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Keywords:

  • refractive index retrieval;
  • collaborative radar;
  • phase unwrapping

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[1] Retrieval of surface layer refractivity via the method of Fabry et al. (1997) is considered. A mathematical framework is constructed, and signal processing algorithms are derived that facilitate refractivity retrieval from the returns from multiple radars viewing a common geographical area. In particular, an approximate discrete model is derived to relate the measured phases to the surface layer refractivity fields, and a modified least squares estimation algorithm is then proposed for the resulting, often ill-conditioned, inversion problem. Because the measurement technique is subject to modulo 2π uncertainties which impact retrievals, a novel algorithm which jointly estimates the unwrapped phases and refractive index (RI) field is also provided. Numerical results indicate the effectiveness of the derived algorithms in both the single- and multiple-radar cases, as well as clearly establishing that having multiple views of the same geographical area from separate radars provides for significant improvement of the RI field estimates.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[2] The refractive index n of a material is the factor by which the phase velocity of electromagnetic radiation is slowed relative to vacuum. Atmospheric refractivity, N, defined as the deviation of the refractive index n from unity in parts per million, or N = (n − 1)106, is strongly affected by temperature, humidity, and atmospheric pressure. At microwave frequencies, refractivity is commonly approximated by Bean and Dutton [1968]:

  • equation image

where P is atmospheric pressure (in mbar), T is temperature (in Kelvin), and e is vapor pressure (in mbar). Pressure, temperature, and water vapor are key inputs to modern numerical weather prediction models; hence the estimation of their values is of significant interest. Equation (1) implies that knowledge of N can aid in the determination of such; hence estimation of the refractivity field is considered in this paper.

[3] The phase velocity of electromagnetic waves in the atmosphere is determined by the refractive index (RI). Hence the round-trip travel time between a radar and a target is determined by the integral of the intervening RI field along the path between the radar and the target at location R,

  • equation image

where c is the speed of light in free space. Thus a measurement of t can provide information on the path integrated refractive index or refractivity field.

[4] However, it is difficult to employ (2) precisely as written. In particular, phase measurements are required to obtain sufficient precision in travel time characterizations, and such measurements are subject to a modulo-2π ambiguity. Significant changes in refractivity over short distances, or subtle changes over long distances can result in phase wrapping in the measurement, requiring a phase unwrapping prior to estimating the refractivity field values from (2).

[5] Recently, Fabry et al. [1997] and Fabry and Petter [2002] proposed a solution to this problem: Instead of trying to determine the absolute RI field, attempt only to determine its variation from a nominal RI field measured at some previous instant t0. Hence, instead of measuring the absolute time of travel t through a phase measurement Φ, measure the difference in travel times between the time of interest t1 and t0 via a differential phase measurement between those times:

  • equation image
  • equation image

where Δt indicates the differential in time, Φ is the phase of the returned radar wave, f is the frequency of the radar and n(x, y, z, t0) is the RI at the reference time, which is found by other means. Thus (3) sets up a relation between the changes of RI and the changes of phase in the returned wave. Since changes in the RI field are much smaller than those of the absolute RI field, this helps to mitigate the phase ambiguity problem. However, even with this modification, the phase ambiguity problem still is one of the most challenging aspects of this approach.

[6] Next, a spatial difference then approximates the required spatial derivative to estimate the refractivity field [Fabry et al., 1997]. In particular, the difference of phase changes between two targets along the same radial line with respect to the radar was employed:

  • equation image

where ΔT indicates the differential between two coradial targets, Δr is the distance between the two targets, and Δn is the average change of n along the path between these two targets. The result of this algorithm is one of the most promising methods of RI field estimation available [Fabry et al., 1997; Fabry and Petter, 2002].

[7] The approach is not without limitations, however. First, and most importantly, is that only two measurements corresponding to targets bookending the RI field area of interest are employed, rather than exploiting all of the information in the radar returns regarding the refractivity at that point. Second, when targets are not well spaced in radial lines, there is an inherent loss due to geometric considerations by forcing such a radial model. Finally, when multiple radars are employed, there is no straightforward method of performing joint RI field estimation given all of the radar returns. These limitations motivate the derivation of a simple discrete model that approximates the radar measurements given the RI field. Once such a model is derived, all of the machinery of modern signal processing is then available for the inversion problem, and the extension to multiple radars is very natural. After recognizing that the least squares (LS) solution with generic regularization for this often ill-conditioned problem is not sufficient, a modified LS algorithm is derived and shown to be effective for both the single-radar scenario or multiple-radar scenario.

[8] The motivation for studying the case where multiple radars scan a common volume is for application in Doppler radar networks where many small radars might be deployed in a relatively dense grid. These small radars may operate at higher microwave frequencies, and hence exacerbate the phase wrapping problem. Phases unwrapping algorithms have been well discussed in the analysis of interferometric synthetic aperture radar (SAR) data, or adaptive optics and speckle imaging [Herraez et al., 1996, 2002]. However, these algorithms are all based on the assumption that the phases are evenly measured on the two-dimensional area. For the application proposed here, this is not achievable since the ground targets cannot be manipulated, and hence a specific phase unwrapping algorithm is necessary for this application. Because of the nature of the phase wrapping problem encountered here, it is difficult to unwrap all of the phases reliably. Hence, rather than trying to unwrap all of the phases and then perform RI field estimation, only those phases where there is great certainty in the wrapping are unwrapped. The remainder of the phase ambiguities are then estimated jointly with the RI field itself.

[9] This paper is organized as follows. In section 2, the system model is described. In section 3, the modified least squares estimation algorithm is discussed. In section 4, a novel algorithm which jointly unwraps phases and estimates the RI field is presented. Numerical results are shown in section 5, and conclusions are drawn in section 6.

2. System Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[10] The system of interest is shown in Figure 1. Several radars scan the same geographical area, which is assumed to contain many stable targets [Fabry et al., 1997]. Suppose K radars are employed and there are J stable targets. Each radar measures a phase from each target at two different times: the reference time and the observation time of interest. Phase differences for the mth radar, {znm}, are calculated, per (3), where znm corresponds to the phase difference of the nth target, 1 ≤ mK, 1 ≤ nJ. Then, all of the phase differences combined form the JK by 1 observation vector y,

  • equation image

Let yi denote the ith element of y. The term “measurements” will be used to indicate the differences of phases as (6) in the remainder of this paper. Temporarily, the frequency of the radars is assumed to be low enough that the differential phase can be reliably unwrapped or that there is no 2π aliasing in the observation measurements. Per section 1, phase unwrapping is indeed a critical topic, which will be addressed in section 4.

image

Figure 1. System overview. There are a large number of targets (triangles) inside the 10 km × 10 km area of interest. Several radars along the borders scan the targets. The darkened region in the top left is enlarged in Figure 2 to illustrate the construction of the discrete model.

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[11] Since, in this paper, no a priori information is assumed known about the RI field, it is impossible to retrieve the exact continuous RI field via (4) through a finite set of phase measurements. Hence the RI field must be parameterized by a finite set of variables. In this paper, the parameterization is as follows. The area of interest is divided into M by M small equal-sized subsquares, and, since the RI field is continuous, the RI value inside each subsquare is assumed to be constant. In other words, the RI value of any point inside a given subsquare is assumed equal to that of the center point, which will be denoted the sampling point. With this gridding scheme, the integral in (4) can be replaced by a finite summation. An example of how the resulting summation is obtained is provided visually per Figures 1 and 2. The region of interest, with the mth radar located at the top left corner, has some large number J of targets inside, and is gridded into M by M subsquares (M equals 40 in Figure 2). The subsquares are numbered sequentially starting in the top left and proceeding row by row (i.e., a raster scan), as shown in Figure 2, which is the enlarged view of the darkened top left region in Figure 1. The line indicates the path from the radar to the target and back to the radar. Hence, under the gridded approach, the phase measurement from target n can be written as

  • equation image
  • equation image
  • equation image

where xj is the change in the RI value of the jth subsquare between the reference time and the time of interest, x is the M2 by 1 RI field vector with elements {xj}, pn,jm is the length of the path between the mth radar and the nth target in the jth subsquare, 0 ≤ pn,jmequation imageδ, δ is the border length of each subsquare, Hm is a J by M2 matrix with elements {Hn,jmequation imagepn,jm}. Note that pn,jm is nonzero only for those j (j = 1, 2,42, 43, 44, 84, 85, 86, 126, 127 in Figure 2) which are intercepted by the straight line from the mth radar to the nth target. Since the RI field is spatially continuous, this summation will approach the integral value as the subsquare size is decreased, per the definition of the Riemann integral.

image

Figure 2. Enlarged view of the darkened region in Figure 1. This illustrates the construction of the discrete model. The whole area is divided into 40 × 40 subsquares, which are numbered sequentially from left to right and row by row. The radar is located at the top left corner. The target under consideration is the white one in Figure 1. The thick line indicates the radio wave path between the radar and the target.

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[12] When the measurements from all K radars are considered, there are J K measurements altogether. Using (8), these can be collected as

  • equation image

where H is a JK by M2 matrix with entries as described above, and e (JK by 1) is the vector of errors.

[13] Efficient inversion algorithms based on this discrete model are developed in the following sections.

3. Modified Least Squares Estimation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[14] A logical first solution to the linear inversion problem implied by (10) is to employ a standard least squares (LS) approach. The LS estimation technique finds the field equation image that minimizes the squared error between the observed vector y and what would have been observed under equation image without noise; that is,

  • equation image

whose solution is given by Scharf [1990, p. 365]:

  • equation image

where (HTH)−1HT is denoted the pseudoinverse.

[15] However, in general, because of the large number of RI field unknowns gathered in x and the path-integrated nature of the phenomenon being exploited, the condition number (the ratio between the largest and the smallest singular value) of matrix H can be very large or infinite (termed column rank deficient or ill conditioned). The pseudoinverse of such a matrix, if it is achievable, leads to error amplification or does not provide a unique solution. An example is shown as following. Suppose the two-dimensional RI field of a 10 km × 10 km area is as shown in Figure 3. The RI in the top left region is 329 (N unit), in the bottom right region, 306, and the change between them is linear. There are 1600 targets evenly distributed in the region. The region is divided into 40 × 40 subsquares, such that there is 1 target in each subsquare, with each target in the center of the subsquare for this simple example. No noise is assumed on the measurements. Although such a distribution of targets can guarantee that the pseudoinverse of the matrix H can be calculated, the error amplification is still too large, as shown in the estimate of the RI field shown in Figure 4, which was obtained with the LS algorithm. It is clear to see that the errors (due to the gridding) are propagated and amplified.

image

Figure 3. Real RI field in a 10 km × 10 km area of interest. The RI is 329 in the top left region and 306 in the bottom right region. The change in RI on the front between the two regions is linear. The nominal RI field at time t0 is 300.

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image

Figure 4. Estimation of the field in Figure 3 via the standard least squares algorithm. The data are collected by one 300 MHz radar located at the top left corner.

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[16] Although the LS method can always provide an accurate match to the observation vector y via (11), often in the case of underdetermined systems, where H lacks full column rank, it cannot provide accurate estimation of the coefficient vector equation image. This is due to the criterion of the LS algorithm, which is to minimize the squared error in the observations. To do so, the LS algorithm projects the observation vector onto the column space of H. If H is not of full column rank, the resulting solution can lie anywhere in a multidimensional space (the solution space), yet still satisfy (11). The widely employed LS algorithm in the case of underdetermined systems, the singular value decomposition (SVD) solution, simply chooses the minimum norm vector in the solution space, which might not be appropriate for this application. Thus additional criteria will be added to find a more suitable solution in the LS solution space.

[17] Expected properties of the RI field can be exploited. For example, the RI field is locally smooth. There are a number of methods to describe spatial smoothness: low spatial bandwidth, low deviation among neighbors, etc. We have found that enforcing a low deviation among neighboring field points on average is an effective criterion; hence that will be employed to choose a solution equation image from the multidimensional LS solution space.

[18] The deviation among neighbors is measured locally by the second differences between a subsquare and its neighbors [Herraez et al., 2002]. The calculation of second differences can be explained with the aid of Figure 5. For a given subsquare centered at (k, l), where (k, l) indicates the two-dimensional position of the subsquare, a smoothness function is defined with the values of its orthogonal and diagonal neighbors. The subsquares (k, l − 1), (k, l + 1), (k − 1, l) and (k + 1, l) are called orthogonal neighboring subsquares. The subsquares (k − 1, l − 1), (k + 1, l + 1), (k + 1, l − 1) and (k − 1, l + 1) are called diagonal neighboring subsquares. The smoothness function is defined as

  • equation image

where

  • equation image

where {fk,l} is a reindexed version of {xj} with respect to its spatial position (k, l):

  • equation image

The expression in (13) can be used for all subsquares in the field except those touching the borders. For subsquares on the borders (excluding the four subsquares at the corners) the smoothness function is defined with the first term (horizontal term) or second term (vertical term) in form (13) only. The smoothness functions for the four corner subsquares are set to zero.

image

Figure 5. Illustration of the smoothness function (13). Each circle indicates the center of a subsquare. The pair (k, l) indicates the two-dimensional position of the subsquare.

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[19] The smoothness quality of the whole field is defined as

  • equation image

Consequently, a lower value of D corresponds to a more locally smooth RI field when measured across the entire area. Then, the modified criterion is

  • equation image

[20] Recall from above how these two criteria work together. The second minimization provides a solution space with multiple degrees of freedom determined by the singular values of H. Then the first minimization chooses one solution in this space.

[21] The detailed solution method to the above problem is as follows. First, apply the singular value decomposition on H:

  • equation image

where U is a JK by JK unitary matrix, V is an M2 by M2 unitary matrix and S is a JK by M2 matrix with (S)i,j = 0, ij, and (S)i,i = σi, where {σi} are known as the singular values of H and are arranged in decreasing order: σ1 ≥ σ2 ≥ … ≥ σM2 ≥ 0. In general, because of the virtual rank deficiency of H, σiequation image 1, for i > I, and σI+12 + σI+22 + … + σM22equation image2, where ε is a small number and I is the numerical rank of H as defined below. Writing SI = diag1, …, σI, 0, 0, …, 0} and

  • equation image

It is easy to see that

  • equation image

where F indicates the Frobenius norm. Then H is said to have numerical rank I and with an accuracy of order ε. Thus H can be approximated by HI, and

  • equation image

Let VTxv = equation image, where v1 is an I by 1 vector and v2 is an (M2I) by 1 vector; thus

  • equation image

[22] From (17), v1 can be calculated, and v2, unlike in the conventional SVD solution where it is set zero without impact on the minimization of the errors in the observations, is left as unknown. Therefore

  • equation image

Each element of x is a linear combination of v2 and clearly this provides a solution space with a dimension of (M2I).

[23] Then the smoothness quality of the whole field is calculated with (14), and it can be written as

  • equation image

where matrices E, F and G are determined by V. E is a positive definite matrix. Clearly this is a concave function of v2. By differentiating the right-hand side of (19), v2 can be optimized:

  • equation image

Finally, using (18), the final solution can be achieved. The performance of this modified LS algorithm will be discussed in section 5.

4. Joint Phase Unwrapping and RI Field Estimation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[24] The discussion in section 3 presumes that either low-frequency radars are employed, or that the phase measurements have been successfully unwrapped via a preprocessing algorithm. The possibility of multiple radars being able to observe a single location implies that such radars will be small in size, and hence will operate at relatively high frequencies. At these frequencies, the 2π alias in the phase measurements can be difficult to unwrap reliably, even for the differential phase measurements contained in y. Let

  • equation image

where yi is the differential phase measurement observed by the radars, which is the modulo 2π result (wrapped phase) of the absolute differential phase yi that is proportional to the differential path-integrated refractive index. Hence Li is an integer multiplier, which is called the measurement ambiguity number. In such a case, the linear model of (10) can be extended to

  • equation image

The dimension of L is JK by 1, the same as that of the measurement vector y.

[25] Phase unwrapping has been well studied for other applications. However, for the application considered here, there are specified properties that need to be considered. First, for most standard phase unwrapping applications, the phase measurements are a good sampling of the measurement space; that is, they are evenly distributed in the area of interest. However, for the case considered here, the sampling is dependent on the availability of stable targets. In general, “good” targets are clustered. In some regions, the targets are sparse, or not available at all. For such cases, previously considered phase unwrapping algorithms are not able to unwrap the whole field successfully. Thus it is desirable to find a more efficient phase unwrapping algorithm. Second, it is important to notice that the phase maps collected by different radars are generated by the same RI field. This property may be exploited, thus helping to solve the difficult phase wrapping problem in these systems.

[26] On the basis of the above observations, a joint phase unwrapping and field estimation algorithm is proposed here. The steps of this algorithm are as follows:

[27] Step 1: Set up a reliability value for each phase measurement yi. The reliability value is a number reflecting the consistency of a given phase measurement with those from nearby targets measured by the same radar. A simple two-valued function for generating these reliability numbers has shown good performance here. If there are other phase measurements from targets within a certain geographical distance of the target corresponding to a given measurement, and the modulo 2π differences of phase measurements from the target of interest and those of neighboring targets are less than a certain threshold (π/4 for example), the reliability value of the phase measurement is set to 1; otherwise, it is set to 0. A more precise reliability function proportional to the reciprocal of the spatial curvature can also be considered, of course.

[28] Step 2: Group geographically adjacent measurements with high reliability values. Adjacent measurements with high reliability values are merged into one group. An identical group ambiguity number Li is assigned to such a group, and an additional modification {0, +2π or −2π} is applied on each measurement yi according to the relative measurement differences. This results in the modified measurement yi. Those phase measurements corresponding to low reliability values, however, are not unwrapped. A separate group and its own group ambiguity number are assigned to each of the latter. Hence yi equals yi for such a case.

[29] Step 3: Estimate the phase unwrapping and the RI field jointly. The last (and critical) step is accomplished as follows. After the second step, the modified measurement, y″, is written as

  • equation image

where y″ is the modified measurement vector, L′ is the group ambiguity number vector formed in step 2, A is a matrix with {0, 1} elements, which maps the group ambiguity number vector to the measurement ambiguity number vector. Note that the dimensions of A and L′ are depending on the result of the “grouping” process of step 2. Also,

  • equation image

The final phase unwrapping and the desired RI field are obtained by running the modified LS algorithm discussed in section 3 on (22).

[30] Note that the first two steps are similar to the region algorithm proposed by Herraez et al. [2002]. However, here the unwrapping procedures stop far before the whole area is unwrapped; only those measurements which can be unwrapped reliably are unwrapped relative to their neighbors.

[31] To illustrate the algorithm, a simple example is presented with the aid of Figure 6. Suppose that there are 6 targets in the field and one radar is employed at the top left corner. The wrapped phase measurements at the radar are shown besides the corresponding targets. Per Figure 6, it can be seen that the modulo 2π difference between {y1, y2} is less than the threshold. So these two measurements are unwrapped relative to each other and merged into one group. A group ambiguity number L1 is then assigned to this group. According to the relative difference, y1 = y1 and y2 = y2 + 2π. Similarly, {y4, y5, y6} are merged into one group, a group ambiguity number L3 is assigned to this group and y4 = y4, y5″ = y5 and y6″ = y6 + 2π. The measurement y3 fails the reliability test. Thus an individual group ambiguity number is assigned to it. So, after the second step, (22) is

  • equation image

and

  • equation image

Then the unknown x and L′ can be estimated by apply (15) to (23).

image

Figure 6. Example illustrating the first two steps of the joint phase unwrapping and RI field estimation algorithm.

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[32] This algorithm is very effective for the application considered in this paper. In particular:

[33] 1. It reduces the errors caused by false unwrapping. It unwraps only measurements where it is certain and lets the modified LS algorithm (15) work on the rest.

[34] 2. It may handle those regions with sparse targets (or no targets at all), while other phase unwrapping algorithms cannot provide the absolute values of the phases in such cases.

[35] In section 5, the performance of this algorithm will be considered.

5. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

5.1. Modified LS Algorithm

[36] The performance of the modified LS algorithm described in section 3 is studied first. For all of the numerical results considered in this paper, the RI field at the reference time t0 is assumed to be 300 N units. In addition, the locations of all targets are assumed to be known exactly by all algorithms, although, of course, these locations would need to be estimated (and hence only known approximately) in practice. Multiple radars are assumed to operate at the same frequency, which will require some sort of time multiplexing, although the proposed algorithms can easily be extended to the case where different radars are at different frequencies (and hence are frequency multiplexed).

[37] To avoid the phase unwrapping problem temporarily, suppose low-frequency (300 MHz) radars are employed on the area shown in Figure 3. Inside this area, 2569 targets are placed randomly according to a uniform distribution, and the SNR of the received signal is 55 dB. The SNR is defined as the aggregated signal-to-noise ratio at the radar receiver front end over many radar pulses. For example, if the one-pulse SNR is 20 dB, then the aggregated SNR over 100 pulses will be 40 dB. To simulate the one-pulse SNR, thermal noise is assumed. Suppose the received signal at the receiver end of the radar is r(t), then

  • equation image

where Es and ϕs are the power and phase of the returned signal respectively, ρejϕe is a complex Gaussian random variable, Es/Varejϕe) = SNRonepulse, and Er and ϕr are the power and phase measured by the radar respectively.

[38] It is important to notice that for such a distribution of targets, the condition number of the matrix H can be as large as 1.3517 × 1018. The pseudoinverse of such a matrix cannot be efficiently calculated, and hence one cannot employ the standard LS algorithm. The estimated fields from the modified LS algorithm are shown in Figure 7. The estimated field in Figure 7a is obtained with data collected by one radar and the estimated field in Figure 7b is obtained with data collected by two radars. Note that, to consider only the geometric advantages of employing two radars, for now, the number of targets is halved from the single-radar case. In reality, even further gain would be obtained via two radars through the effective doubling of the number of observations, which will be shown in section 5.3. One can see from Figure 7 that two radars provide better performance than a single radar. To establish a performance metric, the root mean squared (RMS) error of the estimated field is defined as

  • equation image

The RMS error is expressed in n units if not otherwise noted. For the modified LS algorithm, the RMS error when two radars are used is 9.1413 × 10−7, which is one third of the 2.6565 × 10−6 obtained when only a single radar is used. This comparison between field estimates obtained with one or two radars will be carried out in significantly more detail with higher-frequency radars in section 5.3.

image

Figure 7. Estimates of the field in Figure 3 from the modified LS algorithm. The area is divided into 40 × 40 subsquares. The SNR is 55 dB. (a) Data are collected by one 300 MHz radar located at the top left corner and 2569 targets. (b) Data are collected by two 300 MHz radars located at the top left corner and the bottom right corner, with 1284 targets in the area that are exploited.

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[39] Per Figure 4 the LS algorithm operating on data from one radar cannot provide a quality solution. The error performance of the LS algorithm with two radars is not shown but it is not satisfying either. Hence the modified LS algorithm provides quality RI field estimates in cases where the standard LS algorithm cannot.

5.2. Phase Unwrapping

[40] As the radar frequency increases, even the differential phase measurements suffer modulo 2π wrapping, and hence accurate phase unwrapping is critical for this application. Assume a RI field as shown in Figure 8. An example of a potentially challenging distribution of targets is shown in Figure 9. There are no targets in the top left region, and 2494 targets are placed randomly according to a uniform distribution in the rest of the area. All numerical results from here forward will employ measurements obtained from these targets unless otherwise stated. From the measurements, it is impossible to tell whether there are phases ripples inside the top left region. Actually with the RI field of interest in Figure 8 and radars operating at 3 GHz, there is indeed one phase ripple inside the top left corner, but other phase unwrapping algorithms are unable to determine such, hence failing to provide the absolute phases for these targets. With the algorithm presented in section 4, estimates of the RI field are shown in Figure 10. When one radar is employed and all 2494 targets are exploited (Figure 10a), the RMS error is 1.5535 × 10−6. When two radars are employed and 1254 targets are exploited (Figure 10b), the RMS error is 5.0374 × 10−7.

image

Figure 8. True RI field in a 10 km × 10 km area of interest. The RI is 329 in the top left region and 322 in the bottom right region. Between them are five piecewise linear regions. From the top left region to the bottom right region, the RI changes linearly from 329 to 326, then to 327, 325, 323, and 322. The dashed lines are the contour lines.

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image

Figure 9. Target distribution in the 10 km × 10 km area of interest. There is no target in the top left region, whereas 2494 targets are placed randomly according to a uniform distribution in the rest of the area. When two radars are employed, the algorithms are tested for two cases: (1) approximately half of the targets employed (to preserve the dimension of y between the one radar case and two radar case) and (2) all of the targets employed (to view the full advantages of two radars).

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image

Figure 10. Estimates of the field in Figure 8 from the algorithm proposed. The area is divided into 40 × 40 subsquares. The SNR is 55 dB. (a) Data are collected by one 3 GHz radar located at the top left corner and all 2494 targets in Figure 9. (b) Data are collected by two 3 GHz radars located at the top left corner and the bottom right corner, with only 1254 targets exploited.

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[41] Although the above may appear as a rather contrived example, targets are generally clustered, and thus the phenomenon displayed is quite common. In such cases, it is impossible to tell the absolute phases difference between two widely separated groups of targets, and hence it not possible to unwrap the phase measurements with standard unwrapping algorithms. In contrast, the partial unwrapping algorithm proposed in this paper can handle such cases successfully. Hence this new joint phase unwrapping and RI field estimation algorithm is used for all of the numerical results shown for the proposed algorithm throughout the remainder of the paper.

5.3. Single Radar Versus Multiple Radars

[42] Multiple radars that can view the same geographic area have the potential to greatly improve estimates of the RI field. First, as explored briefly in section 5.1, there is a geometric advantage that improves the invertibility of H. Second, for the same target distribution, there are many more measurements in the vector y. In particular, with the same targets, the number of observations is doubled when going from one radar to two radars. In this subsection, these advantages are explored in detail.

[43] In Figure 11, the estimated RI fields are shown for the case of two radars observing all 2494 targets of Figure 9 (for a total of 5188 measurements). When the SNR is 55 dB (Figure 11a), the RMS error is 1.7389 × 10−7, which is nearly one third of that of the estimated field shown in Figure 10b (5.0374 × 10−7), and almost one order of magnitude better than that of estimated field of Figure 10a (1.5535 × 10−6). When the SNR is reduced to 45 dB (Figure 11b), perhaps to reflect that the multiple-radar system might require smaller radars than the single-radar system, the RMS error of the estimated RI field is 2.6591 × 10−7, which is still better than that of the estimated fields in Figure 10.

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Figure 11. Estimates of the field in Figure 8 from the algorithm proposed with data collected by two 3 GHz radars. The radars are located at the top left corner and the bottom right corner. All 2494 targets in Figure 9 are exploited. The area is divided into 40 × 40 subsquares. (a) SNR is 55 dB. (b) SNR is 45 dB.

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[44] To consider more carefully the SNR gain when multiple radars are employed, consider Figure 12. The top curve in Figure 12 corresponds to a single-radar system employing measurements from all of the targets. The middle curve in Figure 12 corresponds to a two radar system employing measurements from only half of the targets and hence demonstrates only the geometrical advantage of having a second radar. Finally, the bottom curve in Figure 12 is for the case when the two radars exploit all of the targets.

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Figure 12. Error performance versus SNR. The top curve is for the one radar case, for which the 3 GHz radar at the top left corner exploits all the 2494 targets. The middle curve is for the case in which two 3 GHz radars located at the top left corner and the bottom right corner exploit approximately half of the targets in Figure 9. The bottom curve is also for the same two radars but with all 2494 targets in Figure 9 exploited.

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[45] There are a number of things to note from Figure 12. First, there is a potential significant SNR requirement reduction from using multiple radars, and hence the radars can be made more cheaply, as will be required in a dense grid. Second, the performance improves much more rapidly with increasing SNR when two radars rather than one radar are employed; since this improvement is evident in both performance curves where multiple radars are employed, it is attributed to the geometric advantage obtained by employing multiple radars. Finally, all three curves suffer error floors, as expected, due to the inherent loss caused by the gridding in (8) and its amplification through the inversion process. Note, however, the significant improvement of these error floors in the multiple-radar cases.

5.4. Comparison With Fabry's Technique

[46] Per section 1, the motivation for this paper is to improve the estimation of RI fields using differential phase measurements of radar returns from ground targets, a technique pioneered by Fabry et al. [1997] and Fabry and Petter [2002]. In this subsection, the performance of the proposed signal processing approaches introduced here are compared to those employed by Fabry et al. [1997] and Fabry and Petter [2002].

[47] For the RI field of Figure 8 and the target distribution of Figure 9 observed with a single 3 GHz radar, Figure 13 shows the estimated RI field from applying the Fabry technique. Per section 1, the technique estimates the RI field between two targets approximately located on a radial line emanating from the radar, by employing (5), which takes the difference in the differential phase measurements from those two targets. Hence the strong radial characteristics in Figure 13 arise. Generally, the result is then smoothed over some geographical region. Here, a perfect low-pass filter is applied to the estimated RI field of Figure 13; in particular, a sinc(.) window is applied, and the result is shown in Figure 14. Note that the “blank” in the top left region is a symptom of the lack of targets there and results in an unrecoverable area that is 1.73% of the total geographical area. Figures 15 and 16 give the RI field estimates with one radar and two radars, respectively, using the techniques developed in this paper followed by the same sinc(.) smoothing. Note that all of them match the actual RI field very well, in general, although the single-radar case still has difficulty accurately characterizing the RI field in the targetless top left corner.

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Figure 13. Estimate of the RI field (termed path measurements by Fabry et al. [1997]) of Figure 8 with the targets shown in Figure 9 via the original Fabry technique. Each segment is 1° × 500 m. The SNR is 55 dB. The empty (or white) segments are those which Fabry technique cannot recover because of a lack of targets.

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Figure 14. Estimate of the RI field of Figure 8 with the targets shown in Figure 9 via the original Fabry technique followed by sinc(.) smoothing. In particular, the result of Figure 13 has been smoothed by a sinc(.) window, which has a support of 500 m × 500 m and a spatial bandwidth of equation image. The black “dots” in the bottom right region are the distinct errors where targets along the processing paths are not well spaced in radial lines.

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Figure 15. Estimate of the RI field of Figure 8 with the targets shown in Figure 9 via one radar in the top left corner and the techniques proposed in this paper followed by sinc(.) smoothing. The SNR is 55 dB. Note that although the targetless top left corner is recovered here, it is not recovered well. The remainder of the RI field, however, is recovered well.

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image

Figure 16. Estimates of the RI field of Figure 8 with the targets shown in Figure 9 via one radar in the top left corner and one radar in the bottom right corner, using the techniques proposed in this paper followed by sinc(.) smoothing. (a) SNR is 55 dB. Note the very accurate recovery of the entire RI field. (b) SNR is 45 dB. Even at this lower SNR, the recovery of the entire RI field is still accurate.

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[48] Per above, the proposed algorithms match the RI field much better than the Fabry approach, even outside of the top left corner. To quantify this effect, the RMS error is again calculated, but the top left corner is ignored for the result from the Fabry technique. Under this criterion, the RMS error for the estimated RI field of Figure 14 obtained from the Fabry technique is 2.8540 × 10−6, whereas that of the RI field estimates of Figures 15 and 16a are 1.4757 × 10−6 and 1.3580 × 10−7, respectively. At first glance, it seems that the improvement of Figure 15 is not significant. However, this is because the top left corner is not recovered well because of the shortness of targets in this region. If this region is also ignored for Figure 15, the RMS error is 3.7578 × 10−7, which is one order of magnitude better than Fabry's result. Finally, to translate these gains into potential SNR savings, the result from the proposed algorithm operating at an SNR of 45 dB is shown in Figure 16b for the case of two radars. In this case, the RMS error is 2.0314 × 10−7, which is still much better than that of Fabry's original technique.

5.5. Operation at Lower SNRs

[49] Per the numerical results described to this point, the techniques developed in this paper, like the original Fabry technique, are intended for operation at high effective SNRs. In this section, algorithm performance at lower SNRs is investigated to determine the SNR below which the algorithms are no longer able to accurately characterize the RI field. Figure 17 shows the estimated field recovered from the returns of two radars at 35 dB (e.g., 100 pulses at 15 dB/pulse), 30 dB, and 25 dB. The corresponding RMS errors for such cases are 4.2986 × 10−7 (0.42986 N unit), 4.5280 × 10−7 (0.45280 N unit) and 8.8086 × 10−7 (0.88086 N unit), respectively. For the case of an SNR of 25 dB, the fronts can still be seen, but the field is very noisy. At an SNR of 20 dB (not shown), the estimated field is still close to the true field in the sense of RMS error, but the fronts are not recognizable. However, Fabry's technique does not provide reasonable estimates at such low SNRs at all. In particular, the RMS errors of Fabry's technique are 4.8538 × 10−6 (4.8535 N unit) and 5.7840 × 10−6 (5.7840 N unit) when the SNRs are 35 dB and 30 dB, respectively. From Figure 17d, it can be seen that at an SNR of 30 dB, Fabry's technique is unable to provide an estimate from which the fronts can be recognized.

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Figure 17. Performances at lower SNRs: estimates of the RI field of Figure 8 with the targets shown in Figure 9. (a) Proposed algorithms operating at 35 dB. (b) Proposed algorithms operating at 30 dB. (c) Proposed algorithms operating at 25 dB. (d) Fabry's technique operating at 30 dB. In Figures 17a–17c, two radars are employed, one located in the top left corner and the other in the bottom right corner.

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6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[50] In this paper, algorithms have been proposed to recover the surface layer refractive index field from phase measurements of surface clutter targets collected by multiple radars. The techniques leverage the work of Fabry et al. [1997] and Fabry and Petter [2002], who exploits the relation of the observed radar echo phase to the path integrated refractive index between radar and target. A discrete mathematical model has been derived for the problem which allows for the prescription of algorithms with improved performance and a natural extension to multiple radars. In particular, a modified least squares estimation method, which can be employed with either a single radar or multiple radars, is derived. An algorithm that jointly unwraps phase measurements and estimates the refractive field is also presented.

[51] Numerical simulations indicate the performance improvements provided by these algorithms. First, the modified least squares result built on the linear model greatly outperforms previous estimation algorithms for the cases of one radar or multiple radars. Second, the joint phase unwrapping and refractive field estimation algorithm is effective in providing the refractive field for systems operation at higher frequencies, where other algorithms fail. Finally, simulations confirm that multiple views of the same geographical area with multiple radars can provide significant performance improvements.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References

[52] The authors are indebted to F. Fabry for his instructive papers and discussion. This work was supported primarily by the Engineering Research Centers Program of the National Science Foundation under NSF award 0313747 to the Center for Collaborative Adaptive Sensing of the Atmosphere (CASA). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. System Model
  5. 3. Modified Least Squares Estimation
  6. 4. Joint Phase Unwrapping and RI Field Estimation
  7. 5. Numerical Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  • Bean, B. R., and E. J. Dutton (1968), Radio Meteorology, Natl. Bur. Stand. Monogr., vol. 92, 435 pp., Natl. Bur. of Stand., Gaithersburg, Md.
  • Fabry, F., and C. R. Petter (2002), A primer to the interpretation of refractivity imagery during IHOP2002, paper presented at International H2O Project 2002, Natl. Sci. Found., Southern Great Plains, Okla.
  • Fabry, F., C. Frush, I. Zawadzki, and A. Kilambi (1997), On the extraction of near-surface index of refraction using radar phase measurements from ground targets, J. Atmos. Oceanic Technol., 14(4), 978987.
  • Herraez, M. A., D. R. Burton, and D. B. Clegg (1996), Robust, simple, and fast algorithm for phase unwrapping, Appl. Opt., 35, 58475852.
  • Herraez, M. A., D. R. Burton, M. Lalor, and U. Gdeisat (2002), Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path, Appl. Opt., 41, 74377444.
  • Scharf, L. L. (1990), Statistical Signal Processing: Detection, Estimation and Time Series Analysis, Addison-Wesley, Boston, Mass.