}

2.5

-9.3

2.1

85

21

—

δ

3.4

-8.0

2.2

64

22

—

f_{
1
}, f_{
1
}|δ, f_{
2
}

8.2

-4.2

3.3

27

—

0.31

f_{
2
}

17.8

-0.8

5.3

12

37

0.35

f_{
2
}|δ, f_{
1
}

23.9

0.5

5.1

9

36

0.34

Optimal operating parameters for
estimation of the unknown given as θ_{
i
},
by use of heterodyne lidar with wavelength 10 µm (or 2
µm), pulse duration T_{FWHM} =
1 µs (200 ns), range-gate duration T =
1 µs (200 ns), receiver bandwidth
F_{
S
} = 10 MHz (50 MHz) (both equivalent
to ±25 m/s), f_{2} = 0.019,
f_{2}M = 0.19. Also given are the
wideband SNR (obtained from α_{min}) and the number of
pulses n needed to generate on average 100 effective
photocounts. The value of the CRLB is quoted as the relative error
for estimates of SNR (δ) and bandwidth
(f_{2}) and as the standard deviation of the
equivalent velocity estimates in meters per second for Doppler shift
(f_{1}) and bandwidth.

Table 3

Optimal Parameters and Precision for Broadband Return from
Lidara

θ_{
i
}

α_{min}

SNR (dB)

√(F_{
E
})_{min}

n

σ_{BZ}(θ_{
i
})/θ_{
i
} (%)

σ_{BZ}(θ_{
i
}) (m/s)

δ

1.9

-3.4

2.2

22

22

—

f_{
1
}

3.5

-0.9

2.4

13

—

1.11

f_{
2
}

6.4

-1.8

2.9

7

20

0.95

As Table 2, except that return bandwidth
is five times greater than the pulse (f_{2} =
0.094, f_{2}M = 0.94).