1. Introduction

The bistatic Doppler radar system was proposed and developed by Wurman et al. (Wurman et al. 1993; Wurman 1994; Wurman et al. 1994). The system, composed of a single traditional radar (transmitter/receiver) and a remotely located, nontransmitting, passive receiver with a low-gain antenna, can retrieve dual-Doppler wind fields at much lower cost compared to the traditional monostatic Doppler radar pair.

There are some differences in retrieving the velocity vector compared to traditional dual-Doppler system methods. As shown by Protat and Zawadzki (1999), the angle between two Doppler velocities is half the corresponding angle in the dual-Doppler case if the bistatic receiver is replaced by a second monostatic Doppler radar. This can introduce larger error into the synthesized Doppler velocity compared to the traditional dual-Doppler radar system. Furthermore, the resolution volume length changes with scattering angle and approaches infinity along the baseline. This also enhances the error of the Doppler velocity observed by the receiver (Wurman et al. 1993; Protat and Zawadzki 1999; Satoh and Wurman 1999).

Estimation of the error distribution of the synthesized velocity is shown in Fig. 14 of Wurman et al. (1993). However, an explicit mathematical expression for the error was not presented. In addition, the locations of minimum error were not given.

In this paper, rigorous mathematical analysis is made to describe the error estimation as well as the location of the minimum error.

Throughout this paper, the analyses are confined to cases in which the elevation angle of the transmitted beam is small and the vertical component of the wind can be neglected.

Section 2 discusses the two kinds of error in the synthesized Doppler velocity vector. One is the error due to the finite resolution of the Doppler velocity at the transmitter/receiver and the receiver. The other is related to the finite resolution in detecting an azimuth angle at the transmitter/receiver and range at the receiver, and causes an error in determining the scattering angle and the directions of the two Doppler velocities. Section 3 investigates the error due to finite resolution of the Doppler velocity; regions of minimum error are described for the case in which this type of error dominates. Section 4 compares the magnitudes of the two types of error. Discussion and conclusions follow in section 5. Equations for the latter-type error are given in the appendix. These will be useful when using a bistatic radar system with a higher velocity resolution Doppler velocity in which both types of error might be of comparable magnitude.

2. Errors in the synthesized horizontal two-dimensional wind

The synthesized Doppler velocity vector from a bistatic radar system is given by

View Expanded

where V_t and V_b are the Doppler velocities sensed by the transmitter/receiver and the bistatic receiver, respectively, and α is the scattering angle of the transmitted pulse wave and also a crossing angle between two vectors r₁ (from the transmitter/receiver to a target) and r₂ (from the receiver to the same target) (Fig. 1). The vectors e₊ and e₋ are orthogonal unit vectors defined by

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where e_t and e_b are unit vectors that parallel the Doppler velocities of the transmitter/receiver and bistatic receiver, respectively. It can be shown that

αe_te_b(3)

where (e_t · e_b) is a scalar product of two unit vectors e_t and e_b (Protat and Zawadzki 1999; de Elia and Zawadzki 2000).

Doppler velocity measured by the bistatic receiver is given by

View Expanded

where V̂_b is a quantity processed the same way as traditional Doppler velocity (Wurman et al. 1993; Protat and Zawadzki 1999; Satoh and Wurman 1999). The real Doppler velocity V_b is modulated by the resolution volume through a factor of 1/cos(α/2). Then (1) is rewritten as

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The error of this synthesized velocity vector can be written as

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The first two terms represent the errors due to the finite resolution of the measured Doppler velocities at both receivers. These errors depend on the antenna rotation speed and signal processing. We call this error type 1 error. The third and fourth terms represent errors in observing the relevant angles that are determined by measuring two distances and an angle: the length of the baseline, and the distance from the transmitter/receiver to the receiver via the scatterer and the azimuth angle of transmitted beam. The precision of the angles is related to the accuracy of the measured distances and the azimuth angle. The first distance can be determined far more accurately (i.e., with precision of cm by using global positioning system data) than the other distance; thus, the errors in observing the angles are governed by the errors in measuring the second distance and the azimuth angle. The measurement accuracy of the distance is determined by the number of gates between pulses, and is essentially limited by the pulse length. The accuracy of the azimuth angle is limited by the beamwidth. This error is called type 2 error throughout this paper.

The magnitude of the error of the synthesized velocity vector is computed by taking the average value of the square of δV in (6). This averaged value is given by

View Expanded

where and correspond to the type 1 and type 2 error, respectively. Here, is given by

View Expanded

where δV²_t and δV̂²_b are the mean-square error of the Doppler velocities by the traditional Doppler radar and the bistatic receiver, respectively. The correlation between them can be neglected.

The second term in (7) is given by

View Expanded

where V₁ = V_t, V₂ = V_b. Both f_ij and g_ij are complicated functions of scattering angles α and azimuth angle θ whose mathematical expressions will be given in the appendix; δS^*₂ is a nondimensional parameter defined by

View Expanded

where d and δS₂ are the length of the baseline and the range error of the bistatic receiver, respectively. The correlation between δθ and δS₂ can be neglected.

3. The mean-square error of the synthesized Doppler velocity vector

In this section we devote ourselves to an analysis of . The error that arises from the error in measuring the angle α is neglected for the same reason that we neglected If the same signal processor were used at both receivers, it would be reasonable to assume

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Substituting (11) into (8) we can show that the mean square error is

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A geometrical factor in (12) is defined by

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which is independent of the hardware used at the transmitter/receiver and the receiver. In Fig. 2, the geometrical factor is plotted against the scattering angle α and has a minimum value of (3 + 22) when α is cos⁻¹(−3 + 22) ≈ 100°.

Figure 3 plots the horizontal distribution of this geometrical factor. The domain in which the mean-square error is less than twice the minimum value,

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is a crescent-like region defined by two arcs on which the scattering angle takes values of approximately 52° and 142°, respectively. We denote this region by the shaded domain in Fig. 3.

4. Comparison of the two types of errors

In (6) and (9), we showed that depends explicitly on the Doppler velocities V_t and V_b, although is independent on them. Therefore, if we want to compare the magnitudes of these two quantities and , we must use wind field data. But it is not appropriate to evaluate , which should have a general meaning, from a specific wind field. In this paper, we employ the following procedure for the evaluation:

1. We assume that wind speed is constant in place. In this paper we adopt a wind speed of 20 m s⁻¹.

2. At each point by rotating the wind direction from 0° to 180°, we get the maximum value of .

3. We regard the maximum value as the value of at that point.

By employing the above procedure, each point has an equal meaning. If we present the true velocity vector V with its direction e_v and magnitude V by

Ve_vV,(15)

two Doppler velocities, V_t and V_b, are expressed as

View Expanded

where e_t and e_b are unit vectors that appeared in (2).

Then the ratio of the two types of the error is given by

View Expanded

In Fig. 4, we give plots of the ratio r by putting values δV² = 1 m² s⁻², (δS^*₂)² = (100/30 000)², and V = 20 m s⁻¹. into (18). These figures show that the type 2 error is larger along the baseline, especially near the bistatic receiver. In Fig. 5, we plot optimal domains of the measurement for this bistatic radar system, which is the crescent-like region mentioned in the previous section with a domain being cut off in which the ratio r is less than 10. These figures reveal that we must avoid retrieving data from this system near the bistatic receiver as well as the baseline.

5. Conclusions and discussion

The error of the synthesized Doppler velocity measured by a bistatic radar system is estimated. Two sources of error in the synthesized velocity vector are identified. One comes from the error in observing the Doppler velocity at a transmitter/receiver and a bistatic receiver (type 1 error). The magnitude of this error is estimated to be 1 m s⁻¹ for high rpm antennas. The other error is related to measuring the azimuth angle and the distance between the transmitter/receiver and the bistatic receiver via the target (type 2 error). The azimuth angle and the distance measuring errors further affect the synthesized Doppler velocity by introducing errors in the determination of the scattering angle α and the directions of two Doppler velocities. The magnitude of the type 2 error, however, is one order of magnitude smaller than the first error and thus can be neglected except at a domain composed of a circular domain around the receiver and a narrow domain containing the baseline.

The error distribution of type 1 is determined by geometrical factors, but its magnitude is controlled by the resolution of the Doppler velocity at the transmitter/receiver and the receiver.

The optimal domain for this system is a crescent-like region bounded by two arcs on which the scattering angle takes values of approximately 52° and 142°, respectively, with a region being cut off in which the ratio r in (18) satisfies r < 10. The region of minimum error is on an arc on which the scattering angle is approximately 100°. These results are obtained by assuming that the mean-square errors of the Doppler velocity of the transmitter/receiver and the bistatic receiver are equal. If these mean-square errors are mutually different, the angles above will change.

As mentioned in section 1, the narrowness of the angle between the two Doppler velocities yields a larger error than traditional dual-Doppler radar systems. The precision of the synthesized velocity approximately perpendicular to these two velocities might be poor. It is therefore necessary to investigate the directional dependence of the error distribution, which will be discussed in an upcoming paper.