Wind ranging and velocimetry with low peak power and long-duration modulated laser

Eiichi Yoshikawa; Tomoo Ushio

doi:10.1364/OE.25.008845

1. Introduction

A wind lidar transmits laser light, receives backscattering from aerosol particles that move with the wind, and measures delay time and Doppler frequency shift to provide wind ranging and velocimetry respectively [1] (see Chapter 12). This spatio-temporal observation capability is expected to offer an important contribution to civil aviation because aviation is more wind-sensitive than other modes of civil transportation such as railways and shipping. Currently, wind lidars are operated alongside weather radar at some major airports to give complementary all-weather coverage; that is, the weather radar and wind lidar cover areas of precipitation and non-precipitation respectively [2,3]. Wind spatial profiles measured by wind lidars are processed to detect windshear and turbulence which may be hazardous to airplanes taking-off and landing. Furthermore, airborne wind lidars are being developed to enable aircraft to avoid en route clear-air turbulence that cannot otherwise be detected [4,5].

In practice, compared to weather radar, wind lidar has a shorter range and takes longer to acquire reliable measurements. Typical ground-based wind lidars measure up to a distance of 5 km and take 1 sec per observation in each direction [6–8]. In contrast, airport weather radars observe at distances out to tens or hundreds of kilometers depending on local requirements, and acquire data on each direction in tens of milli-seconds [9,10]. These comparatively poor lidar performance characteristics are mainly due to the fact that the power of the transmitted laser light is much lower than microwaves with current technology. Low power output is particularly a problem for high-altitude airborne applications because backscattering from aerosol is much weaker at an altitude of 10 km than at ground level [11]. (On the other hand, the radar reflectivity of precipitation at altitude is comparable to that on the ground.) With current technology, there are also practical problems of weight, size, and power consumption for installing wind lidars on airplanes, and these issues would need to be addressed even if higher power lasers were developed.

Low peak power, long-duration modulated pulse transmission has been utilized in the radar engineering field to achieve high output power from small, lightweight systems with low power consumption. In order to mitigate the coarser range resolution that results from long pulse durations, transmitted pulses are modulated and range-resolving processing is applied to the received signals. Matched filter processing, which is the most fundamental range-resolving technique, achieves a range resolution which is inversely proportional to the frequency bandwidth of the modulation [12] (see Chapter 7). An aerosol-observing lidar using the concept of low peak power, long-duration pulse modulation was first proposed in the mid-1980s [13,14]. The lidar, called a “Random-Modulation Continuous-Wave (RM-CW) lidar”, switched a transmitted continuous wave on and off according to an M sequence, and obtained a resolved range profile by decoding it. Although the RM-CW lidar has ranging function to measure spatial profiles of aerosol density, it lacked a velocimetry capability because the technology to extract phase components from received signals had not yet been realized. Since the 2000s, coherent lasers have become more accurate and laser signals have been arbitrarily modulated, and various lidars applying the concept have been proposed. While many currently existing lidars use the concept to detect hard targets, only a few lidars for aerosol observation have been proposed [15–17], and even these lidars do not realize simultaneous ranging and velocimetry as performed by a weather radar. In 2013, Brinkmeyer and Waterholter have first presented a CW wind lidar with randomly modulated laser, in which simultaneous ranging and velocimetry are achieved by interferometry between received signal and delayed transmitted signal [18].

As has been well discussed in [15], a combination of linear chirp modulation and matched filter processing, which is typical in weather radars, is not applicable to simultaneous ranging and velocimetry in a wind lidar because of the large Doppler shifts due to the high frequency of the transmitted laser light. A matched filter is equivalent to the time integration of an interferometric wave resulting from the mixing between a received signal and a reference signal (normally equal to the transmitted waveform) at an intermediate frequency. In a case with a single target, the interferometric wave is a single-frequency wave at a Doppler frequency by motion of the single target [12] (see Chapter 7). In the typical case of an aerosol particle with a radial velocity of 1.5 m/sec and a 200 THz transmitted laser signal, the single-frequency wave will have a frequency of 2 MHz and will be filtered by time integration with a pulse duration longer than 0.5 μsec. With longer pulse durations, this typical combination will be less tolerant of Doppler shift and so will no longer allow simultaneous ranging and velocimetry. This fact also means that a longer pulse duration does not increase the received power. The gain from broadening the pulse duration is cancelled by narrowing the passband width of the matched filter.

In this paper, a novel approach for simultaneous wind ranging and velocimetry with a low peak power, long-duration pulse modulated laser is proposed. The proposed wind lidar transmits a modulated long-duration pulse (the technique is also applicable to continuous wave transmission). The received signal, which is a sum of signals frequency-shifted due to the radial velocities of aerosol particles, is processed by a multi-reference matched-filter (MRMF) approach that independently applies matched filters between the received signal and several reference signals in parallel. Each reference signal is generated based on the transmitted signal with a presupposed Doppler shift applied. As the delay time of each matched filter result and the presupposed Doppler shift of the filter reference signal correspond respectively to range and velocity, a two-dimensional range-velocity profile of received power is obtained. Although a long pulse duration does not increase the received power in any individual matched filter, the velocity profile of received power in an arbitrary range is finely resolved since a pair of reference signals with a small Doppler shift between them are made independent by the long pulse duration. The finely-resolved velocity profile of the received power results in improved accuracy of both ranging and velocimetry. The proposed wind lidar thus utilizes a long pulse duration instead of a highly-amplified short-duration pulse, and acquires reliable measurements with hardware which is small, lightweight, and has low power consumption.

This paper is organized as follows. In section 2, the principle of the proposed wind lidar is explained. Section 3 derives the theoretical performance of the proposed wind lidar: range and velocity resolution, signal-to-noise ratio (SNR), and accuracies of ranging and velocimetry. Section 4 discusses the relationship between a conventional wind lidar and the proposed wind lidar, and the applicability of modulation types to the proposed wind lidar. Finally, Section 5 concludes this paper.

2. Principle

2.1 Signal transmission and reception

Wind lidar signal theory―laser transmission, backscattering from aerosol particles, and reception―is now formulated. A transmitting signal at an intermediate frequency $s_{T} (t)$ is expressed as a function of time $t$ (sec):

s_{T} (t) = a_{T} s_{M} (t) s_{I} (t),

where

a_{T}

is relative transmitted amplitude.

s_{M} (t)

is a modulation function in both amplitude and frequency applied to transmitting pulse.

s_{I} (t)

is continuous wave at the intermediate frequency. In the proposed wind lidar,

s_{M} (t)

is typically generated by phase shift keying (PSK) based on a random code (sequence of phase) as

s_{M} (t) = {\begin{matrix} \exp (j 2 π \frac{u (t)}{U} t); & 0 \leq t < T \\ 0; & o t h e r w i s e \end{matrix},

where

\begin{matrix} u (t) = u_{k}; & \frac{k}{K} T \leq t < \frac{k + 1}{K} T \end{matrix} .

_$T$ is a pulse duration (sec). Since the phases switch every

T / K

sec, a frequency bandwidth of the modulation is

K / T

Hz.

k

denotes indices of an element of the code that are integers from

0

to

K - 1

.

U

is the number of phases in the PSK modulation, and a PSK modulation with

U

phases is called

U

-PSK.

u_{k}

is an integer for the

k

th element of the code, which is previously designated from

0

to

U - 1

with the uniform random distribution [19].

s_{I} (t)

is expressed as

s_{I} (t) = \exp {j (2 π f_{I} t + θ_{I})},

where

f_{I}

and

θ_{I}

are a frequency (Hz) and an initial phase (rad) of the intermediate-frequency wave, respectively. An emitting signal to air is expressed by mixing the intermediate-frequency transmitting signal and the carrier laser light as

s_{T C} (t) \approx s_{T} (t) \exp (j 2 π f_{C} t + θ_{C}),

where

f_{C}

and

θ_{C}

are a frequency (Hz) and an initial phase (rad) of the carrier laser light, respectively. A signal received from the

i

th aerosol particle in a channel of the laser light propagation is represented as

s_{R C}^{(i)} (t) = s_{T C} (t - τ^{(i)}) x^{(i)} \exp {j 2 π f_{D}^{(i)} (t - τ^{(i)})},

where

x

is a complex amplitude due to an aerosol particle,

τ

is a roundtrip time (sec) of laser light between the wind lidar and the aerosol particle, and

f_{D}

is a Doppler shift (Hz).

x

is defined for each aerosol particle, being represented as

x = \frac{a_{P} g}{l r^{2}},

where

a_{P}

is a complex scattering amplitude of the aerosol particle,

g

is a gain due to spatial radiation pattern of the laser, and

l

is a path attenuation of the roundtrip propagation.

r

is a distance (m) between the wind lidar and the aerosol particle, which is related to

τ

as

r = \frac{c τ}{2},

where

c

is the speed of light (m/sec).

f_{D}

is a function of radial velocity of the aerosol particle (m/sec)

v

as

f_{D} = \frac{2 v}{c} f_{C} .

The signal which is received from the

i

th aerosol particle is converted to the intermediate frequency by a heterodyne detector [20] (see Chapter 10) as

\begin{array}{l} s_{R}^{(i)} (t) \approx s_{R C}^{(i)} (t) \exp {- (j 2 π f_{C} t + θ_{T})} \\ = a_{T} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{I} (t - τ^{(i)}), \end{array}

where

s_{M D} (t, f_{D})

is modulation in addition to a Doppler shift due to the radial velocity of an aerosol particle, expressed as

s_{M D} (t, f_{D}) = s_{M} (t) \exp (j 2 π f_{D} t) .

Here, it is supposed that a high-pass filter eliminates the direct current included in output of the heterodyne detector. A received signal is expressed as the sum of signals received from all the aerosol particles in the channel,

\begin{array}{l} s_{R} (t) = \sum_{i \in Ν} s_{R}^{(i)} (t) \\ = a_{T} \sum_{i \in Ν} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{I} (t - τ^{(i)}), \end{array}

where

Ν

denotes the set of all the aerosol particles in the channel of the laser light propagation. These Eqs. of signal transmission and reception are formulated based on [21] with simplifications assuming that telescope aperture is not considered and that various attenuations including atmospheric refractive turbulence are roughly included in

l

.

A system example which is supposed in this paper is described in Fig. 1. $s_{T} (t)$ converted from digital to analog by an arbitrary signal generator is mixed with the carrier laser light. The modulated laser light is emitted to air via a circulator and a telescope. Signals backscattered from aerosol particles are received by the telescope and sent to a heterodyne detector via the circulator. The heterodyne detector performs an optical to electrical conversion, and the high-path filter removes the direct current and extract intermediate-frequency signals. Finally, $s_{R} (t)$ is digitally sampled.

Fig. 1 Block diagram of system example.

Download Full Size | PDF

2.2 The MRMF approach

The MRMF outputs a time-frequency response of received signal as

c_{R} (r (\hat{τ}), v ({\hat{f}}_{D})) = \int s_{R} (t) {s^{'}}_{T} {(t - \hat{τ}, {\hat{f}}_{D})}^{*} d t,

where

\hat{τ}

and

{\hat{f}}_{D}

correspond to an delay time and Doppler frequency shift. Time-frequency responses are calculated to range-velocity profiles via Eqs. (8) and (9). A reference signal has the form of a Doppler-shifted transmitted signal as

{s^{'}}_{T} (t, f_{D}) = s_{M D} (t, f_{D}) s_{I} (t) .

Substituting Eqs. (12) and (14) into Eq. (13) gives

\begin{array}{l} c_{R} (r (\hat{τ}), v ({\hat{f}}_{D})) \\ \approx a_{T} \sum_{i \in Ν} {x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) \int_{\hat{τ}}^{\hat{τ} + T} s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{M D} {(t - \hat{τ}, {\hat{f}}_{D})}^{*} d t}, \end{array}

where a factor at the intermediate frequency

\exp {- j 2 π f_{I} (τ - τ^{(i)})}

is approximately omitted because it can be considered constant and physically irrelevant over the scattering volume. In an implementation of the MRMF, a designated time interval of duration equal to or greater than

T

is extracted from a received signal and processed. The time interval is normally equivalent to the inter-pulse period (IPP). When

T

equals to length of the IPP, continuous wave transmission results. The MRMF results are obtained for every IPP and are incoherently averaged as in a typical pulsed wind lidar. With respect to a certain range,

{| c_{R} |}^{2}

represents the velocity profile of received power at that range and corresponds to a Doppler spectrum. At an arbitrary range, therefore, received power, radial velocity, and velocity dispersion can be calculated as Doppler spectral moments [22] (see Chapter 2).

It is supposed that the MRMF approach is performed in digital processing in order to make its hardware simple with a single channel. A processing flow of the MRMF approach is described in Fig. 2. $s_{R} (t)$ is input to $M_{v}$ matched filters represented by *, which respectively suppose reference signals with different velocities, $v ({\hat{f}}_{D}^{(1)})$ through $v ({\hat{f}}_{D}^{(M_{v})})$ , as expressed in Eq. (13). $v ({\hat{f}}_{D}^{(1)})$ and $v ({\hat{f}}_{D}^{(M_{v})})$ can arbitrarily be determined. Then it is adequate that $v ({\hat{f}}_{D}^{(1)})$ to $v ({\hat{f}}_{D}^{(M_{v})})$ are uniformly aligned with a step size of velocity resolution which is analytically derived in section 3.1. After conversion (array rearrangement) from the range profiles at the $M_{v}$ velocities to velocity profiles at $M_{r}$ ranges, $r ({\hat{τ}}^{(1)})$ through $r ({\hat{τ}}^{(M_{r})})$ , wind measurements (received power, radial velocity, and velocity dispersion) are calculated at the $M_{r}$ ranges.

Fig. 2 Processing flow of MRMF approach.

Download Full Size | PDF

Computational cost of the MRMF can be estimated analogously to a weather radar using a matched filter. The MRMF consists of multiple matched filters, and each of matched filter is calculated with a condition similar to weather radars with respect to IPP and digital sampling frequency. Since the MRMF approach demands to process a matched filter $M_{v}$ times, its computational cost can be simply estimated $M_{v}$ times larger than that of a typical weather radar using matched filter.

3. Performance

3.1 Resolution

In order to clarify the range and velocity resolution characteristics of the proposed wind lidar, let us consider a range-velocity profile for a single particle, which can be formulated similarly to Eq. (15) as

\begin{array}{l} c_{R}^{(i)} (r (\hat{τ}), v ({\hat{f}}_{D})) \\ = a_{T} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) \int_{\hat{τ}}^{\hat{τ} + T} s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{M D} {(t - \hat{τ}, {\hat{f}}_{D})}^{*} d t . \end{array}

To obtain the range resolution of the proposed wind lidar, we substitute

{\hat{f}}_{D} = f_{D}^{(i)}

into Eq. (16), giving

\begin{array}{l} c_{R}^{(i)} (r (\hat{τ}), v (f_{D}^{(i)})) \\ = a_{T} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) \int_{\hat{τ}}^{\hat{τ} + T} s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{M D} {(t - \hat{τ}, f_{D}^{(i)})}^{*} d t, \end{array}

This Eq. is clearly the time-autocorrelation of

s_{M D} (t, f_{D}^{(i)})

. As is well-known, its range resolution is

Δ r \approx \frac{c}{2 B} .

where

B

is frequency bandwidth of the modulation [12] (see Chapter 7). On the other hand, the velocity resolution is derived by fixing

\hat{τ} = τ^{(i)}

in Eq. (16), giving

\begin{array}{l} c_{R}^{(i)} (r (τ^{(i)}), v ({\hat{f}}_{D})) \\ = a_{T} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) \\ \int_{τ^{(i)}}^{τ^{(i)} + T} {| s_{M} (t - τ^{(i)}) |}^{2} \exp {j 2 π (f_{D}^{(i)} - {\hat{f}}_{D}) (t - τ^{(i)})} d t . \end{array}

Since

{| s_{M} (t - τ^{(i)}) |}^{2}

is a real function, Eq. (19) is equivalent to the Fourier transform applying to a windowed single-frequency wave. As Appendix B of [12] indicates that its resolution corresponds to the discretized step size of the Fourier transform, the velocity resolution is

Δ v \approx \frac{c}{2 T f_{C}} .

Figure 3 shows a simulation example of range and velocity profiles of a signal received from a single particle. This case assumes that a single particle with a radial velocity of 0 m/sec whose complex amplitude equals to 1 exists at a range of 0 m. The supposed configurations of the proposed wind lidar are listed in Table 1. The transmitted amplitude has a relative value of 1. The carrier and intermediate frequencies are at 200 THz and 40 MHz, respectively. The 8-PSK modulation is applied, that consists of eight phases at every π/4. A random code is obtained from a uniform random generator. The phases switch every twenty waves of the intermediate frequency, resulting in a bandwidth of 2 MHz. Three pulse durations of 5, 50, and 500 μsec, that are respectively configured by sequences with 10, 100, and 1000 elements, are examined (e.g. (40 MHz)⁻¹ x 20 waves x 100 elements = 50 μsec.) Plot (a) of Fig. 3 shows the range profiles with a radial velocity of 0 m/sec. As all three configurations have a bandwidth of 2 MHz, the first nulls appear at around 75 m as indicated by Eq. (18). Even with a different random code, the mainlobe (closer than the first nulls) does almost never change. On the other hand, sidelobes (further than the first nulls) fluctuate by changing randomness of codes. See Chapter 8 of [23] in details of the matched filter results with random phase codes. Plot (b) shows velocity profiles on a range of 0 m. As stated above, a velocity profile at an arbitrary range is equivalent to a Doppler spectrum. It is evident that velocity resolution is inversely proportional to pulse duration, which is consistent with Eq. (20). Both plots show that the peak of relative power increases along with pulse duration as $a_{T}^{2} {| x |}^{2} T^{2}$ , which is obvious from either Eqs. (17) or (19). (Note that the number of phases of PSK modulation does not significantly affect these results because characteristics of frequency spectra are similar between PSK modulations with various number of phases.)

Fig. 3 a) Range and b) velocity profiles of a signal received from a single particle.

Download Full Size | PDF

Table 1. Configuration of Proposed Wind Lidar in Range-Velocity Profile Simulation

View Table | View all tables in this article

3.2 SNR

The theoretical SNR is derived by formulating the received signal to include additive noise such as thermal noise and shot noise [20] (see Chapter 10). Equation (12) is rewritten as

\begin{array}{l} s_{R + N} (t) \\ = a_{T} \sum_{i \in Ν} x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) s_{M D} (t - τ^{(i)}, f_{D}^{(i)}) s_{I} (t - τ^{(n)}) + a_{N} s_{N} (t), \end{array}

where

a_{N}

is amplitude of noise and

s_{N} (t)

is approximated to complex standard normal noise. Invoking Eq. (13), the MRMF results in

c_{R + N} (r (\hat{τ}), v ({\hat{f}}_{D})) = c_{R} (r (\hat{τ}), v ({\hat{f}}_{D})) + c_{N} (r (\hat{τ}), v ({\hat{f}}_{D})),

where

c_{N} (r (\hat{τ}), v ({\hat{f}}_{D})) = a_{N} \int_{τ}^{τ + T} s_{N} (t) {s^{'}}_{T} {(t - \hat{τ}, {\hat{f}}_{D})}^{*} d t .

As indicated by Eqs. (15) and (16), $c_{R}$ is the summation of $c_{R}^{(i)}$ . The peak of ${| c_{R}^{(i)} |}^{2}$ is $a_{T}^{2} {| x^{(i)} |}^{2} T^{2}$ as is apparent from Eqs. (17) or (19). Considering a certain range and velocity of $c_{R}$ , aerosol particles which are further than half of the range and velocity resolution in the range-velocity field have a negligible effect on the summation and their contributions can be ignored. (Hereafter, such an area is called “a range-velocity resolution area”.) Moreover, the summation is random among aerosol particles because $\exp (- j 2 π f_{C} τ^{(i)})$ appears in Eq. (21), and a wavelength of the carrier laser light is extremely small compared with the range resolution. Thus, the received power of the signal component at a certain range and velocity is expressed as

E [{| c_{R} (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}] \approx \frac{1}{2} a_{T}^{2} T^{2} \bar{{| x (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}} N (r (\hat{τ}), v ({\hat{f}}_{D})),

where

E [•]

denotes expectation.

\bar{{| x (r (τ), v (f_{D})) |}^{2}}

and

N (r (τ), v (f_{D}))

are respectively the mean square of amplitudes and the number of aerosol particles around an arbitrary range-velocity, respectively defined as

\bar{{| x (r (τ), v (f_{D})) |}^{2}} = \frac{1}{N (r (τ), v (f_{D}))} \sum_{i \in N (r (τ), v (f_{D}))} {| x^{(i)} \exp (- j 2 π f_{C} τ^{(i)}) |}^{2},

and

N (r (τ), v (f_{D})) \approx Δ r Δ v n (r (τ), v (f_{D})),

where

N (r (τ), v (f_{D}))

is a subset of the aerosol particles existing in an arbitrary range-velocity resolution area.

n (r (τ), v (f_{D}))

is the number density of aerosol particles in a unit range and velocity in an arbitrary range-velocity resolution area [24] (See Chapter 2). Since the mean square of the integral in Eq. (23) results in

T

, the received power of the noise component is

E [{| c_{N} (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}] \approx a_{N}^{2} T .

Therefore, the SNR is approximately derived as

\begin{array}{l} S N R (r (\hat{τ}), v ({\hat{f}}_{D})) = \frac{E [{| c_{R} (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}]}{E [{| c_{N} (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}]} \\ \approx \frac{c^{2}}{8} \frac{a_{T}^{2}}{a_{N}^{2} B f_{C}} \bar{{| x (r (\hat{τ}), v ({\hat{f}}_{D})) |}^{2}} n (r (\hat{τ}), v ({\hat{f}}_{D})) . \end{array}

It should be noted here that the SNR is independent of pulse duration. The reason for this is explained by Eq. (20) and Fig. 3(b), which indicate that the increase of relative peak power is cancelled by the sharpened velocity resolution: in other words, the decrease in the number of aerosol particles summed.

3.3 Ranging and velocimetry

The accuracies of ranging and velocimetry are evaluated by analytically estimating the accuracies of received power, radial velocity, and velocity dispersion. All the three estimates are calculated from a velocity profile of received power (a Doppler spectrum) at an arbitrary range. Assuming a Gaussian Doppler spectral model, a Doppler spectrum is formulated as

c_{1 + S N R} = G (s) + n_{c},

where

c_{1 + S N R}

is a vector containing all velocity components of a Doppler spectrum normalized by power of noise:

c_{1 + S N R} \equiv {[\begin{matrix} \frac{{| c_{R + N} (v^{(1)}) |}^{2}}{E [{| c_{N} (v^{(1)}) |}^{2}]} & \frac{{| c_{R + N} (v^{(2)}) |}^{2}}{E [{| c_{N} (v^{(2)}) |}^{2}]} & \dots & \frac{{| c_{R + N} (v^{(M_{v})}) |}^{2}}{E [{| c_{N} (v^{(M_{v})}) |}^{2}]} \end{matrix}]}^{T} .

Note that

c_{R + N}

and

c_{N}

are expressed as functions of only

v

for simplicity.

G (s)

represents a Gaussian Doppler spectral model and is given by

G (s) = {[\begin{matrix} g (s, v^{(1)}) & g (s, v^{(2)}) & \dots & g (s, v^{(M_{v})}) \end{matrix}]}^{T},

where

g (s, v^{(m)}) = \frac{M_{v} Δ v s_{0}}{\sqrt{2 π} s_{2}} \exp {- \frac{{(v^{(m)} - s_{1})}^{2}}{2 s_{2}^{2}}} + 1,

s = {[\begin{matrix} s_{0} & s_{1} & s_{2} \end{matrix}]}^{T} .

_{$s_{0}$},

s_{1}

, and

s_{2}

are respectively estimates of received power (relative to noise), radial velocity, and velocity dispersion. Since fluctuations of measured

c_{1 + S N R}

are denoted by a noise vector

n_{c}

in Eq. (29), errors on

s

given by

n_{c}

are approximately expressed as

n_{s} \approx {(\frac{\partial G}{\partial s})}^{+} n_{c},

where

{(•)}^{+}

denotes a pseudo-inverse matrix. Therefore,

\begin{matrix} σ (n_{s}^{(o)}) = \sqrt{\sum_{m = 1}^{M_{v}} {g^{+}^{(o, m)} σ (n_{c}^{(o)})}^{2}}; & (o = 0, 1, 2), \end{matrix}

where

σ (•)

is standard deviation of a probabilistic variable.

n_{s}^{(o)}

and

n_{c}^{(m)}

are the

o

th element of

n_{s}

and the

m

th element of

n_{c}

, respectively.

g^{+}^{(o, m)}

is the

(o, m)

element of

{(\partial G / \partial s)}^{+}

. According to the central limit theorem, the probabilistic property of

n_{c}^{(m)}

approximately follows a zero-mean Gaussian probabilistic density distribution whose standard deviation is as

σ (n_{c}^{(m)}) \approx \frac{1}{\sqrt{L}} \frac{E [{| c_{R + N} (v^{(m)}) |}^{2}]}{E [{| c_{N} (v^{(m)}) |}^{2}]},

where

L

is the number of Doppler spectra for averaging. This approximation is sufficiently valid for

L = 100

. In a wind lidar, the number is generally hundreds to thousands.

The accuracies of the received power, radial velocity, and velocity dispersion estimates analytically evaluated as above are shown in Fig. 4. In this evaluation, the supposed configurations of the proposed wind lidar are again as in Table 1. $v^{(1)}$ to $v^{(M_{v})}$ are uniformly aligned with a step size of $Δ v$ from –24 to 24 m/sec. Since $G (s)$ is a non-linear function and its partial derivative $\partial G / \partial s$ depends on $s$ , evaluations were carried out with respect to several sets of realistic $s$ as indicated in Table 2. Values of $s_{0}$ in the range −50 – 10 dB were examined to investigate the dependency on received power. $s_{1}$ was fixed at 0 m/sec since $σ (n_{s}^{(1)})$ is explicitly independent on $s_{1}$ . With respect to $s_{2}$ , 1 and 3 m/sec were evaluated. The number of averaged Doppler spectra was supposed to be 400.

Fig. 4 Accuracies of a) received power, b) radial velocity, and c) velocity dispersion estimates. std. denotes standard deviation.

Download Full Size | PDF

Table 2. Wind parameters for evaluation of ranging and velocimetry

View Table | View all tables in this article

Accuracy of ranging is indicated by the standard deviation of received power estimates normalized by received power in dB. The normalized standard deviation is calculated as

\tilde{σ} (n_{s}^{(0)}) = \frac{{σ (n_{s}^{(0)}) |}_{[d B]}}{10 \log (s_{0} + 1)},

where

{σ (n_{s}^{(0)}) |}_{[d B]}

is the standard deviation of received power estimates in dB which is expressed as

{σ (n_{s}^{(0)}) |}_{[d B]} = 10 \log {\exp (\frac{σ (n_{s}^{(0)})}{s_{0} + 1})},

[24] (see Chapter 2). According to Eq. (37), for example, a normalized standard deviation of 1 means that the received power equals to the standard deviation of the received power estimates, and the negative one-sigma of the received power estimates is equal to the mean power of noise (0 dB); thus, the normalized standard deviation can be interpreted as the probability of detection with respect to an SNR. Plot (a) describes the normalized standard deviation, in which lower SNR gives a larger normalized standard deviation. Comparing the three pulse durations, the plot indicates that a longer pulse duration has less normalized standard deviation; that is, a longer pulse duration offers higher probability of detection to signals with lower SNR. The normalized standard deviation is improved in proportion to the square root of pulse duration. Specifically, comparing pulse durations of 500 μsec to 5 μsec, the same normalized standard deviation is obtained with a 10-dB lower SNR. This means that a 100 times longer pulse duration is equivalent to a 10-dB amplification. It is seen in the differences of the two velocity dispersions that a larger velocity dispersion adversely affects the normalized standard deviation in a similar way to shortening the pulse duration. The deterioration resulting from broadening the velocity dispersion from 1 to 3 m/sec is not as great as that resulting from a 10-times shorter pulse duration.

Accuracy of velocimetry is represented by the standard deviation of radial velocity and velocity dispersion estimates, which are shown in plots (b) and (c) respectively. As seen in plot (a), the two standard deviations in plots (b) and (c) decrease with increasing SNR, and improve in proportion to the square root of pulse duration. It seems that velocity dispersion affects velocimetry more severely than ranging as, for example, the accuracy of the 50-μsec pulse duration and 3-m/sec velocity dispersion case (dashed fat line) is worse than that of 5-μsec pulse duration and 1-m/sec velocity dispersion case (solid thin line). Although the standard deviation of radial velocity is very slightly higher than that of velocity dispersion, those are almost equal to each other.

4. Discussion

4.1 Conventional and proposed wind lidars

A typical pulsed wind lidar can be modeled as a certain configuration of the proposed wind lidar. In a conventional wind lidar, $s_{M} (t)$ is ideally a rectangular function. As a result, $s_{M D} (t, f_{D})$ becomes a single-frequency pulse, and Eq. (15) is equivalent to a Fourier transform with a short time window, which is the Doppler spectral processing of a conventional wind lidar. Thus, the MRMF can be understood as a general form that can use various modulations. Its range and velocity resolution can also be derived from Eqs. (18) and (20), respectively. Note that the formulations of the MRMF covers conventional wind lidars in which a time window of the Fourier transform is equal to the pulse duration. Technically, a conventional wind lidar utilizes a time window of the Fourier transform which differs from the pulse duration. Equation (17) reveals that a longer one is applied as $T$ for range resolution, and Eq. (19) indicates that a shorter one is applied for velocity resolution.

Example configurations of the conventional and proposed wind lidars for simulation comparison are shown in Table 3. The configuration of the conventional wind lidar were defined by referring [25]. The simulation comparison is resulted on the assumption that a time window of Fourier transform is equal to the pulse duration in the conventional wind lidars. These two wind lidars have the same theoretical performance. These examples suppose that the conventional wind lidar has an amplifier with a gain of 10 dB, and the proposed wind lidar has no such amplifier. Instead, the proposed wind lidar transmits pulses that are one hundred times longer pulses than the conventional lidar. As mentioned in section 3.1, range resolutions are equivalent for these two wind lidars since their bandwidth are the same. As mentioned in section 3.3, furthermore, their accuracies of the three wind measurements are equivalent (a 100 times longer pulse duration is equivalent to a 10-dB amplification). It is evident that the MRMF allows hardware design to be much easier by eliminating the 10-dB amplifier which increases weight, size, and power consumption. On the other hand, a significant drawback of the proposed wind lidar is range sidelobes. Figure 5 shows the simulated range profiles of the two wind lidars in which a single particle is assumed as in section 3.1 (The range profile of the conventional wind lidar is drawn with y-axis which is offset by −20 dB compared with the proposed wind lidar in order to clearly show correspondence and difference between them). In contrast to the conventional wind lidar, range sidelobes appear in the proposed wind lidar. The range sidelobe appears as a noise-like floor with −25 dB relative to the peak power lasting up to a range of 7500 m which corresponds to a pulse duration of 50 μsec.

Table 3. Configuration Examples of Conventional and Proposed Wind Lidar

View Table | View all tables in this article

Fig. 5 Range profiles of a signal received from a single particle by the conventional and proposed wind lidars.

Download Full Size | PDF

4.2 Modulation types

As [15] discussed that the combination of linear frequency modulation and matched filter processing is not applicable, the MRMF does also not work with linear frequency modulation. That is understood by a characteristic (shift of range) of the combination of linear frequency modulation and matched filter processing that is elaborated in Chapter 8 of [23]. With the linear frequency modulation and matched filter processing, a Doppler frequency shift gives a shift of range $(c T / 2) \times (f_{D} / B)$ . In the configuration of the proposed wind lidar indicated in Table 3, and if it applies a linear frequency modulation, a velocity dispersion of 1.5 m/sec (equivalent to 2-MHz dispersion of Doppler shift) appears as a range dispersion of 7500 m. Broadening the transmitted bandwidth by 200 MHz, the range dispersion is suppressed to 75 m, but such wide band width greatly reduces the SNR (−20 dB) according to Eq. (28).

Moreover, Chapter 8 of [23] describes characteristics of other modulations, and gives insight of applicabilities of other modulation types. A random modulation does not have range dispersion broadening, and this is the main reason for its adoption here. A drawback of random modulation is range sidelobes as stated in the previous subsection, and this will be the toughest problem for the development of the proposed wind lidar system. The characteristics of using non-linear frequency modulation are somewhere between those of using linear frequency modulation and random modulation. Compared to random modulation, non-linear frequency modulation is affected more by the range dispersion broadening but less by range sidelobes.

5. Conclusion

A wind lidar with a low peak power, long pulse duration modulated laser which enables simultaneous ranging and velocimetry was proposed. In the proposed wind lidar, a modulated pulse (which can be extended to a continuous wave) is transmitted. The transmitted signals irradiate aerosol particles that flow with the wind, and backscattered laser light is received. The received signals are converted by heterodyne detection into intermediate-frequency signals that are digitally sampled. The discretized received signals are then processed into a range-velocity profiles of received power by an MRMF approach which performs multiple matched filter processing with reference signals whose frequencies are shifted by various presupposed Doppler frequencies. Theoretical analysis and numerical simulation show that the proposed wind lidar transmitting, for example, 50-μsec pulses, will give wind measurements (received power, mean radial velocity, and velocity dispersion) and range resolution that have accuracies equivalent to a conventional wind lidar transmitting 0.5-μsec pulses with a 10-dB amplifier. Thus, the proposed wind lidar can eliminate the amplifier that has a large impact on hardware design, allowing realization of a small, lightweight system with low power consumption which is useful not only for ground-based observations but also for power, mass and volume-constrained airborne or space platforms.

In the next step of our research, a prototype of the proposed wind lidar will be developed. Three practical issues that will need to be evaluated by experiments with the prototype are raised here. One is range sidelobes. In case that a very long pulse duration or continuous wave is desired, range sidelobes will be a critical problem since return signals from far ranges will be swamped by the sidelobes of return signals from closer ranges. Range sidelobe suppression techniques which have been proposed in the radar engineering field (see Chapter 8 of [23]) would be effective to this issue. Another is accuracy of the modulated laser light shaping. It would be never realized that random PSK modulation is implemented ideally with discontinuous changes and that the laser linewidth is narrow enough for its effects to be negligible as assumed in this paper. A poor shaping accuracy of the modulated laser light would cause loss of mainlobe power and increase the range sidelobes. The discontinuous changes of the random PSK modulation would be solved by using a window function. Laser linewidth could be a practical bottleneck although a typical figure of laser linewidth is 10 to 100 kHz, which is much less than the supposed modulation bandwidth of several MHz. The other is motions of aerosol particles. Although aerosol particles are assumed in this paper to be stationary during being irradiated, their motions might reduce received power and increase power of noise. These three problems will demand a detailed study on modulation types by experiments using the prototype.

References and links

1. C. Weitkamp, Lidar: Range-Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005).

2. P. W. Chan and Y. F. Lee, “Application of Short-Range Lidar in Wind Shear Alerting,” J. Atmos. Ocean. Technol. 29(2), 207–220 (2012). [CrossRef]

3. T. Iijima, N. Matayoshi, and E. Yoshikawa, “Development and Evaluation of Low-Level Turblences Advisory Display for Aircraft Operation,” in Proceedings of 29th Congress of the International Council of the Aeronautical Sciences, (2014).

4. R. Targ, M. J. Kavaya, R. M. Huffaker, and R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30(15), 2013–2026 (1991). [CrossRef] [PubMed]

5. H. Inokuchi, H. Tanaka, and T. Ando, “Development of an Onboard Doppler Lidar for Flight Safety,” J. Aircr. 46(4), 1411–1415 (2009). [CrossRef]

6. K. Asaka, T. Yanagisawa, and Y. Hirano, “1.5-μm Eye-Safe Coherent Lidar System for Wind Velocity Measurement,” Proc. SPIE 4153, 321–328 (2001). [CrossRef]

7. S. M. Hannon, “Pulsed Doppler Lidar for Terminal Area Monitoring of Wind and Wake Hazards,” in Proceedings of 11th Conference on Aviation, Range, and Aerospace (2004).

8. J. P. Cariou, L. Sauvage, L. Thobois, G. Gorju, M. Machta, G. Lea, and M. Doboue, “Long Range Scanning Pulsed Coherent Lidar for Real Time Wind Monitoring in the Planetary Boundary Layer,” in Proceedings of 16th Conference on Coherent Laser Radar (2011).

9. J. E. Evans, “Achieving Higher Integrity in NEXRAD Products through Multi-Sensor Integration,” in Proceedings of 8th Conference on Aviation, Range, and Aerospace Meteorology (1999).

10. M. E. Weber, J. Y. Cho, M. Robinson, and J. E. Evans, “Analysis of Operational Alternatives to the Terminal Doppler Weather Radar (TDWR),” Project Report ATC-332, MIT Lincoln Laboratory (2007).

11. M. Hess, P. Koepke, and I. Schult, “Optical Properties of Aerosols and Clouds: The Software Package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998). [CrossRef]

12. P. Z. Peebles, Jr., Radar Principles, (Wiley Interscience Publications, 1998).

13. N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22(9), 1382–1386 (1983). [CrossRef] [PubMed]

14. N. Takeuchi, H. Baba, K. Sakurai, and T. Ueno, “Diode-laser random-modulation cw lidar,” Appl. Opt. 25(1), 63–67 (1986). [CrossRef] [PubMed]

15. C. J. Karlsson, F. A. A. Olsson, D. Letalick, and M. Harris, “All-fiber multifunction continuous-wave coherent laser radar at 155 µm for range, speed, vibration, and wind measurements,” Appl. Opt. 39(21), 3716–3726 (2000). [CrossRef] [PubMed]

16. M. L. Simpson, M.-D. Cheng, T. Q. Dam, K. E. Lenox, J. R. Price, J. M. Storey, E. A. Wachter, and W. G. Fisher, “Intensity-modulated, stepped frequency cw lidar for distributed aerosol and hard target measurements,” Appl. Opt. 44(33), 7210–7217 (2005). [CrossRef] [PubMed]

17. O. Batet, F. Dios, A. Comeron, and R. Agishev, “Intensity-modulated linear-frequency-modulated continuous-wave lidar for distributed media: fundamentals of technique,” Appl. Opt. 49(17), 3369–3379 (2010). [CrossRef] [PubMed]

18. E. Brinkmeyer and T. Waterholter, “Continuous wave synthetic low-coherence wind sensing Lidar: motionless measurement system with subsequent numerical range scanning,” Opt. Express 21(2), 1872–1897 (2013). [CrossRef] [PubMed]

19. A. Goldsmith, Wireless Communications, (Cambridge University, 2005).

20. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th edition, (Oxford University, 2006).

21. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30(36), 5325–5352 (1991). [CrossRef] [PubMed]

22. V. Banakh and I. Smalikho, Coherent Doppler Wind Lidars in a Turbulent Atmosphere, (Artech House, 2013).

23. M. I. Skolnik, Radar Handbook, 3rd edition, (McGraw Hill, 2008).

24. J. W. Goodman, Statistical Optics, (John Wiley & Sons, Inc., 1985).

25. S. Kameyama, T. Ando, K. Asaka, Y. Hirano, and S. Wadaka, “Compact all-fiber pulsed coherent Doppler lidar system for wind sensing,” Appl. Opt. 46(11), 1953–1962 (2007). [CrossRef] [PubMed]

Parameter	Value
Received power (relative to noise); $s_{0}$	−50 – 10 dB
Radial velocity; $s_{1}$	0 m/sec
Velocity dispersion; $s_{2}$	1, 3 m/sec

Parameter	Value
Received power (relative to noise); $s_{0}$	−50 – 10 dB
Radial velocity; $s_{1}$	0 m/sec
Velocity dispersion; $s_{2}$	1, 3 m/sec

Wind ranging and velocimetry with low peak power and long-duration modulated laser

Abstract

1. Introduction

2. Principle

2.1 Signal transmission and reception

2.2 The MRMF approach

3. Performance

3.1 Resolution

3.2 SNR

3.3 Ranging and velocimetry

4. Discussion

4.1 Conventional and proposed wind lidars

4.2 Modulation types

5. Conclusion

References and links

Cited By

Figures (5)

Tables (3)

Equations (38)

Optics Express

Parameter	Value
Transmitted amplitude; $a_{T}$	1
Carrier frequency; $f_{C}$	200 THz
Intermediate frequency; $f_{I}$	40 MHz
Modulation	8-PSK
Bandwidth; $B$	2 MHz
Pulse duration; $T$	5, 50, and 500 μsec

Parameter	Conventional	Proposed
Transmitted amplitude; $a_{T}$	$\sqrt{10}$	1
Carrier frequency; $f_{C}$	200 THz
Intermediate frequency; $f_{I}$	40 MHz
Modulation	Pulsed single frequency	8-PSK
Bandwidth; $B$	2 MHz
Pulse duration; $T$	0.5 μsec	50 μsec