Effect of speckle on lidar pulse–pair ratio statistics

Edward P. MacKerrow, Mark J. Schmitt, and David C. Thompson

Author Affiliations

Edward P. MacKerrow, Mark J. Schmitt, and David C. Thompson

^{}E. P. MacKerrow and M. J. Schmitt are with the Los Alamos National Laboratory, Applied Theoretical Physics Division, X-CM, Mail Stop E543, Los Alamos, New Mexico 87545.

^{}D. C. Thompson is with the Los Alamos National Laboratory, Chemical Science and Technology Division, CST-1, Mail Stop E543, Los Alamos, New Mexico 87545.

The ratio of temporally adjacent lidar pulse returns is commonly used in differential absorption lidar (DIAL) to reduce correlated noise. These pulses typically are generated at different wavelengths with the assumption that the dominant noise is common to both. This is not the case when the mean number of laser speckle integrated per pulse by the lidar receiver is small (namely, less than 10 speckles at each wavelength). In this case a large increase in the standard deviation of the ratio data results. We demonstrate this effect both theoretically and experimentally. The theoretical value for the expected standard deviation of the pulse–pair ratio data compares well with the measured values that used a dual CO_{2} laser-based lidar with a hard target. Pulse averaging statistics of the pulse–pair data obey the expected
${\mathrm{\sigma}}_{1}/\sqrt{N}$ reduction in the standard deviation, σ_{N}, for N-pulse averages. We consider the ratio before average, average before ratio, and log of the ratio before average methods for noise reduction in the lidar equation. The implications of our results are discussed in the context of dual-laser versus single-laser lidar configurations.

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The single-wavelength values were calculated previously12 and are considered more accurate than the values of M calculated in this paper. The previous, single-wavelength results included target depolarization, detector noise, laser noise, and target albedo variations. The values calculated in this paper take into consideration only speckle noise. Generally, the values of M obtained in this paper, by curve fitting Eq. (21) to the histograms in Figs. 6, are larger than the values obtained previously.12

Table 2

Standard Deviation for Single-Wavelength Data and Pulse–Pair Ratio Data^{a}

σ_{1} (Single Wavelength)

σ_{1} (Ratio)

M

L0(0)

L0(1)

L1(0)

L1(1)

00/01

10/11

00/11

01/10

1.03

0.818

0.787

0.86

0.86

∞ (4.16)

∞ (7.77)

∞ (6.84)

∞ (6.96)

1.5

0.75

0.72

0.87

1.04

3.4

∞ (4.95)

∞ (7.51)

∞ (5.61)

2.7

0.63

0.62

0.83

0.72

0.97

1.17

1.13

1.10

4.7

0.44

0.45

0.58

0.59

0.68

1.13

0.85

0.90

5.1

0.42

0.42

0.48

0.46

0.62

0.69

0.66

0.71

5.5

0.41

0.41

0.47

0.48

0.62

0.71

0.67

0.67

9.0

0.34

0.34

0.35

0.34

0.49

0.52

0.52

0.51

14.8

0.26

0.27

0.31

0.30

0.36

0.44

0.39

0.40

Note the singular values for the standard deviation of pulse–pair ratio data when a small number of speckles were integrated by the lidar receiver. The values in parentheses, associated with the infinite values for the pulse–pair ratio data, are the standard deviation of the pulse–pair ratio data after the singular values were removed from the data. The pulse–pair ratio, 00/11, is defined as 00/11 ≡ L0(10P16)/L1(10P20), 01/10 ≡ L0(10P20)/L1(10P16), etc.

Table 3

Number of Infinite Data Values Over the Total Number of Pulses Measured for Each Value of M^{a}

M Ratio Taken →

Number of Infinite Pulse–Pair Ratio Values/Total Number of Pulses

L0(0)/L0(1)

L1(1)/L1(0)

L0(0)/L1(1)

L0(1)/L1(0)

1.03

7/6010

22/6010

22/6010

31/6010

1.50

0/4845

4/4845

4/4845

17/4845

2.70

0/4878

0/4878

0/4878

0/4878

4.70

0/4874

0/4874

0/4874

1/4874

5.10

0/4947

0/4947

0/4947

0/4947

5.50

0/4948

0/4948

0/4948

0/4948

9.0

0/4394

0/4394

0/4394

0/4394

14.8

0/4432

0/4432

0/4432

0/4432

Data values are datum where the denominator of the pulse–pair ratio was set to zero by the analog-to-digital converter.
The pulse–pair ratios taken are identified as L0(0)/L0(1) is equivalent to L0(10P16)/L0(10P20) and similarly for the other wavelength/laser combinations.

Table 4

Number of Negative Values for the Pulse–Pair Ratio That Were Removed from the Data before Using the Log of the Ratio Before Average Method, Eq. (5)^{a}

M Ratio Taken →

Number of Negative Pulse–Pair Ratio Values/Total Number of Pulses

L0(0)/L0(1)

L1(1)/L1(0)

L0(0)/L1(1)

L0(1)/L1(0)

1.03

155/6010

704/6010

409/6010

452/6010

1.50

2/4845

234/4845

62/4845

167/4845

2.70

0/4878

1/4878

1/4878

0/4878

4.70

0/4874

5/4874

1/4874

3/4874

5.10

0/4947

6/4947

2/4947

4/4947

5.50

0/4948

0/4948

0/4948

0/4948

9.0

0/4394

0/4394

0/4394

0/4394

14.8

0/4432

0/4432

0/4432

0/4432

The pulse–pair ratios taken are identified as L0(0)/L0(1), equivalent to L0(10P16)/L0(10P20), and similarly for the other wavelength–laser combinations.

Table 5

Standard Deviation for Pulse–Pair Ratio Data with All the Negative Ratio Values Removed^{a}

σ_{1}(I_{z} = I_{x}/I_{y} ≥ 0)

σ_{1}(ln[Ratio])

M

00/01

10/11

00/11

01/10

ln(00/01)

ln(10/11)

ln(00/11)

ln(01/10)

1.03

2.99

2.93

3.03

2.96

1.25

1.27

1.28

1.24

1.5

1.49

2.72

3.74

3.12

1.08

1.25

1.18

1.17

2.7

0.97

1.20

1.13

1.07

0.83

0.93

0.88

0.89

4.7

0.68

1.02

0.85

0.89

0.62

0.83

0.73

0.75

5.1

0.66

0.86

0.80

0.79

0.58

0.77

0.70

0.69

5.5

0.62

0.70

0.67

0.67

0.57

0.65

0.61

0.60

9.0

0.49

0.52

0.52

0.51

0.46

0.48

0.48

0.48

14.8

0.36

0.44

0.39

0.40

0.35

0.41

0.38

0.38

This was done for a fair comparison between the log of the ratio data and the ratio data. Negative values for the pulse–pair ratio result from fluctuations in the boxcar integrator baseline value; the baseline was subtracted from the return signal values. The pulse–pair ratios are defined here as 00/11 ≡ L0(10P16)/L1(10P20), 01/10 ≡ L0(10P20)/L1(10P16), etc.

Tables (5)

Table 1

Number of Speckle Integrated by the Lidar Receiver as a Function of the Receiver Diameter^{a}

The single-wavelength values were calculated previously12 and are considered more accurate than the values of M calculated in this paper. The previous, single-wavelength results included target depolarization, detector noise, laser noise, and target albedo variations. The values calculated in this paper take into consideration only speckle noise. Generally, the values of M obtained in this paper, by curve fitting Eq. (21) to the histograms in Figs. 6, are larger than the values obtained previously.12

Table 2

Standard Deviation for Single-Wavelength Data and Pulse–Pair Ratio Data^{a}

σ_{1} (Single Wavelength)

σ_{1} (Ratio)

M

L0(0)

L0(1)

L1(0)

L1(1)

00/01

10/11

00/11

01/10

1.03

0.818

0.787

0.86

0.86

∞ (4.16)

∞ (7.77)

∞ (6.84)

∞ (6.96)

1.5

0.75

0.72

0.87

1.04

3.4

∞ (4.95)

∞ (7.51)

∞ (5.61)

2.7

0.63

0.62

0.83

0.72

0.97

1.17

1.13

1.10

4.7

0.44

0.45

0.58

0.59

0.68

1.13

0.85

0.90

5.1

0.42

0.42

0.48

0.46

0.62

0.69

0.66

0.71

5.5

0.41

0.41

0.47

0.48

0.62

0.71

0.67

0.67

9.0

0.34

0.34

0.35

0.34

0.49

0.52

0.52

0.51

14.8

0.26

0.27

0.31

0.30

0.36

0.44

0.39

0.40

Note the singular values for the standard deviation of pulse–pair ratio data when a small number of speckles were integrated by the lidar receiver. The values in parentheses, associated with the infinite values for the pulse–pair ratio data, are the standard deviation of the pulse–pair ratio data after the singular values were removed from the data. The pulse–pair ratio, 00/11, is defined as 00/11 ≡ L0(10P16)/L1(10P20), 01/10 ≡ L0(10P20)/L1(10P16), etc.

Table 3

Number of Infinite Data Values Over the Total Number of Pulses Measured for Each Value of M^{a}

M Ratio Taken →

Number of Infinite Pulse–Pair Ratio Values/Total Number of Pulses

L0(0)/L0(1)

L1(1)/L1(0)

L0(0)/L1(1)

L0(1)/L1(0)

1.03

7/6010

22/6010

22/6010

31/6010

1.50

0/4845

4/4845

4/4845

17/4845

2.70

0/4878

0/4878

0/4878

0/4878

4.70

0/4874

0/4874

0/4874

1/4874

5.10

0/4947

0/4947

0/4947

0/4947

5.50

0/4948

0/4948

0/4948

0/4948

9.0

0/4394

0/4394

0/4394

0/4394

14.8

0/4432

0/4432

0/4432

0/4432

Data values are datum where the denominator of the pulse–pair ratio was set to zero by the analog-to-digital converter.
The pulse–pair ratios taken are identified as L0(0)/L0(1) is equivalent to L0(10P16)/L0(10P20) and similarly for the other wavelength/laser combinations.

Table 4

Number of Negative Values for the Pulse–Pair Ratio That Were Removed from the Data before Using the Log of the Ratio Before Average Method, Eq. (5)^{a}

M Ratio Taken →

Number of Negative Pulse–Pair Ratio Values/Total Number of Pulses

L0(0)/L0(1)

L1(1)/L1(0)

L0(0)/L1(1)

L0(1)/L1(0)

1.03

155/6010

704/6010

409/6010

452/6010

1.50

2/4845

234/4845

62/4845

167/4845

2.70

0/4878

1/4878

1/4878

0/4878

4.70

0/4874

5/4874

1/4874

3/4874

5.10

0/4947

6/4947

2/4947

4/4947

5.50

0/4948

0/4948

0/4948

0/4948

9.0

0/4394

0/4394

0/4394

0/4394

14.8

0/4432

0/4432

0/4432

0/4432

The pulse–pair ratios taken are identified as L0(0)/L0(1), equivalent to L0(10P16)/L0(10P20), and similarly for the other wavelength–laser combinations.

Table 5

Standard Deviation for Pulse–Pair Ratio Data with All the Negative Ratio Values Removed^{a}

σ_{1}(I_{z} = I_{x}/I_{y} ≥ 0)

σ_{1}(ln[Ratio])

M

00/01

10/11

00/11

01/10

ln(00/01)

ln(10/11)

ln(00/11)

ln(01/10)

1.03

2.99

2.93

3.03

2.96

1.25

1.27

1.28

1.24

1.5

1.49

2.72

3.74

3.12

1.08

1.25

1.18

1.17

2.7

0.97

1.20

1.13

1.07

0.83

0.93

0.88

0.89

4.7

0.68

1.02

0.85

0.89

0.62

0.83

0.73

0.75

5.1

0.66

0.86

0.80

0.79

0.58

0.77

0.70

0.69

5.5

0.62

0.70

0.67

0.67

0.57

0.65

0.61

0.60

9.0

0.49

0.52

0.52

0.51

0.46

0.48

0.48

0.48

14.8

0.36

0.44

0.39

0.40

0.35

0.41

0.38

0.38

This was done for a fair comparison between the log of the ratio data and the ratio data. Negative values for the pulse–pair ratio result from fluctuations in the boxcar integrator baseline value; the baseline was subtracted from the return signal values. The pulse–pair ratios are defined here as 00/11 ≡ L0(10P16)/L1(10P20), 01/10 ≡ L0(10P20)/L1(10P16), etc.