### Abstract

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

- Top of page
- Abstract
- 1. Introduction
- 2. System Model
- 3. Modified Least Squares Estimation
- 4. Joint Phase Unwrapping and RI Field Estimation
- 5. Numerical Results
- 6. Conclusions
- Acknowledgments
- 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)10^{6}, is strongly affected by temperature, humidity, and atmospheric pressure. At microwave frequencies, refractivity is commonly approximated by *Bean and Dutton* [1968]:

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*,

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 *t*_{0}. 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 *t*_{1} and *t*_{0} via a differential phase measurement between those times:

where Δ_{t} 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*, *t*_{0}) 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:

where Δ_{T} indicates the differential between two coradial targets, Δ*r* is the distance between the two targets, and Δ*n* 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

- Top of page
- Abstract
- 1. Introduction
- 2. System Model
- 3. Modified Least Squares Estimation
- 4. Joint Phase Unwrapping and RI Field Estimation
- 5. Numerical Results
- 6. Conclusions
- Acknowledgments
- 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 *m*th radar, {*z*_{n}^{m}}, are calculated, per (3), where *z*_{n}^{m} corresponds to the phase difference of the *n*th target, 1 ≤ *m* ≤ *K*, 1 ≤ *n* ≤ *J*. Then, all of the phase differences combined form the *JK* by 1 observation vector **y**,

Let *y*_{i} denote the *i*th 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.

[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 *m*th 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

where *x*_{j} is the change in the RI value of the *j*th subsquare between the reference time and the time of interest, **x** is the *M*^{2} by 1 RI field vector with elements {*x*_{j}}, *p*_{n,j}^{m} is the length of the path between the *m*th radar and the *n*th target in the *j*th subsquare, 0 ≤ *p*_{n,j}^{m} ≤ δ, δ is the border length of each subsquare, **H**^{m} is a *J* by *M*^{2} matrix with elements {**H**_{n,j}^{m} ≜ *p*_{n,j}^{m}}. Note that *p*_{n,j}^{m} 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 *m*th radar to the *n*th 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.

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

where **H** is a *JK* by *M*^{2} 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

- Top of page
- Abstract
- 1. Introduction
- 2. System Model
- 3. Modified Least Squares Estimation
- 4. Joint Phase Unwrapping and RI Field Estimation
- 5. Numerical Results
- 6. Conclusions
- Acknowledgments
- 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 that minimizes the squared error between the observed vector **y** and what would have been observed under without noise; that is,

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

where (**H**^{T}**H**)^{−1}**H**^{T} 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.

[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 . 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 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

where

where {*f*_{k,l}} is a reindexed version of {*x*_{j}} with respect to its spatial position (*k*, *l*):

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.

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

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

[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**:

where U is a *JK* by *JK* unitary matrix, V is an *M*^{2} by *M*^{2} unitary matrix and S is a *JK* by *M*^{2} matrix with (*S*)_{i,j} = 0, *i* ≠ *j*, and (*S*)_{i,i} = σ_{i}, where {σ_{i}} are known as the singular values of **H** and are arranged in decreasing order: σ_{1} ≥ σ_{2} ≥ … ≥ σ_{M2} ≥ 0. In general, because of the virtual rank deficiency of **H**, σ_{i} 1, for *i* > *I*, and σ_{I+1}^{2} + σ_{I+2}^{2} + … + σ_{M2}^{2} ≤ ^{2}, where ε is a small number and I is the numerical rank of **H** as defined below. Writing *S*_{I} = *diag*{σ_{1}, …, σ_{I}, 0, 0, …, 0} and

It is easy to see that

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 **H**_{I}, and

Let *V*^{T}**x** ≜ **v** = , where **v**_{1} is an *I* by 1 vector and **v**_{2} is an (*M*^{2} − *I*) by 1 vector; thus

[22] From (17), **v**_{1} can be calculated, and **v**_{2}, 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

Each element of **x** is a linear combination of **v**_{2} and clearly this provides a solution space with a dimension of (*M*^{2} − *I*).

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

where matrices *E*, *F* and *G* are determined by *V*. *E* is a positive definite matrix. Clearly this is a concave function of **v**_{2}. By differentiating the right-hand side of (19), **v**_{2} can be optimized:

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

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

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

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 *y*_{i}. 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 *L*′_{i} is assigned to such a group, and an additional modification {0, +2π or −2π} is applied on each measurement *y*_{i} according to the relative measurement differences. This results in the modified measurement *y*″_{i}. 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 *y*″_{i} equals *y*_{i} 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

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,

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 {*y*_{1}, *y*_{2}} is less than the threshold. So these two measurements are unwrapped relative to each other and merged into one group. A group ambiguity number *L*′_{1} is then assigned to this group. According to the relative difference, *y*″_{1} = *y*_{1} and *y*″_{2} = *y*_{2} + 2π. Similarly, {*y*_{4}, *y*_{5}, *y*_{6}} are merged into one group, a group ambiguity number *L*′_{3} is assigned to this group and *y*″_{4} = *y*_{4}, *y*_{5}″ = *y*_{5} and *y*_{6}″ = *y*_{6} + 2π. The measurement *y*_{3} fails the reliability test. Thus an individual group ambiguity number is assigned to it. So, after the second step, (22) is

and

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

[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.