Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

TL;DR

This paper analyzes the mathematical properties of SAR imaging operators with moving transmitter and receiver at different speeds, revealing artifact behaviors and bifurcations in image quality.

Contribution

It classifies the forward operator as a Fourier integral operator with singularities and characterizes the normal operator's structure in different motion scenarios.

Findings

Artifacts are as strong as the true image in both cases.

Normal operator belongs to specific classes of Fourier integral operators.

A bifurcation in artifact types occurs when source and receiver move in opposite directions.

Abstract

In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when the transmitter and receiver are both moving with different speeds along a single line parallel to the ground in the same direction or in the opposite directions. In both cases, we classify the forward operator $\Fc$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\Fc^*\Fc$ in both cases (where $\Fc^{*}$ is the $L^{2}$-adjoint of $\Fc$). When the transmitter and receiver move in the same direction, we prove that $\Fc^*\Fc$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (\Delta, C_1)$. When they move in opposite directions, $\Fc^*\Fc$ is a sum of such operators. In both cases artifacts appear and we show that they are, in general, as strong as the bona-fide part of the image. Moreover, we demonstrate that as soon as the source and receiver start to move in opposite directions, there is an interesting bifurcation in the type of artifact that appears in the image.

Equations

$${\gamma }_T(s)=(\alpha s,0,h){\gamma }_R(s)=(s,0,h)$$

$$\int e^{-\mathrm{i}\omega \left(t-\frac{1}{c_0}R(s,x)\right)}{a}_0(s,x,\omega )V(x)\mathrm{d}x\mathrm{d}\omega$$

$$R(s,x)=\Vert {\gamma }_T(s)-x\Vert +\Vert x-{\gamma }_R(s)\Vert$$

$$a_0(s,x,\omega )=\frac{{\omega }^{2}p(\omega )}{16{\pi }^2\Vert {\gamma }_T(s)-x\Vert \Vert {\gamma }_R(s)-x\Vert }.$$

$$t<\sqrt{(\alpha -1{)}^2{s}^2+4h^2},$$

$$\left\{(s,t):s>0,0<t\le \sqrt{(\alpha -1{)}^2{s}^2+4h^2}\right\}.$$

$$\{(s,t):s>0,\sqrt{(\alpha -1{)}^2{s}^2+4h^2}<t<\infty \}.$$

$$\mathcal{F}V(s,t):=\int e^{-i\omega \left(t-\Vert x-{\gamma }_T(s)\Vert -\Vert x-{\gamma }_R(s)\Vert \right)}a(s,t,x,\omega )V(x)\mathrm{d}x\mathrm{d}\omega .$$

$$A$$

$$B$$

$$E(s,t)=\left\{x\in {\mathbb{R}}^2:A(s,x)+B(s,x)=t\right\}$$

$$|{\partial }_{\omega }^{\delta }{\partial }_s^{{\beta }_1}{\partial }_t^{{\beta }_2}{\partial }_x^{\lambda }a(s,t,x,\omega )|\le c(1+|\omega |{)}^{2-\delta }.$$

$$\displaystyle\mathcal{C}_{\mathcal{F}}=\Bigg{\{}$$

$$\displaystyle x_{1},x_{2},\omega\left(\frac{x_{1}-\alpha s}{A}+\frac{x_{1}-s}{B}\right),\omega\left(\frac{x_{2}}{A}+\frac{x_{2}}{B}\right)\Big{)}$$

$$\displaystyle\,:\,\omega\neq 0,\ t=A+B\Bigg{\}}.$$

$${\Sigma }_1=\left\{(s,x_1,x_2,\omega )\in {\mathcal{C}}_{\mathcal{F}}:x_2=0\right\}.$$

$$C_1=\{(x_1,x_2,{\xi }_1,{\xi }_2;x_1,-x_2,{\xi }_1,-{\xi }_2):(x,\xi )\in T^{\ast }X\setminus 0\}$$

$$x_1^{+}=\frac{2\alpha s}{\alpha +1}+\sqrt{-\alpha \frac{(\alpha -1{)}^2}{(\alpha +1{)}^2}{s}^2-h^2},$$

$$x_1^{-}=\frac{2\alpha s}{\alpha +1}-\sqrt{-\alpha \frac{(\alpha -1{)}^2}{(\alpha +1{)}^2}{s}^2-h^2}$$

$$t_s^{\pm }=A\left(s,(x_1^{\pm },0)\right)+B\left(s,(x_1^{\pm },0)\right)$$

$$\mathcal{G}V(s,t):=\int e^{-i\omega \left(t-\Vert x-{\gamma }_T(s)\Vert -\Vert x-{\gamma }_R(s)\Vert \right)}a(s,t,x,\omega )V(x)\mathrm{d}x\mathrm{d}\omega ,$$

$${\Sigma }_2$$

$$\displaystyle=\Bigg{\{}(s,x,\omega)\in\mathcal{C}_{\mathcal{G}}\,:\,\left(x_{1}-\frac{2\alpha s}{\alpha+1}\right)^{2}+x_{2}^{2}=-\alpha s^{2}\frac{(\alpha-1)^{2}}{(\alpha+1)^{2}}-h^{2},\ x_{2}\neq 0\Bigg{\}}$$

$${\Sigma }_{2,X}(s)=\left\{x:\frac{\alpha }{A^2(s,x)}+\frac{1}{B^2(s,x)}=0,x_2\ne 0\right\}.$$

$$s_0=\frac{h(\alpha +1)}{\sqrt{-\alpha }(\alpha -1)}$$

$${\mathcal{G}}^{\ast }\mathcal{G}\in I^{3,0}(\Delta ,C_1)+I^{3,0}(\Delta ,C_2)+I^{3,0}(C_1,C_2)$$

$$|{\partial }_z^{\alpha }{\partial }_{\xi }^{\beta }{\partial }_{\sigma }^{\delta }a(z,\xi ,\sigma )|\le c_{K,\alpha ,\beta }(1+|\xi |{)}^{p-|\beta |}(1+|\sigma |{)}^{l-|\delta |}$$

$$u_2(x,y)=\int e^{i((x^{\prime }-y^{\prime })\cdot {\xi }^{\prime }+(x_n-y_n-s)\cdot {\xi }_n+s\cdot \sigma )}a(x,y,s;\xi ,\sigma )d\xi d\sigma ds$$

$$\mathcal{T}V(s,t):=\int e^{-i\omega \left(t-\Vert x-{\gamma }_T(s)\Vert -\Vert x-{\gamma }_R(s)\Vert \right)}a(s,t,x,\omega )V(x)\mathrm{d}x\mathrm{d}\omega ,$$

$${\gamma }_T(s)=\sqrt{(x_1+s{)}^2+x_2^2+h^2}\text{mbox}and{\gamma }_R(s)=\sqrt{(x_1-s{)}^2+x_2^2+h^2}.$$

$${\mathcal{T}}^{\ast }\mathcal{T}\in I^{2m,0}(\Delta ,{\Lambda }_1)+I^{2m,0}(\Delta ,{\Lambda }_2)+I^{2m,0}({\Lambda }_1,{\Lambda }_3)+I^{2m,0}({\Lambda }_2,{\Lambda }_3).$$

$${\Lambda }_2=\{(x_1,x_2,{\xi }_1,{\xi }_2;-x_1,x_2,-{\xi }_1,{\xi }_2):(x,\xi )\in T^{\ast }X\setminus 0\}$$

$${\Lambda }_3=\{(x_1,x_2,{\xi }_1,{\xi }_2;-x_1,-x_2,-{\xi }_1,-{\xi }_2):(x,\xi )\in T^{\ast }X\setminus 0\}.$$

Full text

{AMS}

Primary 35S30, 35R30; Secondary 50J40

1 Introduction

Synthetic Aperture Radar (SAR) is a high-resolution imaging technology that uses antennas on moving platforms to send electromagnetic waves to objects of interest and measures the scattered echoes. These are then processed to form an image of the objects. For a good overview of SAR imaging, especially from a mathematical point of view, we refer the reader to [6, 7]. In monostatic SAR imaging, the moving transmitter also acts as a receiver, whereas in bistatic SAR imaging, the transmitter and receiver are located on different platforms.

Our focus in this article is on a bistatic SAR imaging setup, where the transmitter and receiver move along a straight line parallel to the ground, in the same direction or in the opposite directions, and with different speeds (see (2.1)). Here and in the rest of the article we assume that the ground is represented by a plane. The SAR imaging task is then mathematically equivalent to recovering the ground reflectivity function $V$ from the measured data $\mathcal{F}V$ for a certain period of time along each point of the receiver trajectory. Since exact reconstruction of $V$ from $\mathcal{F}V$ is an extremely difficult task, a reasonable compromise (acceptable in practice for most applications) is to find the singularities of $V$ from $\mathcal{F}V$. In particular, this will allow to see the edges (and hence shapes) of the objects on the ground. Unfortunately even that simpler task may not be completely accomplishable, since in certain setups $\mathcal{F}V$ may not have enough information for correct recovery of singularities of $V$. In these cases, the best possible reconstruction of $V$ may miss certain parts of the original singularities, or have added “fake” singularities, called artifacts.

In this article, the reconstructed images, including artifacts, are analyzed using the calculus of singular Fourier integral operators (FIOs). The forward operator $\mathcal{F}$ which maps singularities in the scene to those in the data is an FIO. It is conventional to reconstruct the image of an object by using the backprojection operator, $\mathcal{F}^{*}$ applied to the data $\mathcal{F}V$. We study the normal operator $\mathcal{F}^{*}\mathcal{F}$ and the artifacts which appear by using this method.

The current article is a continuation of our prior work [2], where, motivated by certain multiple scattering scenarios, we considered the case when the transmitter and receiver move at equal speeds away from a common midpoint along a straight line. The main result of that article made precise the added singularities and their strengths (in comparison to the true singularities) when reconstruction is done using the backprojection method mentioned above. We showed that the backprojection method introduces three additional singularities for each true singularity with potentially no way of avoiding them if the transmitter and receiver are assumed omnidirectional. One of our main motivations in studying the case of different speeds for the transmitter and receiver was to remove some of these artifacts. However, when the transmitter and receiver move away from each other, the backprojection method still introduces additional artifacts, which in the limiting case (when the speeds are equal) gives the artifacts considered in [2].

The microlocal analysis of the normal operator in the study of generalized Radon transforms and in imaging problems has a long history. Guillemin and Sternberg were the first to study integral geometry problems from the FIO and microlocal analysis point of view, and made fundamental contributions [21, 20]. Later, paired Lagrangian calculus introduced by Melrose-Uhlmann [27] and Guillemin-Uhlmann [22], and also studied in Antoniano-Uhlmann [4] was used by Greenleaf-Uhlmann in several of their highly influential works on the study of generalized Radon transforms [17, 18]. Microlocal techniques have also been very useful in the context of seismic imaging [5, 31, 38, 29, 37, 8, 12]), in sonar imaging see [11, 13, 33]), in X-ray Tomography; in addition to works mentioned above also see [32, 24, 15, 14, 16]), and in tensor tomography [34, 35, 39].

The microlocal analysis of linearized SAR imaging operators (both monostatic and bistatic) was done in [30, 40, 10, 2, 36]. Bistatic SAR imaging problems, due to the fact that the transmitter and receiver are spatially separated, naturally lead to the study of elliptical Radon transforms, which are also of independent interest. These have been studied in the literature as well [3, 1, 28].

The article is organized as follows: In Section 2 we state the main facts and results: the positions of the transmitter and receiver that we consider (2.1), the forward operator $\mathcal{F}$ (2.2), the canonical relation of $\mathcal{F}$ and the properties of the projections from its canonical relation ($\pi_{L}$ and $\pi_{R}$) (Theorems 2.1 and 2.4). Then, we describe the composition calculus results (Theorems 2.2 and 2.6).

In Section 3 we recall briefly the definition of the fold/blowdown singularities and the properties of the $I^{p,l}$ classes we need in this article, and Section 4 briefly summarizes the main result of [2].

In Section 5 we consider the case when the transmitter and receiver are moving in the same direction along a line parallel to the ground. We show that the normal operator $\mathcal{F}^{*}\mathcal{F}$ has a distribution kernel belonging to the paired Lagrangian distribution class $I^{2m,0}(\Delta,C_{1})$ where $\Delta$ is the diagonal relation and $C_{1}$ is the graph of a simple reflection map about the $x$-axis. This result is valid even if the transmitter is stationary, for example, when the transmitter is a fixed radio tower and the receiver is a drone.

In Section 6 we consider the case when the transmitter and receiver move in opposite directions along the line, and the analysis becomes considerably more complicated. To distinguish this case from the case in Section 5, we denote the forward operator by $\mathcal{G}$. First of all, the projections drop rank by one along two smooth disjoint hypersurfaces $\Sigma_{1}\cup\Sigma_{2}$. Then, Theorem 2.6 shows that the backprojection adds two sets of artifacts and the normal operator $\mathcal{G}^{*}\mathcal{G}$ is a sum of operators belonging to $I^{p,l}$ classes: $I^{2m,0}(\Delta,C_{1})+I^{2m,0}(\Delta,C_{2})+I^{2m,0}(C_{1},C_{2})$, where $C_{1}$ causes the same artifact which appears for $\alpha\geq 0$ in Section 5 and $C_{2}$ is a two-sided fold (Def. 3.2). Finally, in section 6.4 we consider spotlighting, in which certain portions of the ground are selectively illuminated. In this case, we show that the normal operator $\mathcal{G}^{*}\mathcal{G}$ belongs to $I^{2m,0}(\Delta,C_{2})$ where $C_{2}$ is a two-sided fold canonical relation (see Theorem 6.15).

In Appendix A, we prove Theorem 2.2 using the iterated regularity method and in Appendix B, we give a geometric explanation of the points we cannot image in the case considered in Section 6.

In all these situations, we show that additional artifacts (coming from $C_{1}$ and $C_{2}$) could be just as strong as the bona-fide part of the image, in other words, singularities related to $\Delta$. In this article, for $\mathcal{F}^{*}\mathcal{F}\in I^{p,l}(\Delta,C)$, the strength of the artifact ($C$) means the order of $\mathcal{F}^{*}\mathcal{F}$ on $C\setminus\Delta$. We find the order of $\mathcal{F}^{*}\mathcal{F}$ on both $\Delta\setminus C$ and $C\setminus\Delta$ and, if they are the same, then we conclude that in general, the artifact is as strong (see Remarks 2.3 and 2.7).

An obvious but perhaps important observation that follows from the results of this paper is that if one has a choice of having the source and receiver platforms moving in the same or opposite directions along the same straight line, then it is highly preferable to have them move in the same direction in order to avoid additional set of artifacts, other than the usual left-right ambiguities.

2 Statements of the main results

2.1 The linearized scattering model

For simplicity, we assume that both the transmitter and receiver are at the same height $h>0$ above the ground at all times and that the transmitter and receiver move at constant but different speeds along a line parallel to the $x$ axis. Let

[TABLE]

for $s\in(0,\infty)$ be the trajectories of the transmitter and receiver respectively.

The case $\alpha=-1$ corresponds to the common midpoint problem, which was fully analyzed in [2]. Therefore we will assume $\alpha\neq-1$. We also assume $\alpha\neq 1$, since $\alpha=1$ corresponds to the monostatic case (where the same device serves as both a transmitter and a receiver) and has also been fully analyzed in earlier works [30, 10].

We are aware that there are other cases for the transmitter and receiver to be considered, like moving along parallel lines at different heights or along skew lines at different speeds or along intersecting lines in a plane parallel to the ground. At this point we can only say that in those cases the left-right ambiguity which appears in the case considered in this article will, in general, be lost. However, we will limit the analysis of $\mathcal{F}$ and $\mathcal{F}^{*}\mathcal{F}$ only to the case mentioned in (2.1) since it is already leads to interesting analysis. We point out that, in practice, the flight paths can be more complicated because of turbulence and other factors.

The linearized model for the scattered signal we will use in this article is

[TABLE]

for $(s,t)\in(0,\infty)\times(0,\infty)$, where $V(x)=V(x_{1},x_{2})$ is the function modeling the object on the ground, and

[TABLE]

is the bistatic distance–the sum of the distance from the transmitter to the scatterer and from the scatterer to the receiver, $c_{0}$ is the speed of electromagnetic wave in free-space and the amplitude term $a_{0}$ is given by

[TABLE]

This function includes terms that take into account the transmitted waveform and geometric spreading factors.

From now on, we denote the $(s,t)$ space by $Y=(0,\infty)^{2}$ and the $(x_{1},x_{2})$ space by $X=\mathbb{R}^{2}$.

For simplicity, we will assume that $c_{0}=1$. Because the ellipsoidal wavefronts do not meet the ground for

[TABLE]

there is no signal for such $t$. As we will see, our method cannot image the point on the ground directly “between” the transmitter and receiver (see the proof of Theorem 2.1 in Section 5). Given transmitter and receiver positions $\alpha s$ and $s$ respectively, such a point on the ground has coordinates $\left(\frac{(\alpha+1)s}{2},0\right)$. Note that this point on the $x$-axis corresponds to $t=\sqrt{(\alpha-1)^{2}s^{2}+4h^{2}}$. For these two reasons, we multiply $a_{0}$ by a cutoff function $f$ that is zero in a neighborhood of

[TABLE]

In addition, to be able to compose our forward operator and its adjoint, we further assume that $f$ is compactly supported and equal to $1$ in a neighborhood of a suitably large compact subset of

[TABLE]

We let $f\cdot a_{0}=a$, and this gives us the data

[TABLE]

We require additional cutoffs for our analysis to work for the case of $\alpha<0$ (see Remarks 2.5 and 6.6).

Throughout the article we use the following notation

[TABLE]

and we define the ellipse

[TABLE]

We assume that the amplitude function $a\in S^{2}$, that is, it satisfies the following estimate: For every compact $K\subset Y\times X$ and for every non-negative integer $\delta$ and for every $2$-index $\beta=(\beta_{1},\beta_{2})$ and $\lambda$, there is a constant $c$ such that

[TABLE]

This assumption is satisfied if the transmitted waveform from the antenna is approximately a Dirac delta distribution. The qualitative features predicted by the approach based on microlocal analysis are consistent with practical reconstructions, including for example, the well-known right-left ambiguity artifact in low-frequency SAR images [7].

2.2 Transmitter and receiver moving in the same direction:

$\alpha\geq 0$

The case $\alpha\geq 0$ corresponds to the situation when the transmitter and receiver are traveling in the same direction along a line parallel to the ground or when the transmitter is stationary ($\alpha=0$) on that line. For $\alpha\geq 0$, we refer to the forward operator by $\mathcal{F}$. We show that for the case $\alpha\geq 0$, the operator $\mathcal{F}$ in (2.2) is a FIO of order $\frac{3}{2}$ and study the properties of the natural projection maps from the canonical relation of $\mathcal{F}$. We have the following results.

Theorem 2.1**.**

Let $\mathcal{F}$ be the operator in (2.2) for $\alpha\geq 0$.

$\mathcal{F}$* is an FIO of order $3/2$.* 2. 2.

The canonical relation $\mathcal{C}_{\mathcal{F}}\subset T^{*}Y\setminus\mathbf{0}\times T^{*}X\setminus\mathbf{0}$ associated to $\mathcal{F}$ is given by

[TABLE]

where $A=A(s,x)$ and $B=B(s,x)$ are defined in (2.3). Furthermore, $(s,x_{1},x_{2},\omega)$ is a global parameterization of $\mathcal{C}_{\mathcal{F}}$. 3. 3.

Denote the left and right projections from $\mathcal{C}_{\mathcal{F}}$ to $T^{*}Y\setminus\mathbf{0}$ and $T^{*}X\setminus\mathbf{0}$ by $\pi_{L}$ and $\pi_{R}$ respectively. Then $\pi_{L}$ and $\pi_{R}$ drop rank simply by one on the set

[TABLE] 4. 4.

$\pi_{L}$* has a fold singularity along $\Sigma_{1}$ and $\pi_{R}$ has a blowdown singularity along $\Sigma_{1}$ (see Def. 3.1).*

We next analyze the imaging operator $\mathcal{F}^{*}\mathcal{F}$.

Theorem 2.2**.**

Let $\mathcal{F}$ be as in Theorem 2.1. Then $\mathcal{F}^{*}\mathcal{F}\in I^{3,0}(\Delta,C_{1})$, where

[TABLE]

*which is the graph of $\chi_{1}(x,\xi)=(x_{1},-x_{2},\xi_{1},-\xi_{2})$. *

Remark 2.3**.**

*Since $\mathcal{F}^{*}\mathcal{F}\in I^{3,0}(\Delta,C_{1})$, using the properties of the $I^{p,l}$ classes [22], we have that microlocally away from $C_{1}$, $\mathcal{F}^{*}\mathcal{F}$ is in $I^{3}(\Delta\setminus C_{1})$ and microlocally away from $\Delta$, $\mathcal{F}^{*}\mathcal{F}\in I^{3}(C_{1}\setminus\Delta)$. This means that $\mathcal{F}^{*}\mathcal{F}$ has the same order on both $\Delta\setminus C_{1}$ and $C_{1}\setminus\Delta$ which implies that the artifacts caused by $C_{1}\setminus\Delta$ will, in general, have the same order as the reconstruction of the singularities in $V$ that cause them (see the comments below Def. 3.7). However, more complicated behavior can occur including smoothing or cancellation of artifacts. *

2.3 Transmitter and receiver moving in opposite directions:

$\alpha<0$

When $\alpha<0$, the transmitter and receiver travel away from each other, and we refer to the forward operator by $\mathcal{G}$.

2.3.1 Further preliminary modifications of the scattered

data

In the case when $\alpha<0$, we further modify the operator $\mathcal{F}V$ considered in Section 2.1.

Our method cannot image a neighborhood of two points on the ground for a given transmitter and receiver positions in addition to the points muted by the cutoff function $f$ in Section 2.1. Therefore we modify or pre-process the receiver data further such that the contribution to it from a neighborhood of these two points is [math]. The two points on the $x_{1}$-axis that we would like to avoid are of the form $\left(x_{1}^{\pm},0\right)$, where

[TABLE]

as explained in Remark 2.5. We define a smooth mute function $g(s,t)$ that is identically [math] if the ellipse $E(s,t)$ is near one of the points $(x_{1}^{\pm},0)$; for each $s$, the corresponding values of $t$ are

[TABLE]

where $A$ and $B$ are given by (2.3). The points $t^{\pm}_{s}$ are given explicitly in Appendix B.

With the function $g$, we modify $\mathcal{F}$ in (2.2) by replacing $a$ by $g\cdot a$ and call it $a$ again. Throughout this section, corresponding to the case $\alpha<0$, we will designate the operator as $\mathcal{G}$. That is, we have

[TABLE]

where $a$ takes into account the cutoff functions $f$ in Section 2.1 and the function $g$ defined in the last paragraph.

Theorem 2.4**.**

Let $\mathcal{G}$ be the operator given in (2.12) for $\alpha<0$. Then

$\mathcal{G}$* is an FIO of order $\frac{3}{2}$* 2. 2.

The canonical relation $\mathcal{C}_{\mathcal{G}}$ associated to $\mathcal{G}$ is given by (2.6) with global parameterization $(s,x_{1},x_{2},\omega)$. 3. 3.

The left and right projections $\pi_{L}$ and $\pi_{R}$ respectively from $\mathcal{C}_{\mathcal{G}}$ drop rank simply by one on the set $\Sigma=\Sigma_{1}\cup\Sigma_{2}$ where $\Sigma_{1}$ is given by (2.7) and

[TABLE] 4. 4.

$\pi_{L}$* has a fold singularity along $\Sigma$ (see Def. 3.1).* 5. 5.

$\pi_{R}$* has a blowdown singularity along $\Sigma_{1}$ and a fold singularity along $\Sigma_{2}$ (see Def. 3.1).*

For convenience, we denote, for each $s$, the projection of the part of $\Sigma_{2}$ above $s$ to $\mathbb{R}^{2}$ (the projection to the base of \pi_{R}\left({{\Sigma_{2}}{\big{|}_{s}}}\right)) by $\Sigma_{2,X}(s)$, and this is the circle described in (2.14) and in Appendix B. It can be written

[TABLE]

Remark 2.5**.**

From Equation (2.14) we have that $\Sigma_{2,X}(s)$ is a circle of radius $\sqrt{-\alpha s^{2}\frac{(\alpha-1)^{2}}{(\alpha+1)^{2}}-h^{2}}$ and centered at $(2\alpha s/(\alpha+1),0)$.

*Now we can explain why we need to cutoff the data for ellipses near the two points given by (2.9)-(2.10). Since $\pi_{R}(\Sigma_{1})$ intersects $\pi_{R}(\Sigma_{2})$ above these two points, $\pi_{R}$ drops rank by two above these points. So, we mute data near $(s,t_{s}^{\pm})$ given by (2.11). We will precisely describe this mute, $g$ in Remark 6.6. *

We now analyze the imaging operator $\mathcal{G}^{*}\mathcal{G}$. Unlike the case $\alpha\geq 0$, this case is more complicated and we consider several restricted transforms.

Theorem 2.6**.**

Let $\alpha\leq 0$ and $\alpha\neq-1$. Let $\mathcal{G}$ be the operator in (2.12) and let

[TABLE]

Then the following hold:

Let $O_{1}=\{(s,t):0<s<s_{0}$ and $0<t<\infty\}$ and let $r_{1}$ be a smooth cutoff function that is compactly supported in $O_{1}$. Consider the operator $\mathcal{G}$ defined in (2.12) with the amplitude function $a$ replaced by $r_{1}\cdot a$. Then $\mathcal{G}^{*}\mathcal{G}\in I^{3,0}(\Delta,C_{1})$ where $C_{1}$ is defined in (2.8). 2. 2.

Let $O_{2}=\{(s,t):s_{0}<s<\infty$ and $t_{s}^{-}<t<t_{s}^{+}\}$ where $t_{s}^{\pm}$ is defined in (2.11). Let $r_{2}$ be a smooth cutoff function and compactly supported in $O_{2}$. Consider the operator $\mathcal{G}$ defined in (2.12) with the amplitude function $a$ replaced by $r_{2}\cdot a$. Then $\mathcal{G}^{*}\mathcal{G}\in I^{3,0}(\Delta,C_{1})+I^{3,0}(\Delta,C_{2})+I^{3,0}(C_{1},C_{2})$ where $C_{2}$ is a two-sided fold given by (6.1). 3. 3.

Let $O_{3}=\{(s,t):s_{0}<s<\infty$ and $t<t_{s}^{-}$ or $t>t_{s}^{+}\}$ with $t_{s}^{\pm}$ defined in (2.11). Let $r_{3}$ be a smooth cutoff function compactly supported in $O_{3}$. Consider the operator $\mathcal{G}$ defined in (2.12) with the amplitude function $a$ replaced by $r_{3}\cdot a$. Then $\mathcal{G}^{*}\mathcal{G}\in I^{3,0}(\Delta,C_{1})$.

Remark 2.7**.**

Using the properties of the $I^{p,l}$ classes for case 2 of the theorem,

[TABLE]

*implies that artifacts in the reconstruction could show up because of $C_{1}$ (reflection in the $x_{1}$ axis) and because of $C_{2}$ (a 2-sided fold). Furthermore, from the discussion below Definition 3.8, we have that $\mathcal{G}^{*}\mathcal{G}\in I^{3}(C_{i}\setminus\Delta),\ i=1,2$, $\mathcal{G}^{*}\mathcal{G}\in I^{3}(C_{i}\setminus C_{j}),\ i,j=1,2,i\neq j$ and $\mathcal{G}^{*}\mathcal{G}\in I^{3}(\Delta\setminus C_{i}),\ i=1,2$ and thus these artifacts could be, in general, as strong as the bona-fide part of the image (corresponding to $I^{3}(\Delta)$). *

3 Preliminaries: Singularities and $I^{p,l}$

classes

Here we introduce the classes of distributions and singular FIO we will use to analyze the forward operators $\mathcal{F}$ and $\mathcal{G}$ and the normal operators $\mathcal{F}^{*}\mathcal{F}$ and $\mathcal{G}^{*}\mathcal{G}$.

Definition 3.1**.**

[19]** Let $M$ and $N$ be manifolds of dimension $n$ and let $f:M\to N$ be $C^{\infty}$. Define $\Sigma=\{m\in M\,:\,\det(df)_{m}=0\}$.

$f$* drops rank by one simply on $\Sigma$ if for each $m_{0}\in\Sigma$, *

rank $(df)_{m_{0}}=n-1$ and $d(\det(df)_{m_{0}})\neq 0$. 2. 2.

$f$* has a Whitney fold along $\Sigma$ if $f$ is a local diffeomorphism away from $\Sigma$ and $f$ drops rank by one simply on $\Sigma$, so that $\Sigma$ is a smooth hypersurface and $\ker\>(df)_{m_{0}}\not\subset T_{m_{0}}\Sigma$ for every $m_{0}\in\Sigma$.* 3. 3.

$f$* is a blowdown along $\Sigma$ if $f$ is a local diffeomorphism away from $\Sigma$ and $f$ drops rank by one simply on $\Sigma$, so that $\Sigma$ is a smooth hypersurface and $\ker(df)_{m_{0}}\subset T_{m_{0}}(\Sigma)$ for every $m_{0}\in\Sigma$.*

Definition 3.2** ([26]).**

*A smooth canonical relation $C$ for which both projections $\pi_{L}$ and $\pi_{R}$ have only (Whitney) fold singularities, is called a two-sided fold or a folding canonical relation. *

This notion was first introduced by Melrose and Taylor [26], who showed the existence of a normal form in $T^{*}{\mathbb{R}}^{n}\times T^{*}{\mathbb{R}}^{n}$.

Theorem 3.3** ([26]).**

*If dim  $X=n$ dim  $Y=n$ and $C\subset(T^{*}X\setminus\mathbf{0})\times(T^{*}Y\setminus\mathbf{0})$ is a two-sided fold, then microlocally there are homogeneous canonical transformations, $\chi_{1}:T^{*}X\to T^{*}{\mathbb{R}}^{n}$ and $\chi_{2}:T^{*}Y\to T^{*}{\mathbb{R}}^{n}$, such that $(\chi_{1}\times\chi_{2})(C)\subseteq C_{0},$ near $\xi_{2}\neq 0$ where, $C_{0}=N^{*}\{x_{2}-y_{2}=(x_{1}-y_{1})^{3};\ x_{i}=y_{i},\ 3\leq i\leq n\}$. *

We now define $I^{p,l}$ classes. They were first introduced by Melrose and Uhlmann [27], Guillemin and Uhlmann [22] and Greenleaf and Uhlmann [18] and they have been used in the study of SAR imaging [30, 10, 25, 2].

Definition 3.4**.**

*Two submanifolds $M$ and $N$ intersect cleanly if $M\cap N$ is a smooth submanifold and $T(M\cap N)=TM\cap TN$. *

Consider two spaces $X$ and $Y$ and let $\Lambda_{0}$ and $\Lambda_{1}$ and $\tilde{\Lambda}_{0}$ and $\tilde{\Lambda}_{1}$ be Lagrangian submanifolds of the product space $T^{*}X\times T^{*}Y$. If they intersect cleanly, $(\tilde{\Lambda}_{0},\tilde{\Lambda}_{1})$ and $(\Lambda_{0},\Lambda_{1})$ are equivalent in the sense that there is, microlocally, a canonical transformation $\chi$ which maps $(\Lambda_{0},\Lambda_{1})$ into $(\tilde{\Lambda}_{0},\tilde{\Lambda}_{1})$ and $\chi(\Lambda_{0}\cap\Lambda_{1})=(\tilde{\Lambda}_{0}\cap\tilde{\Lambda}_{1})$. This leads us to the following model case.

Example 3.5**.**

*Let $\tilde{\Lambda}_{0}=\Delta_{T^{*}\mathbb{R}^{n}}=\{(x,\xi;x,\xi)\,:\,x\in\mathbb{R}^{n},\ \xi\in\mathbb{R}^{n}\setminus\mathbf{0}\}$ be the diagonal in $T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}$ and let $\tilde{\Lambda}_{1}=\{(x^{\prime},x_{n},\xi^{\prime},0;x^{\prime},y_{n},\xi^{\prime},0)\,:\,x^{\prime}\in\mathbb{R}^{n-1},\ \xi^{\prime}\in\mathbb{R}^{n-1}\setminus\mathbf{0}\}$. Then, $\tilde{\Lambda}_{0}$ intersects $\tilde{\Lambda}_{1}$ cleanly in codimension $1$. *

Now we define the class of product-type symbols $S^{p,l}(m,n,k)$.

Definition 3.6**.**

$S^{p,l}(m,n,k)$* is the set of all functions $a(z;\xi,\sigma)\in C^{\infty}(\mathbb{R}^{m}\times(\mathbb{R}^{n}\setminus\mathbf{0})\times\mathbb{R}^{k})$ such that for every $K\subset\mathbb{R}^{m}$ and every $\alpha\in\mathbb{Z}^{m}_{+},\beta\in\mathbb{Z}^{n}_{+},\delta\in\mathbb{Z}^{k}_{+}$ there is $c_{K,\alpha,\beta}$ such that*

[TABLE]

*for all $(z,\xi,\tau)\in K\times(\mathbb{R}^{n}\setminus\mathbf{0})\times\mathbb{R}^{k}$. *

Since any two sets of cleanly intersecting Lagrangians are equivalent, we first define $I^{p,l}$ classes for the case in Example 3.5.

Definition 3.7** ([22]).**

Let $I^{p,l}(\tilde{\Lambda}_{0},\tilde{\Lambda}_{1})$ be the set of all distributions $u$ such that $u=u_{1}+u_{2}$ with $u_{1}\in C^{\infty}_{0}$ and

[TABLE]

*with $a\in S^{p^{\prime},l^{\prime}}$ where $p^{\prime}=p-\frac{n}{2}+\frac{1}{2}$ and $l^{\prime}=l-\frac{1}{2}$. *

This allows us to define the $I^{p,l}(\Lambda_{0},\Lambda_{1})$ class for any two cleanly intersecting Lagrangians in codimension $1$ using the microlocal equivalence with the case in Example 3.5.

Definition 3.8**.**

[22*]** Let $I^{p,l}(\Lambda_{0},\Lambda_{1})$ be the set of all distributions $u$ such that $u=u_{1}+u_{2}+\sum v_{i}$ where $u_{1}\in I^{p+l}(\Lambda_{0}\setminus\Lambda_{1})$, $u_{2}\in I^{p}(\Lambda_{1}\setminus\Lambda_{0})$, the sum $\sum v_{i}$ is locally finite and $v_{i}=Aw_{i}$ where $A$ is a zero order FIO associated to $\chi^{-1}$, the canonical transformation from above, and $w_{i}\in I^{p,l}(\tilde{\Lambda}_{0},\tilde{\Lambda}_{1})$. *

This class of distributions is invariant under FIOs associated to canonical transformations which map the pair $(\Lambda_{0},\Lambda_{1})$ to itself, whilst also preserving the intersection. By definition, $F\in I^{p,l}(\Lambda_{0},\Lambda_{1})$ if its Schwartz kernel belongs to $I^{p,l}(\Lambda_{0},\Lambda_{1})$. If $F\in I^{p,l}(\Lambda_{0},\Lambda_{1})$ then $F\in I^{p+l}(\Lambda_{0}\setminus\Lambda_{1})$ and $F\in I^{p}(\Lambda_{1}\setminus\Lambda_{0})$ [22]. Here by $F\in I^{p+l}(\Lambda_{0}\setminus\Lambda_{1})$, we mean that the Schwartz kernel of $F$ belongs to $I^{p+l}(\Lambda_{0})$ microlocally away from $\Lambda_{1}$.

To show that a distribution belongs to $I^{p,l}$ class we use the iterated regularity property:

Theorem 3.9** ([18, Proposition 1.35]).**

*If $u\in\mathcal{D}^{\prime}(X\times Y)$ then $u\in I^{p,l}(\Lambda_{0},\Lambda_{1})$ if there is an $s_{0}\in\mathbb{R}$ such that for all first order pseudodifferential operators $P_{i}$ with principal symbols vanishing on $\Lambda_{0}\cup\Lambda_{1}$, we have $P_{1}P_{2}\dots P_{r}u\in H^{s_{0}}_{loc}$. *

In section 6, we will use the following theorem.

Theorem 3.10** ([10, 29]).**

*If $F$ is a FIO of order m whose canonical relation is a two-sided fold then $F^{*}F\in I^{2m,0}(\Delta,\tilde{C})$ where $\tilde{C}$ is another two-sided fold. *

4 Summary of the main result for the case $\alpha=-1$

Recall that in the statement of Theorem 2.6, we assumed that $\alpha\neq-1$. In fact, as already mentioned in the introduction, the case when $\alpha=-1$ in the context of Theorem 2.6 was analyzed in our earlier paper [2], and the results obtained in this work can be considered as a bifurcation of the singularities that appear for the case when $-1\neq\alpha<0$, as $\alpha\to-1$. With this is mind, we state the main result obtained in [2], and briefly explain how our earlier result fits into the framework of the current article.

Let $\mathcal{T}$ denote the operator

[TABLE]

where

[TABLE]

Theorem 4.1**.**

[2]** Let $\mathcal{T}$ be the operator in (4.1). Then the normal operator $\mathcal{T}^{*}\mathcal{T}$ can be decomposed as a sum:

[TABLE]

Here $\Lambda_{1},\Lambda_{2}$ and $\Lambda_{3}$ denote the additional singularities (artifacts) caused due to reflection about the $x_{1}$ axis, reflection about the $x_{2}$ axis and rotation by $\pi$ about the origin, respectively. In other words, $\Lambda_{1}$ is the same as the one defined in (2.8) and $\Lambda_{2}$ and $\Lambda_{3}$ are defined as:

[TABLE]

and

[TABLE]

The theorem above is a limiting case of the result in Theorem 2.6, and in the limit, there is a bifurcation of the singularities, due to the presence of $4$ Lagrangians in the result above compared to $3$ in Theorem 2.6. From (2.13), we see that when $\alpha=-1$, the circle given by the equation

[TABLE]

becomes the straight line $x_{1}=0$ regardless of the value $s$, and as noted in Remark 2.3 of [2], the canonical relation associated to the operator $\mathcal{T}$ is a 4-1 relation due to the symmetries with respect to both the $x_{1}$ and $x_{2}$ axes. When $\alpha\neq-1$, the additional symmetry (about the $x_{2}$ axis) in the canonical relation of $\mathcal{T}$ is broken. This was one of our main motivations for the results in this paper.

5 Analysis of the operator $\mathcal{F}$ and the imaging operator

$\mathcal{F}^{*}\mathcal{F}$ for $\alpha\geq 0$

In this section, we prove Theorems 2.1 and 2.2.

Proof 5.1** (Proof of Theorem 2.1).**

We first prove that

[TABLE]

is a non-degenerate phase function. We have that $\varphi$ is a phase function in the sense of Hörmander [23] because $\nabla_{x}\varphi\neq 0$ at points where the amplitude of the operator $\mathcal{F}$ is elliptic. The differential $\nabla_{x}\varphi$ vanishes at a point on the ground directly “between” the source and receiver and this point is given by $\left({\frac{(\alpha+1)s}{2},0}\right)$. However, in a neighborhood of such points the amplitude $a$ vanishes due to the cutoff function $f$ in the definition of $\mathcal{F}$ given in (2.2). Also we have that $\nabla_{s,t}\varphi$ is nowhere [math] since $\omega\neq 0$. The same reason that $\nabla_{x}\varphi$ is non-vanishing at points where the amplitude $a$ is elliptic also gives that $\varphi$ is non-degenerate. Since $a$ satisfies an amplitude estimate, $\mathcal{F}$ is an FIO [9]. Finally since $a$ is of order $2$, the order of the FIO is $3/2$ [9, Definition 3.2.2]. By definition [23, Equation (3.1.2)]

[TABLE]

This establishes (2.6). Furthermore, it is easy to see that $(x_{1},x_{2},s,\omega)$ is a global parametrization of $\mathcal{C}_{\mathcal{F}}$.

Now we prove the claims about the canonical left and right projections from $\mathcal{C}_{\mathcal{F}}$, the final parts of Theorem 2.1. In the parameterization of $\mathcal{C}_{\mathcal{F}}$, we have

[TABLE]

and the derivative is

[TABLE]

Then

[TABLE]

The third term would be zero when the unit vectors $(\gamma_{T}(s)-x)/A$ and $(\gamma_{R}(s)-x)/B$ point in opposite directions, but this cannot happen since the transmitter and receiver are above the plane of the Earth. Also since $\alpha>0$, $\left(\frac{\alpha}{A^{2}}+\frac{1}{B^{2}}\right)\neq 0$. Hence, this determinant vanishes to first order when $x_{2}=0$. This corresponds to $\Sigma_{1}$ given in (2.7).

On $\Sigma_{1}$ the kernel of $\left(d\pi_{L}\right)$ is spanned by $\frac{\partial}{\partial x_{2}}\not\subset T\Sigma_{1}$. So $\pi_{L}$ has a fold singularity along $\Sigma_{1}$.

Similarly, we have,

[TABLE]

Then

[TABLE]

has the same determinant as $\left(d\pi_{L}\right)$.Therefore $\pi_{R}$ drops rank simply by one on $\Sigma_{1}$. On $\Sigma_{1}$, the kernel of $\pi_{R}$ is spanned by $\frac{\partial}{\partial\omega}$ and $\frac{\partial}{\partial s}$ which are tangent to $\Sigma_{1}$. Thus $\pi_{R}$ has a blowdown singularity along $\Sigma_{1}$.

Next we analyze the imaging operator $\mathcal{F}^{*}\mathcal{F}$. We have the following integral representation for $\mathcal{F}^{*}\mathcal{F}$:

[TABLE]

where

[TABLE]

After an application of the method of stationary phase in $t$ and $\tilde{\omega}$, the Schwartz kernel of this operator becomes

[TABLE]

where

[TABLE]

Note that $\tilde{a}\in S^{4}$ since we have assumed that $a\in S^{2}$.

Proposition 5.2**.**

Let $\alpha\geq 0$. The wavefront relation of the kernel $K$ of $\mathcal{F}^{*}\mathcal{F}$ satisfies

[TABLE]

*where $\Delta$ is the diagonal in $T^{*}X\times T^{*}X$, and $C_{1}$ is given by (2.8). We have that $\Delta$ and $C_{1}$ intersect cleanly in codimension 2 in $\Delta$ or $C_{1}$. *

Proof 5.3**.**

Let $(s,t,\sigma,\tau;y,\eta)\in\mathcal{C}_{\mathcal{F}}$. Then we have

[TABLE]

and $(x,\xi;s,t,\sigma,\tau)\in C^{t}$ implies

[TABLE]

From the first two relations in (5.4) and (5.3), we have

[TABLE]

and

[TABLE]

We will use prolate spheroidal coordinates with foci $\gamma_{R}(s)$ and $\gamma_{T}(s)$ to solve for $x$ and $y$. We let

[TABLE]

with $\rho$ and $\rho^{\prime}$ positive, $\phi$ and $\phi^{\prime}$ in the interval $[0,\pi]$ and $\theta$ and $\theta^{\prime}$ in $[0,2\pi]$. In this case $x_{3}=0$ and we use it to solve for $h$. Hence

[TABLE]

and

[TABLE]

Noting that $s>0$ and $\cosh\rho\pm\cos\phi>0$, the first relation given by (5.6) in these coordinates becomes

[TABLE]

from which we get

[TABLE]

The second relation, given by (5.7), becomes

[TABLE]

After simplification we get

[TABLE]

which implies either that

[TABLE]

(since we can assume $\phi\in[\pi,2\pi]$ for points on the ground) or that

[TABLE]

where $\tilde{A}$ and $\tilde{B}$ are defined as in (2.3) but evaluated at $(s,y)$ and the third term in the equality is equivalent to the first term.

We consider Conditions (5.13) and (5.14) separately. First, assume Condition (5.13) holds. Then we have $\phi=\phi^{\prime}$. In this case,

[TABLE]

and note that $x_{3}=0$ implies that $\sin\phi\neq 0$. We also remark that it is enough to consider $\cos\theta=\cos\theta^{\prime}$ as no additional relations are introduced by considering $\cos\theta=-\cos\theta^{\prime}$ over the relations we now address. Now we go back to $x$ and $y$ coordinates. If $\theta=\theta^{\prime}$ then $x_{1}=y_{1},\ x_{2}=y_{2},\xi_{i}=\eta_{i}$ for $i=1,2$. In this case, the composition, $\mathcal{C}_{\mathcal{F}}^{t}\circ\mathcal{C}_{\mathcal{F}}\subset\Delta=\{(x,\xi;x,\xi)\}$. If $\theta^{\prime}=\pi-\theta$ then $x_{1}=y_{1},\ -x_{2}=y_{2},\xi_{1}=\eta_{1},-\xi_{2}=\eta_{2}$. For these points the composition, $\mathcal{C}_{\mathcal{F}}^{t}\circ\mathcal{C}_{\mathcal{F}}$ is a subset of $C_{1}$ in (2.8). Note that (5.14) has no solutions for $\alpha\geq 0$. The statements about clean intersection are the same as the one given in [2]. This concludes the proof of Proposition 5.2.

Proof 5.4** (Proof of Theorem 2.2).**

We will use the iterated regularity method (Theorem 3.9) to show that the kernel of $\mathcal{F}^{*}\mathcal{F}\in I^{3,0}(\Delta,C_{1})$. We consider the generators of the ideal of functions that vanish on $\Delta\cup C_{1}$ [10]. These are given by

[TABLE]

We show in Appendix A that each $\tilde{p}_{i}$ can be expressed as sums of products of $\partial_{\omega}{\Phi}$ and $\partial_{s}{\Phi}$ with smooth functions. Let $p_{i}=q_{i}\tilde{p}_{i}$, for $1\leq i\leq 6$, where $q_{1},q_{2}$ are homogeneous of degree $1$ in $(\xi,\eta)$, $q_{3},q_{4}$ and $q_{5}$ are homogeneous of degree [math] in $(\xi,\eta)$ and $q_{6}$ is homogeneous of degree $-1$ in $(\xi,\eta)$. Let $P_{i}$ be pseudodifferential operators with principal symbols $p_{i}$ for $1\leq i\leq 6$. The $\tilde{p}_{i}$ and arguments using iterated regularity are similar to those used in [10, Thm. 1.6] and in [25, Thm. 4.3].

*We use the same arguments as in [2] to show that the orders $p,l$ from $I^{p,l}(\Delta,C_{1})$ are $p=3$ and $l=0$. *

6 Analysis of the forward operator $\mathcal{G}$ and the imaging operator

$\mathcal{G}^{*}\mathcal{G}$ for $\alpha<0$

In this section, we analyze the operator $\mathcal{G}$ in (2.12). In [2] we analyzed the case $\alpha=-1$. For the case with $\alpha<0$, we make another simplification:

[TABLE]

If $\alpha\in(-1,0)$, then we can reduce it to the case $\alpha<-1$ by using the diffeomorphisms $(x_{1},x_{2})\mapsto(-x_{1},x_{2})$ and $(s,t)\mapsto(s/|\alpha|,t)$.

We first prove Theorem 2.4.

Proof 6.1** (Proof of Theorem 2.4).**

In the proof of this theorem, most of the statements are already proved in Theorem 2.1. We just prove the statements regarding the properties of the projection maps $\pi_{L}$ and $\pi_{R}$. Recall from the proof of Theorem 2.1 that

[TABLE]

and

[TABLE]

Clearly this determinant drops rank when the first term, $x_{2}=0$. This corresponds to $\Sigma_{1}$ given by (2.7).

The determinant also drops rank when the second term is zero, which can occur when $\alpha<0$; this corresponds to $\Sigma_{2}$ given by (2.13). Note that $\pi_{L}$ drops rank by 2 at the intersection points of $\Sigma_{1}$ and $\Sigma_{2}$ (where $x_{2}=0$) but we exclude them using the cutoff function $g$ described preceding (2.11).

On $\Sigma_{2}$, using the first, second, and fourth row of $d\pi_{L}$, the kernel is $\frac{\partial}{\partial x_{1}}-\frac{\frac{x_{1}-\alpha s}{A}+\frac{x_{1}-s}{B}}{x_{2}(\frac{1}{A}+\frac{1}{B})}\frac{\partial}{\partial x_{2}}$ which applied to $\Sigma_{2}$ gives us $\frac{2s(\alpha-1)}{(\alpha+1)(\frac{1}{A}+\frac{1}{B})}(\frac{\alpha}{A}-\frac{1}{B})$. We have that $s\neq 0$ and $\alpha+1\neq 0$. If $\frac{\alpha}{A}-\frac{1}{B}=0$ then using $\frac{\alpha}{A^{2}}+\frac{1}{B^{2}}=0$ we get $\frac{\alpha(\alpha+1)}{A^{2}}=0$ which is a contradiction. Thus $\ker\left({\mathrm{d}\pi_{L}}\right)\not\subset T\Sigma_{2}$ which implies that $\pi_{L}$ has a fold singularity along $\Sigma_{2}$.

Similarly,

[TABLE]

*has the same determinant up to sign and so $\pi_{R}$ drops rank by one on $\Sigma$. On $\Sigma_{2}$, using the last row of $d\pi_{R}$, the kernel is $\frac{\partial}{\partial s}-\omega\frac{\frac{x_{1}-\alpha s}{A^{3}}\alpha+\frac{x_{1}-s}{B^{3}}}{\frac{1}{A}+\frac{1}{B}}\frac{\partial}{\partial\omega}$ which applied to $\Sigma_{2}$ gives $2\alpha(s-\frac{2x_{1}}{\alpha+1})$. If $s=\frac{2x_{1}}{\alpha+1}$ then from $\frac{\alpha}{A^{2}}+\frac{1}{B^{2}}=0$ we obtain $s^{2}(\frac{\alpha-1}{2})^{2}+x_{2}^{2}+h^{2}=0$ which is a contradiction. Hence $\ker\left({\mathrm{d}\pi_{R}}\right)\not\subset T\Sigma_{2}$ which implies that $\pi_{R}$ has a fold singularity along $\Sigma_{2}$ as well. This completes the proof of Theorem 2.4. *

Proposition 6.2**.**

For $\alpha<0$, the wavefront relation of the kernel $K$ of $\mathcal{G}^{*}\mathcal{G}$ satisfies,

[TABLE]

where $\Delta$ is the diagonal in $T^{*}X\times T^{*}X$, $C_{1}$ is given by (2.8) and $C_{2}$ is defined as

[TABLE]

*where $A=A(s,x)$ and $\widetilde{A}=A(s,y)$, $B=B(s,x)$ and $\widetilde{B}=B(s,y)$ and $E(s,t)$ is given by (2.4). Furthermore, $\Delta$ and $C_{1}$ intersect cleanly in codimension 2, $\Delta$ and $C_{2}$ intersect cleanly in codimension 1, $C_{2}$ and $C_{1}$ intersect cleanly in codimension 1, and $\Delta\cap C_{1}\cap C_{2}=\emptyset$. *

Proof 6.3**.**

In fact, this proposition is already proved in Proposition 5.2. Here, unlike the situation in Proposition 5.2, there is a nontrivial contribution to the wavefront of the composition from (5.14). Hence for $\alpha<0$, we have that

[TABLE]

To show that no point in $C_{2}$ has $x_{2}=0=y_{2}$, one uses (5.14) and that $x_{3}=0=y_{3}$ in (5.11). Finally, note that $\Delta\cap C_{1}\cap C_{2}=\emptyset$ since we exclude the points of intersection of $\Sigma_{1}$ and $\Sigma_{2}$ due to the cutoff function $g$ defined in Section 2.3. One can show that $C_{2}$ is an immersed conic Lagrangian manifold that is a two-sided fold using Definition 3.2 and the proof of Theorem 6.13 part (b)).

Using Def. 3.4 and the calculations above, one can also show that these manifolds intersect in the following ways:

1. (a)

$\Delta$* intersects $C_{1}$ cleanly in codimension 2,*

[TABLE]

This is part of Proposition 5.1 in **[2]**. 2. (b)

$\Delta$* intersects $C_{2}$ cleanly in codimension 1,*

[TABLE]

Note that the condition that $x_{1}=y_{1}$ in $C_{2}$ implies $x_{2}=\pm y_{2}$ and so the condition $x_{2}=y_{2}$ does not increase the codimension of the intersection. Using **[26]**, one can show the intersection is clean. 3. (c)

$C_{1}$* intersects $C_{2}$ cleanly in codimension 1,*

[TABLE]

Using **[26]**, one can show the intersection is clean. 4. (d)

$\Delta\cap C_{1}\cap C_{2}=\emptyset$* since we exclude the points of intersection of $\Sigma_{1}$ and $\Sigma_{2}$.*

*This completes the proof of the proposition. *

For the rest of the proof, we focus on $C_{2}$. Let $\beta=\sqrt{-\alpha}$, then $\beta>1$. Let $(x,\xi,y,\xi^{\prime})\in C_{2}$. then, by (6.1) there is an $(s,t)$ such that $x$ and $y$ are both in $E(s,t)$ and

[TABLE]

where $A,\widetilde{A},B,\widetilde{B}$ are given below (6.1). Therefore, if $(x,\xi,y,\xi^{\prime})\in C_{2}$ then

[TABLE]

A calculation shows that if $k\neq\beta$ then

[TABLE]

If $k=\beta$, then $\beta B/A=k$ is the equation of a vertical line with $x_{1}$ intercept $(1-\beta^{2})s/2$.

We first use this characterization of $C_{2}$ to prove Statements (1) and (3) of Theorem 2.6. As already mentioned, the diagonal relation $\Delta$ and $C_{1}$ given by (2.8) intersect cleanly in codimension 2 on either submanifold. Hence there is a well-defined $I^{p,l}$ class associated to $\Delta$ and $C_{1}$.

6.1 Proof of Theorem 2.6**,**

Statement (1)

Recall from statement (1) of this theorem that the function $r_{1}$ is a cutoff function compactly supported in

[TABLE]

where $s_{0}$ (see (2.16)) can be written in terms of $\beta$ as

[TABLE]

We show that for $(s,t)\in O_{1}$, there are no $x$ and $y$ satisfying (6.2). Therefore, $C_{2}\cap WF^{\prime}(K_{1})=\emptyset$ where $K_{1}$ is the Schwartz kernel of $\mathcal{G}^{*}r_{1}\mathcal{G}$.

So, assume for some $(s,t)\in O_{1}$ (6.2) holds. Then, the right-hand side of (6.3) can be estimated by

[TABLE]

Since $\beta>1$, if $0<k\leq 1$, this calculation and (6.3) shows that $\frac{\beta B}{A}=k$ has no solution. Therefore there are no solutions to (6.2) if $0<k\leq 1$. Now assume for some $k>1$, $\frac{\beta B(s,x)}{A(s,x)}=k$, then for (6.2) to have a solution that means that there must be a point $y\in E(s,t)$ with $\frac{\beta B(s,y)}{A(s,y)}=1/k$. However, this is impossible since $0<1/k\leq 1$. This shows that for $(s,t)\in O_{1}$, there is no solution to (6.2).

Now following the proof of Theorem 2.2, we achieve the result of Statement (1).

6.2 Proof of Theorem 2.6**,**

Statement (3)

Recall that the cutoff function $r_{3}(s,t)$ is compactly supported in

[TABLE]

where $t_{s}^{\pm}$ is defined in (2.11). The operator we analyze in this part of the proof is $\mathcal{G}^{*}r_{3}\mathcal{G}$.

We define the following set

[TABLE]

Note that $C(s,1)=\Sigma_{2,X}(s)$, $C(s,\beta)$ is the vertical line $x_{1}=(1+\alpha)s/2=(1-\beta^{2})s/2$, and if $k$ is small enough, $C(s,k)=\emptyset$. By (6.3), when $k\neq\beta$ and $C(s,k)\neq\emptyset$ then $C(s,k)$ is the circle centered at $\left({\frac{\beta^{2}s(1+k^{2})}{\beta^{2}-k^{2}},0}\right)$ and of radius

[TABLE]

Let $(s,t)\in O_{3}$. If there were a solution $(x,y)$ to (6.2) for some $k$, then $x\in E(s,t)\cap C(s,k)$ (as $\beta B/A=k$ on $C(s,k)$) and $y\in E(s,t)\cap C(s,1/k)$. If $t>t_{s}^{+}$ then the ellipse $E(s,t)$ encloses $\Sigma_{2,X}(s)$ by a calculation. Therefore, by the final statement of Lemma 6.4, $E(s,t)$ meets no circle $C(s,k)$ for $k\in(0,1]$ and so there is no solution to (6.2). Now, if $t<t_{s}^{-}$ then the ellipse $E(s,t)$ is enclosed by $\Sigma_{2,X}(s)$ and, by the final statement of Lemma 6.4, $E(s,t)$ meets no $C(s,k)$ for $k\in(1,\infty)$ and so there is no solution to (6.2) in this case, too. Therefore, $C_{2}\cap WF^{\prime}(K_{3})=\emptyset$ where $K_{3}$ is the Schwartz kernel of $\mathcal{G}^{*}r_{3}\mathcal{G}$. Now proceeding as in the proof of Theorem 2.2, we complete the proof of Statement (3) of Theorem 2.6.

The rest of this section is devoted to the proof of Statement (2) of Theorem 2.6.

6.3 Proof of Theorem 2.6,

Statement (2)

The reconstruction operator we consider in statement (2) of Theorem 2.6 is $\mathcal{G}^{*}r_{2}\mathcal{G}$ where the mute $r_{2}$ has compact support in

[TABLE]

where $t_{s}^{\pm}$ is defined in (2.11) and where $s_{0}$ is defined by (6.5).

Recall that the canonical relation of $\mathcal{G}$ drops rank on the union of two sets, $\Sigma_{1}$ and $\Sigma_{2}$. Accordingly, we decompose $\mathcal{G}$ into components such that the canonical relation of each component is either supported near a subset of the union of these two sets, one of these two sets or away from both these sets. To do this, we define several cutoff functions.

6.3.1 The primary cutoff functions $\psi_{1}$ and

$\psi_{2}$

The cutoff $\psi_{1}(x)$ will be equal to $1$ near the $x_{1}$-axis and zero away from it, and $\psi_{2}(s,x)$ will be equal to one near $\Sigma_{2,X}(s)$ and equal to zero away from it as in Figure 1.

To define these functions precisely, we need to set up some preliminary relations. Because the mute function $r_{2}$ is zero near $s_{0}$ and has compact support, there is an $s_{1}>s_{0}$ such that $r_{2}$ is zero for $s\leq s_{1}$ and all $t$. Because the radius $r(s_{1},1)>0$ and the function $r$ is continuous, there is a $k_{1}\in(0,1)$ such that $r(s_{1},k_{1})>0$. Since $r$ (see (6.8)) is an increasing function in $s$ and $k$ separately, we can choose $\epsilon>0$ such that

[TABLE]

Without loss of generality, we can assume

[TABLE]

Now, let $\psi_{1}$ be an infinitely differentiable function defined as follows:

[TABLE]

and we extend this function smoothly between [math] and $1$.

For $s>s_{0}$, let $k_{0}(s)$ be defined by

[TABLE]

Note that $k_{0}(s)$ can be explicitly calculated using (6.8). So, if $k>k_{0}(s)$, $C(s,k)$ is a nontrivial circle. Finally, note that if $s\geq s_{1}$, then $k_{1}>k_{0}(s)$; this is true because $r(s,k_{1})\geq r(s_{1},k_{1})>0$ for such $s$.

To define $\psi_{2}$ we first prove a lemma about the circles $C(s,k)$.

Lemma 6.4**.**

Let $s\geq s_{1}$.

If $k>\beta$ then $C(s,k)$ is to the left of the vertical line $C(s,\beta)$ which is to the left of $C(s,\ell)$ for any $\ell\in(k_{0}(s),\beta)$. 2. 2.

If $k_{0}(s)<j<k<\beta$ then $C(s,j)$ is contained inside $C(s,k)$, and these circles do not intersect. 3. 3.

For any $\delta\in(0,6\epsilon)$,

[TABLE]

is an open set containing $\Sigma_{2,X}(s)=C(s,1)$.

Proof 6.5**.**

Statement (1) of the lemma is a straightforward calculation.

Now, fix $s\geq s_{1}$. Let $k\in(k_{0}(s),\beta)$, then the endpoints of $C(s,k)$ on the $x_{1}$-axis are

[TABLE]

Clearly the functions ${x_{\ell}}$ and ${x_{r}}$ are smooth for $k\in(k_{0}(s),\beta)$. It is straightforward to see that $k\mapsto{x_{r}}(k)$ is a strictly increasing smooth function for $k\in(k_{0}(s),\beta)$.

We prove that the function ${x_{\ell}}$ is strictly decreasing by showing ${x_{\ell}}^{\prime}$ is always negative. A somewhat tedious calculation shows that

[TABLE]

By replacing the square root in this expression by the upper bound $\frac{\beta sk(\beta^{2}+1)}{(\beta^{2}-k^{2})}$, we see that

[TABLE]

and the right-hand side of this expression is clearly negative.

The circles $C(s,\cdot)$ are symmetric about the $x_{1}$-axis, so if $j$ and $k$ are points in $(k_{0}(s),\beta)$ with $j<k$, since ${x_{\ell}}(k)<{x_{\ell}}(j)<{x_{r}}(j)<{x_{r}}(k)$, the circle $C(s,j)$ is strictly inside the circle $C(s,k)$. This proves (2).

*By the choice of $\epsilon$ in (6.11), $1-6\epsilon>k_{1}$ and $1+6\epsilon<\beta$. Because ${x_{\ell}}(k)$ and ${x_{r}}(k)$ are smooth strictly monotonic functions with nonzero derivatives, $(1-6\epsilon,1+6\epsilon)\ni k\mapsto C(s,k)$ is a foliation of an open, connected region containing $C(s,1)=\Sigma_{2,X}(s)$, and this proves (3). *

We define

[TABLE]

and we extend smoothly between (which is possible by Lemma 6.4, statement (3)). By the lemma, $\psi_{2}(s,\cdot)$ is equal to $1$ on an open neighborhood of $\Sigma_{2,X}(s)$ and zero away from $\Sigma_{2,X}(s)$.

We assume, without loss of generality, that $\psi_{1}$ and $\psi_{2}$ are symmetric about the $x_{1}$-axis.

Remark 6.6**.**

We now can define the function $g(s,t)$ in Remark 2.5. We let

[TABLE]

The set $D(s,4\epsilon)$ is represented by the shaded set in Figure 2 that is near $C(s,1)=\Sigma_{2,X}(s)$ and the $x_{1}$-axis. Let $g$ be a smooth function of $(s,t)$ that is zero if the ellipse $E(s,t)$ given in (2.4) intersects $D(s,4\epsilon)$ and is equal to $1$ if $E(s,t)$ does not meet $D(s,5\epsilon)$.

6.3.2 Properties of $\mathcal{G}^{*}\mathcal{G}$ and end of proof

We now write $\mathcal{G}=\mathcal{G}_{0}+\mathcal{G}_{1}+\mathcal{G}_{2}+\mathcal{G}_{3}$ where $\mathcal{G}_{i}$ are given in terms of their kernels

[TABLE]

where $\varphi$ is the phase function of $\mathcal{G}$. The supports of the $\mathcal{G}_{i}$ are given in Figure 3.

Now we consider $\mathcal{G}^{*}\mathcal{G}$, which using the decomposition of $\mathcal{G}$ as above can be written as $\mathcal{G}^{*}\mathcal{G}=\mathcal{G}^{*}_{0}\mathcal{G}+(\mathcal{G}_{1}+\mathcal{G}_{2})^{*}\mathcal{G}_{0}+\mathcal{G}^{*}_{1}\mathcal{G}_{1}+\mathcal{G}^{*}_{2}\mathcal{G}_{2}+\mathcal{G}^{*}_{1}\mathcal{G}_{2}+\mathcal{G}^{*}_{2}\mathcal{G}_{1}+\mathcal{G}^{*}_{1}\mathcal{G}_{3}+\mathcal{G}^{*}_{2}\mathcal{G}_{3}+\mathcal{G}^{*}_{3}\mathcal{G}$

The theorem now follows from Lemmas 6.7-6.11, and Theorem 6.13, which we now state and prove. In the lemmas, we analyze the compositions above.

Recall that $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are operators defined as follows:

[TABLE]

and

[TABLE]

Lemma 6.7**.**

*The operators $\mathcal{G}_{1}^{*}\mathcal{G}_{2}$ and $\mathcal{G}_{2}^{*}\mathcal{G}_{1}$ are smoothing. *

Proof 6.8**.**

We show that $\mathcal{G}_{1}^{*}\mathcal{G}_{2}$ is smoothing. The proof for the case of $\mathcal{G}_{2}^{*}\mathcal{G}_{1}$ is similar.

We have

[TABLE]

where $\psi_{1}(x)$ and $\psi_{2}$ are defined in (6.12) and (6.14) respectively. The Schwartz kernel of $\mathcal{G}_{1}^{*}\mathcal{G}_{2}$ is

[TABLE]

where $\widetilde{a}(x,y,s,\omega)$ has the following products of cutoff functions as an additional factor:

[TABLE]

Here $t$ is determined from $s$ and $x$ as the value for which $x\in E(s,t)$. For this reason, in trying to understand the propagation of singularities, we need only to restrict ourselves, for each fixed $s\geq s_{1}$, to those base points $x$ and $y$ for which

[TABLE]

We use this setup to show $\mathcal{G}_{1}^{*}\mathcal{G}_{2}$ is smoothing by showing its symbol is zero for covectors in $\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}$ (note that our argument shows that the symbol of the operator is zero in a neighborhood in $\left({T^{*}(X)\setminus\mathbf{0}}\right)^{2}$ of $\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}$). Let $(x,\xi,y,\xi^{\prime})\in\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}$. Then, there is an $(s_{2},t_{2},\eta)\in T^{*}(Y)\setminus\mathbf{0}$ such that $(x,\xi,s_{2},t_{2},\eta)\in\mathcal{C}_{\mathcal{G}}^{t}$ and $(s_{2},t_{2},\eta,y,\xi^{\prime})\in\mathcal{C}_{\mathcal{G}}$. For the rest of the proof, we fix this $s_{2}$. (If there are other values of $s$ associated to the composition, we repeat this proof for those values of $s$.)

Because $\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}\subset\Delta\cup C_{1}\cup C_{2}$, we consider three cases separately.

1. I.

Covectors $(x,\xi,y,\xi^{\prime})\in\Delta\cap\left({\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}}\right)$:* In this case, $x=y$ and $x$ is in $\operatorname{supp}(\psi_{1})\cap\operatorname{supp}(\psi_{2})\subset D(s_{2},4\epsilon)$. By the choice of the function $g(s,t)$ in Remark 6.6, the symbol of ${\mathcal{G}_{1}^{*}\mathcal{G}_{2}}$ is zero above such $x$.* 2. II.

Covectors $(x,\xi,y,\xi^{\prime})\in C_{1}\cap\left({\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}}\right)$:* In this case, $(x_{1},x_{2})=(y_{1},-y_{2})$ and the argument in case I shows that the symbol of ${\mathcal{G}_{1}^{*}\mathcal{G}_{2}}$ is zero for such $x$ and $y$* 3. III.

Covectors $(x,\xi,y,\xi^{\prime})\in C_{2}\cap\left({\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}}\right)$:* If $(x,\xi,y,\xi^{\prime})\in C_{2}\cap\left({\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}}\right)$, then for some $(s_{2},t_{2})$ above, there is a $k_{2}>k_{0}(s_{2})$, such that*

[TABLE]

Using (6.17), the fact that $k_{2}=\beta B(s_{2},x)/A(s_{2},x)$, we see that $\left|{x_{2}}\right|<2\epsilon$ and $\left|{1-k_{2}}\right|>\epsilon$. Now, using the restriction on $1/k_{2}$ in (6.18) and the fact that $\epsilon<1/4$, we see $\left|{1-k_{2}}\right|<4\epsilon$. Putting this together shows that

[TABLE]

Since $\left|{x_{2}}\right|<2\epsilon$, this shows that $x\in D(s_{2},4\epsilon)$. Therefore $E(s_{2},t_{2})\cap D(s_{2},4\epsilon)\neq\emptyset$ and $g(s_{2},t_{2})=0$ by Remark 6.6. Therefore, the symbol of ${\mathcal{G}_{1}^{*}\mathcal{G}_{2}}$ is zero near $(x,\xi,y,\xi^{\prime})$ so ${\mathcal{G}_{1}^{*}\mathcal{G}_{2}}$ is smoothing near $(x,\xi,y,\xi^{\prime})$.

*This finishes the proof that ${\mathcal{G}_{1}^{*}\mathcal{G}_{2}}$ is smoothing. *

Lemma 6.9**.**

*The operator $\mathcal{G}_{0}$ is smoothing. *

Proof 6.10**.**

Recall that the Schwartz kernel of $\mathcal{G}_{0}$ is

[TABLE]

*For each fixed $s\geq s_{1}$, the support of $\psi_{1}(\cdot)\psi_{2}(s,\cdot)$ is inside $D(s,4\epsilon)$ and by the choice of the function $g(s,t)$ in Remark 6.6, the symbol of $\mathcal{G}_{0}$ is zero above such $(s,x)$. *

Lemma 6.11**.**

*The operators $\mathcal{G}^{*}_{1}\mathcal{G}_{3}$, $\mathcal{G}^{*}_{2}\mathcal{G}_{3}$ and $\mathcal{G}^{*}_{3}\mathcal{G}$ can be decomposed as a sum of operators belonging to the space $I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$. *

Proof 6.12**.**

Each of these compositions is covered by the transverse intersection calculus.

We decompose $\mathcal{G}_{1}$, $\mathcal{G}_{2}$, and $\mathcal{G}_{3}$ into a sum of operators on which the compositions will be easier to analyze. This is represented in Figure 4.

For $\mathcal{G}_{1}$, note that $\Sigma_{2,X}(s)$ divides $\left\{{\left|{x_{2}}\right|<2\epsilon}\right\}$ in three regions since $r(s,1)>12\epsilon$ by (6.10). Let $H_{1}(s)$ be the part of $\left\{{\left|{x_{2}}\right|<2\epsilon}\right\}\setminus\Sigma_{2,X}(s)$ to the left of $\Sigma_{2,X}(s)$ and let $H_{2}(s)$ be the part inside $\Sigma_{2,X}(s)$ and $H_{3}(s)$ the part to the right of $\Sigma_{2,X}(s)$. Define our partitioned operators as follows $\mathcal{G}_{1}=\mathcal{G}_{1}^{1}+\mathcal{G}_{1}^{2}+\mathcal{G}_{1}^{3}$ where

[TABLE]

for $i=1,2,3$. Note that the symbols are all smooth because

[TABLE]

is a smooth cutoff function in $(s,x)$ since the support of $\psi_{1}$ is inside $\left\{{\left|{x_{2}}\right|<2\epsilon}\right\}$ and the support of $(1-\psi_{2}(s,\cdot))$ does not meet $\Sigma_{2}(s)$.

We decompose $\mathcal{G}_{2}$ into two operators in a similar way. Let $I_{1}$ be the open upper half plane and let $I_{2}$ be the open lower half plane. Define

[TABLE]

for $j=1,2$ Because the functions $(1-\psi_{1}(\cdot))\psi_{2}(s,\cdot)$ are supported away from the $x_{1}$ axis, these symbols are smooth.

We decompose $\mathcal{G}_{3}$ into four operators in a similar way using Figure 4: $\Sigma_{2,X}(s)$ divides $\left\{{x_{2}\neq 0}\right\}$ into four regions $J_{1}(s)$, the unbounded region above the $x_{1}$-axis, $J_{2}(s)$, it’s mirror image in the $x_{1}$-axis, $J_{3}(s),$ the bounded region inside $\Sigma_{2,X}(s)$ and above the $x_{1}$-axis, and its mirror image, $J_{4}(s)$. We define

[TABLE]

for $k=1,2,3,4$, and because of the cutoffs used, these are all FIO with smooth symbols.

To find the canonical relation of ${\mathcal{G}_{1}^{j}}^{*}\mathcal{G}_{3}^{k}$, we consider $(x,\xi,y,\xi^{\prime})\in\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}$ and let $(s,t)\in Y$ such that $(x,\xi,s,t,\eta)\in\mathcal{C}_{\mathcal{G}}^{t}$ and $(s,t,\eta,y,\xi^{\prime})\in\mathcal{C}_{\mathcal{G}}$. In any case, $(\mathcal{G}_{1}^{i})^{*}\mathcal{G}_{3}^{j}$ has canonical relation a subset of $\mathcal{C}_{\mathcal{G}}^{t}\circ\mathcal{C}_{\mathcal{G}}\subset\Delta\cup C_{1}\cup C_{2}$. To find which subset, we consider the restriction that the supports of the $\mathcal{G}_{i}^{j}$ put on $x$ and $y$. We use the fact that $x$ and $y$ are on $E(s,t)$ plus the following rules to understand the canonical relations of these operators:

1. $\left(i\right)$

If the supports exclude $x$ and $y$ from being equal, then the canonical relation ($WF^{\prime}$) of the composed operator does not include $\Delta$. 2. $\left(ii\right)$

if the supports exclude $x$ and $y$ from being reflections in the $x_{1}$ axis then the canonical relation of the composed operator does not include $C_{1}$. 3. $\left(iii\right)$

If the supports exclude $x$ from being outside $\Sigma_{2,X}(s)$ and $y$ being inside or vice versa, then the canonical relation of the composed operator does not include $C_{2}$.

We first consider $(\mathcal{G}_{1})^{*}\mathcal{G}_{3}$. To do this, we partition $\mathcal{G}_{1}$ further. Let $u$ be a smooth cutoff function supported in $[-\epsilon,\epsilon]$ and equal to one on $[-\epsilon/2,\epsilon/2]$ and let $\operatorname{\sigma^{+}}=\chi_{[0,2\epsilon]}(1-u)\psi_{1}(1-\psi_{2})$, $\sigma^{o}=\chi_{[-\epsilon,\epsilon]}u\psi_{1}(1-\psi_{2})$, and $\operatorname{\sigma^{-}}=\chi_{[-2\epsilon,0]}(1-u)\psi_{1}(1-\psi_{2})$ where the characteristic functions and $u$ are functions of $x_{2}$. Note that, for each fixed $s$ and functions of $x$, $\operatorname{supp}(\operatorname{\sigma^{+}})\subset\mathbb{R}\times[\epsilon/2,2\epsilon]$, $\operatorname{supp}(\operatorname{\sigma^{o}})\subset\mathbb{R}\times[-\epsilon,\epsilon]$, $\operatorname{supp}(\operatorname{\sigma^{-}})\subset\mathbb{R}\times[-2\epsilon,-\epsilon/2]$. All these functions are smooth and $\psi_{1}(1-\psi_{2})=\operatorname{\sigma^{+}}+\operatorname{\sigma^{o}}+\operatorname{\sigma^{-}}$. This allows us to divide up each $\mathcal{G}_{1}^{j}$ ($j=1,2,3)$ into the sum of three operators where $\mathcal{G}_{1}^{j+}(V)$ has symbol equal to the symbol of $\mathcal{G}$ but multiplied by $\operatorname{\sigma^{+}}H_{j}$, $\mathcal{G}_{1}^{jo}(V)$ has symbol equal to the symbol of $\mathcal{G}$ but multiplied by $\operatorname{\sigma^{o}}H_{j}$, and $\mathcal{G}_{1}^{j-}(V)$ has symbol equal to the symbol of $\mathcal{G}$ but multiplied by $\operatorname{\sigma^{-}}H_{j}$. Note that $\mathcal{G}_{1}^{j}=\mathcal{G}_{1}^{j+}+\mathcal{G}_{1}^{jo}+\mathcal{G}_{1}^{j-}$.

We now analyze the composition ${\mathcal{G}_{1}^{1}}^{*}{\mathcal{G}_{3}^{1}}$ using this partition of ${\mathcal{G}_{1}^{1}}$. Consider the composition $({\mathcal{G}_{1}^{1+}})^{*}{\mathcal{G}_{3}^{1}}$. Because both operators are supported in $x$ above the $x_{1}$ axis, the canonical relation of this composition cannot intersect $C_{1}$ (see (ii)). Because they are both supported outside $\Sigma_{2,X}(s)$, it cannot intersect $C_{2}$ (since $C_{2}$ associates points inside $\Sigma_{2,X}(s)$ only with points outside $\Sigma_{2,X}(s)$ and vice versa by (iii)). So this shows $({\mathcal{G}_{1}^{1+}})^{*}{\mathcal{G}_{3}^{1}}\in I(\Delta\setminus C_{1})$.

Note that we use the transverse intersection calculus to show $({\mathcal{G}_{1}^{1+}})^{*}{\mathcal{G}_{3}^{1}}$ and each of the other operators in this lemma are regular FIO.

Now, we consider $({\mathcal{G}_{1}^{1o}})^{*}{\mathcal{G}_{3}^{1}}$. Note that ${\mathcal{G}_{1}^{1o}}$ is supported in $x$ in $\left|{x_{2}}\right|<\epsilon$ and ${\mathcal{G}_{3}^{1}}$ is supported in $x_{2}>\epsilon$. Therefore, the canonical relation of the composition can include neither $\Delta$ nor $C_{1}$ by (i), (ii). Furthermore, because they are both supported outside $\Sigma_{2,X}(s)$, it does not contain $C_{2}$ by (iii). Therefore, $({\mathcal{G}_{1}^{1o}})^{*}{\mathcal{G}_{3}^{1}}$ is smoothing.

Next, we consider $({\mathcal{G}_{1}^{1-}})^{*}{\mathcal{G}_{3}^{1}}$. The argument is similar to the case $({\mathcal{G}_{1}^{1+}})^{*}{\mathcal{G}_{3}^{1}}$, but this canonical relation is contained in $C_{1}\setminus\Delta$.

This shows that $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{3}^{1}}$ is a sum of operators in $I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))$.

The proof that $(\mathcal{G}_{1}^{1})^{*}\mathcal{G}_{3}^{2}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))$ follows using the same arguments but the roles of ${\mathcal{G}_{1}^{1-}}$ and ${\mathcal{G}_{1}^{1+}}$ are switched because ${\mathcal{G}_{3}^{2}}$ has support in $x$ below the $x_{1}$-axis and below $\Sigma_{2,X}(s)$.

Now we consider $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{3}^{3}}$. Because the support in $x$ of ${\mathcal{G}_{1}^{1}}$ is to the left of $\Sigma_{2,X}(s)$ and the support of ${\mathcal{G}_{3}^{3}}$ is inside, the canonical relation of $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{3}^{3}}$ cannot intersect $\Delta$ (since there are no points $(x,\xi,x,\xi)$ in that canonical relation by the support condition and (i) and it cannot intersect $C_{1}$ for a similar reason by (ii). So $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{3}^{3}}\in I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

A similar argument using symmetry of support of ${\mathcal{G}_{3}^{3}}$ and ${\mathcal{G}_{3}^{4}}$ in the $x_{1}$ axis shows that $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{3}^{4}}\in I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

Putting these together, we see that $({\mathcal{G}_{1}^{1}})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

The proof that $({\mathcal{G}_{1}^{2}})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$ is similar but here we use the partition of ${\mathcal{G}_{1}^{2}}$: ${\mathcal{G}_{1}^{2+}}$, ${\mathcal{G}_{1}^{2o}}$ and ${\mathcal{G}_{1}^{2-}}$. In a similar way, $({\mathcal{G}_{1}^{3}})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

Thus, $(\mathcal{G}_{1})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2})+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

Now we consider $(\mathcal{G}_{2})^{*}\mathcal{G}_{3}$. Here we partition $\mathcal{G}_{2}^{j}$, $j=1,2$ into three operators with smooth symbols as we did for $\mathcal{G}_{1}$:

• •

$\mathcal{G}_{2}^{j+}$* will have support in $x$ for fixed $s$ in the union of circles *

$\cup_{k\in[1+\epsilon,1+2\epsilon]}C(s,k)$* (outside of $\Sigma_{2,X}(s)$),*

• •

$\mathcal{G}_{2}^{jo}$* will have support in $x$ for fixed $s$ in the union of circles $\cup_{k\in[1-\epsilon,1+\epsilon]}C(s,k)$ (surrounding $\Sigma_{2,X}(s)$) and be equal to the symbol of $\mathcal{G}_{2}$ in *

$\cup_{k\in[1-\epsilon/2,1+\epsilon/2]}C(s,k)$, and

• •

$\mathcal{G}_{2}^{j-}$* will have support in $x$ for fixed $s$ in the union of circles *

$\cup_{k\in[1-2\epsilon,1-\epsilon]}C(s,k)$* (inside $\Sigma_{2,X}(s)$).*

The proof follows similar arguments as for $(\mathcal{G}_{1})^{*}\mathcal{G}_{3}$ and it shows $(\mathcal{G}_{2})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2})+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$.

Finally, we consider $(\mathcal{G}_{3})^{*}\mathcal{G}$. By symmetry of the conditions (i), (ii), (iii), we justify $(\mathcal{G}_{3})^{*}\mathcal{G}_{1}$ and $(\mathcal{G}_{3})^{*}\mathcal{G}_{2}$ are in $I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$. So, the only composition to consider is $(\mathcal{G}_{3})^{*}\mathcal{G}_{3}$, and by analyzing all combinations, we see $(\mathcal{G}_{3})^{*}\mathcal{G}_{3}\in I^{3}(\Delta\setminus(C_{1}\cup C_{2}))+I^{3}(C_{1}\setminus(\Delta\cup C_{2}))+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$. This finishes the proof.

We are left with the analysis of the compositions $\mathcal{G}_{1}^{*}\mathcal{G}_{1}$ and $\mathcal{G}_{2}^{*}\mathcal{G}_{2}$. This is the content of the next theorem:

Theorem 6.13**.**

Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be as above. Then

1. (a)

$\mathcal{G}_{1}^{*}\mathcal{G}_{1}\in I^{3,0}(\Delta,C_{1})+I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$. 2. (b)

$\mathcal{G}_{2}^{*}\mathcal{G}_{2}\in I^{3,0}(\Delta,C_{2})+I^{3,0}(C_{1},C_{2})$.

Proof 6.14**.**

Consider the intersections of $\Delta,\ C_{1},\ C_{2}$. We have that $\Delta$ intersects $C_{1}$ cleanly in codimension $2$; $\Delta$ intersects $C_{2}$ cleanly in codimension $1$ and $C_{1}$ intersects $C_{2}$ cleanly in codimension $2$.

For part (a) we decompose $\mathcal{G}_{1}={\mathcal{G}_{1}^{1}}+{\mathcal{G}_{1}^{2}}+{\mathcal{G}_{1}^{3}}$. Now, we consider the compositions that $(\mathcal{G}_{1}^{j})^{*}\mathcal{G}_{1}^{j}$ for $j=1,2,3$. Using (i), (ii), and (iii), we have that $WF^{\prime}((\mathcal{G}_{1}^{j})^{*}\mathcal{G}_{1}^{j})\subset\Delta\cup C_{1}$. Then, using a proof similar to the one for Theorem 2.2, we see that $(\mathcal{G}_{1}^{j})^{*}\mathcal{G}_{1}^{j}\in I^{3,0}(\Delta,C_{1})$.

Arguments using (i), (ii), and (iii) show that the cross terms $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{1}^{2}}$, $({\mathcal{G}_{1}^{2}})^{*}{\mathcal{G}_{1}^{1}}$, $({\mathcal{G}_{1}^{2}})^{*}{\mathcal{G}_{1}^{3}}$, and $({\mathcal{G}_{1}^{3}})^{*}{\mathcal{G}_{1}^{2}}$ are in $I^{3}(C_{2}\setminus(\Delta\cup C_{1}))$ and $({\mathcal{G}_{1}^{3}})^{*}{\mathcal{G}_{1}^{1}}$ and $({\mathcal{G}_{1}^{1}})^{*}{\mathcal{G}_{1}^{3}}$ are smoothing.

Now, we consider part (b) and the operator $\mathcal{G}_{2}^{*}\mathcal{G}_{2}$.

We recall that $\Sigma_{1}$ and $\Sigma_{2}$ are disjoint, $\Sigma_{2}\in C\setminus\Sigma_{1}$ thus $C\setminus\Sigma_{1}$ is a two sided fold. Next we use [26] to get that $(C\setminus\Sigma_{1})^{t}\circ(C\setminus\Sigma_{1})=\Delta\cup C_{2}$, and that $C_{2}$ is a two sided fold.

We use the decomposition (6.19) $\mathcal{G}_{2}=\mathcal{G}_{2}^{1}+\mathcal{G}_{2}^{2}$ where $\mathcal{G}_{2}^{1}$ is supported in the upper part of $\Sigma_{2}$ and $\mathcal{G}_{2}^{2}$ is supported in the lower part of $\Sigma_{2}$. Note that the support in $x$ of $\mathcal{G}_{2}^{1}$ and $\mathcal{G}_{2}^{2}$ are disjoint.

Then using Theorem 3.10 we have that

[TABLE]

Consider the operator $R$ defined as follows:

[TABLE]

This is a Fourier integral operator of order [math] and it is easy to check its canonical relation is $C_{1}$. Let $\hat{\mathcal{G}}=\mathcal{G}_{2}^{2}\circ R$. We have $\hat{\mathcal{G}}^{*}\mathcal{G}_{2}^{1}\in I^{3,0}(\Delta,C_{2})$. Note that $C_{1}\circ\Delta=C_{1}$, $C_{1}\circ C_{2}=C_{2}$ and $C_{1}\times\Delta$ (as well as $C_{1}\times C_{2}$) intersects $T^{*}X\times\Delta_{T^{*}{X}}\times T^{*}X$ transversally. Using [22, Proposition 4.1], this implies that $R^{*}\tilde{\mathcal{G}}^{*}\mathcal{G}_{2}^{1}\in I^{3,0}(C_{1},C_{2})$. Since $\tilde{\mathcal{G}}^{*}=R^{*}(\mathcal{G}_{2}^{2})^{*}$ and $(R^{*})^{2}=\mbox{Id}$ we have $(\mathcal{G}_{2}^{2})^{*}\mathcal{G}_{2}^{1}\in I^{3,0}(C_{1},C_{2})$

Similarly, we show that

[TABLE]

This concludes the proof of Statement (2) of Theorem 2.6.

6.4 Spotlighting

This is equivalent to assuming the scatterer $V$ has support in either the open half-plane $x_{2}>0$ or $x_{2}<0$. In this case, $C_{1}$ does not appear in the analysis.

Theorem 6.15**.**

Let $\mathcal{G}$ be as in (2.2) of order $\frac{3}{2}$. Assume the amplitude of $\mathcal{G}$ is nonzero only on a subset of either the upper half-plane $(x_{2}>0)$ or the lower half plane $x_{2}<0$ and bounded away from the $x_{1}$ axis. Then $\mathcal{G}^{*}\mathcal{G}\in I^{3,0}(\Delta,C_{2})$, where $C_{2}$ is given by (6.1).

Proof 6.16**.**

*We assume $x_{2}>0$ (the other case is similar), $\Sigma_{1}$ is empty and $\pi_{L}$ and $\pi_{R}$ have fold singularities along $\Sigma_{2}$ as proved in Proposition 6.2. Thus $C^{t}\circ C=\Delta\cup C_{2}$ where $C_{2}$ is a two-sided fold. Using the results in Felea [10] and Nolan [29], we have that $\mathcal{G}^{*}\mathcal{G}\in I^{3,0}(\Delta,C_{2})$. *

In this case, $C_{2}$ does contribute to the added singularities and this is discussed in Remark 2.7.

7 Acknowledgements

All authors thank The American Institute of Mathematics (AIM) for the SQuaREs (Structured Quartet Research Ensembles) award, which enabled their research collaboration, and for the hospitality during the authors’ visits to AIM in 2011, 2012, and 2013. Most of the results in this paper were obtained during the last two visits. Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged by Krishnan, Nolan, and Quinto.

The authors thank the referees for their thorough, thoughtful, and insightful comments that made the article clearer.

Ambartsoumian was supported in part by NSF grants DMS 1109417 and DMS 1616564, and by Simons Foundation grant 360357.

Felea was supported in part by Simons Foundation grant 209850.

Krishnan was supported in part by NSF grants DMS 1109417 and DMS 1616564. He also benefited from the support of Airbus Corporate Foundation Chair grant titled “Mathematics of Complex Systems” established at TIFR CAM and TIFR ICTS, Bangalore, India.

Quinto was partially supported by NSF grants DMS 1311558 and DMS 1712207, and a fellowship from the Otto Mønsteds Fond during fall 2016 at the Danish Technical University as well from the Tufts University Faculty Research Awards Committee.

Appendix A Proofs of iterated regularity for $\mathcal{F}$ ($\alpha\geq 0$)

In this section, we prove that each of the $\tilde{p}_{i}$ given in (5.15) is a sum of products of derivatives of $\Phi$ and smooth functions. This will finish the proof that $\mathcal{F}^{*}\mathcal{F}\in I^{3,0}(\Delta,C_{1})$.

A.1 Expression for $x_{1}-y_{1}$

We will use the same prolate spheroidal coordinates (5.11) with foci $\gamma_{R}(s)$ and $\gamma_{T}(s)$ to solve for $x$ and $y$. We have

[TABLE]

We have

[TABLE]

Therefore in (A.1), it is enough to express $\cos\phi-\cos\phi^{\prime}$ in terms of $\partial_{\omega}\Phi$ and $\partial_{s}\Phi$. We obtain:

[TABLE]

Combining the first and the third term, and second and the fourth term above and then simplifying, we get

[TABLE]

where $\times$ indicates multiplication with the expression in the previous line. Now denote

[TABLE]

Note that since $\alpha>0$, $P_{+}>0$. Therefore we have

[TABLE]

Now using this expression for the difference of cosines in (A.1), we are done.

A.2 Expression for $x_{2}^{2}-y_{2}^{2}$

We have

[TABLE]

Since $x_{3}=y_{3}=0$, we have that the last term in (A.2) is [math].

Now we can write

$\sinh^{2}\rho\sin^{2}\phi-\sinh^{2}\rho^{\prime}\sin^{2}\phi^{\prime}=(\cosh^{2}\rho-\cosh^{2}\rho^{\prime})\sin^{2}\phi-(\cos^{2}\phi-\cos^{2}\phi^{\prime})\sinh^{2}\rho^{\prime}=\\ (\cosh\rho-\cosh\rho^{\prime})(\cosh\rho+\cosh\rho^{\prime})\sin^{2}\phi-(\cos\phi-\cos\phi^{\prime})(\cos\phi+\cos\phi^{\prime})\sinh^{2}\rho^{\prime}.$

Since $\cosh\rho-\cosh\rho^{\prime}$ and $\cos\phi-\cos\phi^{\prime}$ can be expressed in terms of $\partial_{\omega}\Phi$ and $\partial_{s}\Phi$ as above, we are done.

A.3 Expression for $\xi_{1}-\eta_{1}$

We have

[TABLE]

In prolate spheroidal coordinates, we have

[TABLE]

As before we get the terms $\cosh\rho-\cosh\rho^{\prime}$ and $\cos\phi-\cos\phi^{\prime}$ which can be expressed in terms of $\partial_{\omega}\Phi$ and $\partial_{s}\Phi$.

A.4 Expression for $(x_{2}-y_{2})(\xi_{2}+\eta_{2})$

We have

$\xi_{2}=-\omega\left(\frac{x_{2}}{\sqrt{(x_{1}-\alpha s)^{2}+x_{2}^{2}+h^{2}}}+\frac{x_{2}}{\sqrt{(x_{1}-s)^{2}+x_{2}^{2}+h^{2}}}\right)$   and

$\eta_{2}=-\omega\left(\frac{y_{2}}{\sqrt{(y_{1}-\alpha s)^{2}+y_{2}^{2}+h^{2}}}+\frac{y_{2}}{\sqrt{(y_{1}-s)^{2}+y_{2}^{2}+h^{2}}}\right)$

Thus

[TABLE]

Now

[TABLE]

Next we use again the expressions for $\cosh\rho-\cosh\rho^{\prime}$ and $\cos\phi-\cos\phi^{\prime}$ as before and for $x_{2}^{2}-y_{2}^{2}$ we use (A.2).

A.5 Expression for $(x_{2}+y_{2})(\xi_{2}-\eta_{2})$

We have

[TABLE]

Now we are in a similar situation as in the previous case.

A.6 Expression for $\xi_{2}^{2}-\eta_{2}^{2}$

We have

[TABLE]

This part is complete as well.

Appendix B Expressions for $t^{-}_{s}$ and

$t^{+}_{s}$

Recall that $\Sigma_{2}$ is defined in (2.13) as

[TABLE]

Recall that $s_{0}$ is defined by (6.5) and for $s>s_{0}$ $\Sigma_{2}$ is nonempty and not trivial.

We assume in this section that the cutoff function $f$ in Section 2 is chosen so it is zero for $s\leq s_{0}$.

The radius and the $x_{1}$-coordinate of the center of circle $\Sigma_{2}$ are

[TABLE]

Let $e(s)=(\alpha+1)s/2<0$ denote the $x_{1}$-coordinate of the center of ellipses in the plane. Then the distance between $e(s)$ and $c(s)$ can be written as

[TABLE]

For a fixed $s$ let $t^{-}_{s}$ and $t^{+}_{s}$ denote correspondingly the smallest and the largest values of $t$, for which the ellipsoid intersects $\Sigma_{2}$. Notice, that since the normal to an ellipse at a point $P$ bisects the angle from the $P$ to the foci, the condition $\widetilde{\gamma_{R}}(s)\leq\gamma_{R}(s)<c(s)$ implies that our ellipses on the ground can not intersect the circle $\Sigma_{2}$ at more than two points. Here $\widetilde{\gamma_{R}}(s)$ denotes the right focus of the ellipse on the ground. Figure 5 shows the setup for $t_{0}$, where the ellipsoid passes through $x_{1}^{-}$, the closest to $e(s)$ point of $\Sigma_{2}$. The setup for $t^{+}_{s}$ is similar, with the ellipsoid passing through $x_{1}^{+}$, which is the farthest from $e(s)$ point of $\Sigma_{2}$.

A straightforward computation shows that

[TABLE]

where $\beta=\sqrt{-\alpha}$.