Fourier integral operators with open umbrellas and seismic inversion for cusp caustics

0.27 MB·English

by  Raluca Felea

Fourier integral operators with open umbrellas
0
and seismic inversion for cusp caustics
1
0
2

n Raluca Felea and Allan Greenleaf
a
J

7
2
Abstract

]
P In general the composition of Fourier integral operators (FIOs)
A
need not be an FIO. Motivated by the problem of linearized seismic
h. inversion in the presence of cusp caustics for the background sound
t speed, we consider FIOs whose canonical relations have certain two-
a
m sided cusp degeneracies, and show that the resulting compositions
[ have wave-front relations in the union of the diagonal and an open

umbrella, the simplest type of singular Lagrangian manifold.
1
v
8
4 1 Introduction
8
4
. A fundamental problem concerning Fourier integral operators (FIOs) is that,
1
0 outside of the standard transverse [19] and clean intersection [5, 32] calculus,
0
a composition of two FIOs is typically not another FIO. Describing the oper-
1
: ators resulting from the composition of completely general FIOs and placing
v
i them in a usable class, with a symbol calculus, Sobolev space estimates, and
X
the possibility of constructing parametrices under suitable ellipticity con-
r
a ditions, is at this point a distant prospect. Some progress has been made
for specific geometries arising in integral geometry [16] and inverse problems
[6, 7, 21, 28]. In all of these works, the compositions were shown to belong
to an existing class, the pseudodifferential operators with singular symbols,
which are not FIOS and were introduced to construct parametrices for op-
erators of real or complex principal type [24, 18, 26]. Such operators have
wave front relation, i.e., the wave front set of the Schwartz kernel, contained
in the union of the diagonal and another smooth canonical relation which
intersects the diagonal cleanly.
1
In the present paper, motivated by a linearized inverse problem from
seismology for the acoustic wave equation, we analyze FIOs having a special
type of degeneracy, which we call a flat two-sided cusp, and show that their
composition results in operators which are fundamentally more singular than
those in works referred to above.
FIOs with two-sided cusps, i.e., those for which the projections from the
canonical relation both to the left and the right are cusps, arise naturally
when considering generalized Radon transforms. They have previously been
studied in terms of their L2 Sobolev regularity properties; it was shown in
[3, 13] that there is a loss of 1/4 derivative compared with the nondegen-
erate case of local canonical graphs. L2 Lq estimates for FIOs with
−→
one-sided cusps have also been obtained, with T∗T making an appearance
throughStrichartz estimate type arguments [12]; however, such compositions
have not been studied in their own right. We show that for the subclass of
flat two-sided cusps which we formulate, the normal operator F∗F is not a
standard pseudodifferential operator with singular symbol, as it is in the case
of fold caustics [28, 6, 7]. This includes the canonical relations underlying
the linearized forward scattering operators F in the seismology problem in
the presence of caustics of cusp type. While there is still a large part of its
wave front relation, WF(KF∗F)′, which is contained in the diagonal, ∆, the
remaining portion, C, is now no longer a smooth canonical relation. Rather,
C has the structure of the simplest kind of singular Lagrangian manifold, an
e
open umbrella [9]. The diagonal and this open umbrella intersect in codi-
e
mension one, and we quantize the variety ∆ C by associating to it a class
∪
of generalized Fourier integral operators, Im(∆ C). This is in the spirit of
e∪
the paired Lagrangian (or Ip,l) distributions associated to a pair of cleaning
e
intersecting smooth Lagrangians; however, rather than nondegenerate phase
functions and product type symbols as in [24, 18, 26], we use a combination
of a degenerate phase function (whose gradient exhibits normal crossings)
and standard symbols. We remark that operators with wave front relation
in the union of ∆ and a canonical relation having a conical singularity were
considered by Melrose and Uhlmann [25].
As applied to the seismic imaging problem, the results here have nega-
tive implications (as do the results of [28, 6] in the case of fold caustics):
even microlocally away from ∆ C, the non-pseudodifferential part of the
∩
normal operator has the same order as the pseudodifferential part, resulting
e
in a strong, nonremovable artifact. However, our work provides a precise
2
description of the imaging artifacts resulting from cusp caustics, observed by
Nolan and Symes [29] in a model 2D case. It would be of interest to obtain
Sobolev estimates for operators associated with ∆ C, similar to the results
∪
in [8] for certain Ip,l classes, including those arising from fold caustics.
e
We begin in Sec. 2 with a simple example from harmonic analysis of
a FIO having two-sided cusps of the type that is relevant for the seismol-
ogy problem, recall some basic singularity theory, and examine the corre-
sponding normal operator. In Sec. 3, we describe the linearized inverse
problem from seismology, and show that the canonical relation underly-
ing the linearized forward scattering operator F is (i) associated with a
two-sided cusp, but with the additional (highly nongeneric) features that
(ii)thecusppointsfortheleftandrightprojectionsareequal, and(iii)theim-
ages of the cusp points on both the left and right are involutive (coisotropic).
In Sec. 4 we define a general class of canonical relations having this struc-
ture, and then derive a weak normal form, very close to the model examined
in Sec. 2, for a general flat two-sided cusp. Finally, in Sec. 5, we use the
oscillatory representations found in Sec. 4 to analyze the composition B∗A
for two FIOs associated with such a canonical relation. We point out that
the technique of deriving weak normal forms for degenerate FIOs and using
them to prove composition theorems has its origins in [16] and was continued
in [7]; different notions of weak normal forms were used to obtain estimates
in [11, 12].
2 Example of an FIO with a flat two-sided
cusp
2.1 A generalized Radon transform
We start with a simple example of a generalized Radon transform that in-
corporates the features of the linearized seismic inversion problem in which
we are ultimately interested. In fact, we will later show that perturbations
of this example serve as weak normal forms for the general class of operators
with flat two-sided cusps.
The operator averaging over translates in R4 of a curve γ(t) such that
...
γ˙, γ¨, γ, γ(4) are linearly independent (1)
3
is an FIO having a canonical relation which is a two-sided cusp [12], but the
same holds for the curve γ(t) = (t,t2,t4) in R3 [13, 14], and this is the model
we will use.
Thus, consider the generalized Radon transform R : ′(R3) ′(R3),
0
D −→ D
R f(x) = f x (t,t2,t4) χ(t)dt (2)
0
Z −
(cid:0) (cid:1)
= ei(x2−y2−(x1−y1)2))θ2+(x3−y3−(x1−y1)4)θ3)f(y)χ(x y )1(θ)dθ dθ dy,
1 1 2 3
ZR2 −
where χ C∞(R) is a fixed cutoff function. The associated canonical re-
∈ 0
lation, the conormal bundle C = N∗Z′ T∗R3 0 T∗R3 0 , where
0
⊂ \ × \
Z = supp(K ) R3 R3 and(x,ξ;y,η)′ =(cid:0)(x,ξ;y, η(cid:1))is(cid:0)standard(cid:1)notation
R0 ⊂ × −
for the twist map, is
C = x ,x ,x , 2(x y )θ 4(x y )3θ ,θ ,θ ;
0 1 2 3 1 1 2 1 1 3 2 3
− − − −
n(cid:16)
y ,y ,y , 2(x y )θ 4(x y )3θ ,θ ,θ : (3)
1 2 3 1 1 2 1 1 3 2 3
− − − −
(cid:17)
x y (x y )2 = x y (x y )4 = 0 .
2 2 1 1 3 3 1 1
− − − − − −
o
We will show that C has several properties, which remarkably also hold in
0
thetotallyunrelatedseismic imaging problem: (i)the projectionsbothtothe
leftandright,π ,π : C T∗R3 0,have(Whitney)cuspdegeneracies; (ii)
L R 0
−→ \
the cusp points for the two projections are the same, Σ (π ) = Σ (π ) :=
1,1 L 1,1 R
Σ ; and (iii) the images of the cusp points, π (Σ ) and π (Σ ), are
1,1 L 1,1 R 1,1
coisotropic (involutive) submanifolds of T∗R3. Conditions (ii) and (iii) are
unstable and quite special among two-sided cusps, i.e., canonical relations
satisfying (i); see the Remarks in Sec. 2.5 below.
2.2 Singularity classes
We first recall some basic facts about cusps and refer to [33, 27, 10] for
more details. Let f : RN RN be a smooth function. We say that f
→
drops rank simply at p if rank (df) = N 1 and if (d(detdf)) = 0, so
p − p 6
that Σ (f) := x RN : det(df(x)) = 0 , the corank one critical set of f,
1
{ ∈ }
is a smooth hypersurface near p. If kerdf T Σ (f), then f has a fold
p p 1
6⊂
singularity, and f is an immersion. Considering the more degenerate
|Σ1(f)
case when kerdf T Σ (f), one may choose a nonzero vector field v along
p p 1
⊂
4
Σ (f) such that v kerdf , so that v is tangent to Σ (f) at p. Let g be a
1 p 1
∈
smooth function such that g = 0 and dg = 0, e.g., g = detdf. Thus,
|Σ1(f) p 6
dg(v) has a zero at p.
Definition 2.1. f has a (Whitney) cusp at p if dg(v) has a simple zero at p.
Σ (f), the cusp set of f, is then codimension 2, and f is an im-
1,1 |Σ1,1(f)
mersion. One may use adapted coordinates to clarify this. These are lo-
cal coordinates such that f(x ,x ,...,x ) = (x ,x ,...,x ,h(x)) and
1 2 N 1 2 N−1
h(0) = 0 [27]. Then, Σ (f) = x : ∂h (0) = 0 , and f has a cusp singu-
1 { ∂xN }
larity at 0 iff ∂2h(0) = 0, ∂3h(0) = 0 and rank [d ( ∂h ),d (∂2h)] = 2.
∂x2N ∂x3N 6 x ∂xN x ∂x2N
These conditions are adapted coordinate-independent, and the notion of a
cusp makes sense for any smooth mapping between N-dimensional man-
ifolds. In suitable smooth coordinates on the domain and range spaces,
any map with a cusp singularity can be put into the local normal form,
f(x ,x ,...x ) = (x ,x ,...,x ,x x +x3 ).
1 2 N 1 2 N−1 N−1 N N
ForthecanonicalrelationC , theprojectiontotheleft, π : C T∗R3,
0 L 0
−→
is π (x ,x ,x ,y ,θ ,θ ) = (x ,x ,x , 2(x y )θ 4(x y )3θ ,θ ,θ );
L 1 2 3 1 2 3 1 2 3 1 1 2 1 1 3 2 3
− − − −
hence, letting α = 2θ +12(x y )2θ , β = 2(x y ) andγ = 4(x y )3,
2 1 1 3 1 1 1 1
− − − − −
one has
1 0 0 0 0 0
 0 1 0 0 0 0 
0 0 1 0 0 0
dπ =  
L  α 0 0 α β γ 
 − 
 0 0 0 0 1 0 
 
 0 0 0 0 0 1 
 
which has detdπ = 2θ +12(x y )2θ and kerπ = R ∂ at Σ (π ) =
L 2 1 − 1 3 L · ∂y1 1 L
2θ +12(x y )2θ = 0 . Since ∂ (2θ +12(x y )2θ ) = 24(x y )θ
{ 2 1− 1 3 } ∂y1 2 1− 1 3 − 1− 1 3
and ∂2 (2θ +12(x y )2θ ) = 24θ = 0, the cusp set is
∂y12 2 1 − 1 3 3 6
Σ (π ) = 2θ +12(x y )2θ = x y = 0 = θ = x y = 0 .
1,1 L 2 1 1 3 1 1 2 1 1
{ − − } { − }
Noting that 2θ +12(x y )2θ and x y have linearly independent gra-
2 1 1 3 1 1
− −
dients, we see that π has a cusp singularity.
L
Similarly, the projection to the right is
π (x,y ,θ ,θ )=(y ,x (x y )2,x (x y )4, 2(x y )θ 4(x y )3θ ,θ ,θ ),
R 1 2 3 1 2 1 1 3 1 1 1 1 2 1 1 3 2 3
− − − − − − − −
5
so that detdπ = 2θ 12(x y )2θ , and kerπ = ∂ at Σ (π ) = 2θ +
R − 2− 1− 1 3 R ∂x1 1 R { 2
12(x y )2θ = 0 . Onealsohas ∂ (2θ +12(x y )2θ ) = 24(x y )θ and
1− 1 3 } ∂x1 2 1− 1 3 1− 1 3
∂2 (2θ +12(x y )2θ ) = 24θ = 0, so that Σ (π ) = θ = x y = 0 ,
∂x21 2 1− 1 3 3 6 1,1 R { 2 1− 1 }
and thus π also has a cusp singularity.
R
Note that Σ (π ) = Σ (π ) and Σ (π ) = Σ (π ) =: Σ . The first
1 R 1 L 1,1 R 1,1 L 1,1
of these is true for any canonical relation [19], but the second is a very
strong condition; indeed, for general two-sided cusps, there is no relationship
between Σ (π ) and Σ (π ). Furthermore, for the images of these sets,
1,1 L 1,1 R
one has
27 27
π (Σ ) = ξ3 = ξ2ξ , π (Σ ) = η3 = η2η ,
L 1 { 2 − 8 1 3} R 1 { 2 − 8 1 3}
with π (Σ ) = ξ = ξ = 0 , π (Σ ) = η = η = 0 , resp., their
L 1,1 1 2 R 1,1 1 2
{ } { }
cuspidal edges. π (Σ ) and π (Σ ) are codimension two coisotropic (or
L 1,1 R 1,1
involutive) submanifolds ofT∗R3 0[19]; again, thisis a nongeneric situation.
\
Consider next the composition of canonical relations,
Ct C = (x,ξ;y,η) : (z,ζ) s.t. (x,ξ;z,ζ) Ct and (z,ζ;y,η) C .
0 ◦ 0 ∃ ∈ 0 ∈ 0
(cid:8) (cid:9)
For such a triple (x,ξ;z,ζ;y,η), one obtains the following equations:
z x = (z x )2, z x = (z x )4,
2 2 1 1 3 3 1 1
− − − −
ξ = ζ = 2(z x )ζ 4(z x )3ζ , ξ = ζ , ξ = ζ ,
1 1 1 1 2 1 1 3 2 2 3 3
− − − −
z y = (z y )2, z y = (z y )4,
2 2 1 1 3 3 1 1
− − − −
η = ζ = 2(z y )ζ 4(z y )3ζ , η = ζ , η = ζ .
1 1 1 1 2 1 1 3 2 2 3 3
− − − −
Using 2(z x )ξ +4(z x )3ξ = 2(z y )ξ +4(z y )3ξ , after sim-
1 1 2 1 1 3 1 1 2 1 1 3
− − − −
plification one obtains:
(y x )ξ +2[(z x )3 (z y )3]ξ = 0,
1 1 2 1 1 1 1 3
− − − −
(y x )[ξ +2(3z2 3z (x +y )+x2 +y2 +x y )ξ ] = 0
1 − 1 2 1 − 1 1 1 1 1 1 1 3
6
It follows that the contribution to Ct C from y x = 0 is contained
0 ◦ 0 { 1 − 1 }
in ∆, and that from y x = 0 is contained in C , where
1 1 0
{ − 6 }
e
C = x ,x ,x , 2(z x )θ 4(z x )3θ ,θ ,θ ;
0 1 2 3 1 1 2 1 1 3 2 3
− − − −
n
(cid:0)
e y ,y ,y , 2(z x )θ 4(z x )3θ , θ , θ : (4)
1 2 3 1 1 2 1 1 3 2 3
− − − −
x R3, y ,z R, θ R 0, y = x +(cid:1)(y x )(2z x y ),
1 1 3 2 2 1 1 1 1 1
∈ ∈ ∈ \ − − −
y = x +(y x )(2z x y )((z x )2 +(z y )2),
3 3 1 1 1 1 1 1 1 1 1
− − − − −
θ = 2(3z2 3z (x +y )+x2 +y2+x y )θ
2 − 1 − 1 1 1 1 1 1 1 3
o
Notice that ξ = η = 4θ (z x )(z y )(2z x y ) and that C
1 1 3 1 1 1 1 1 1 1 0
− − − −
intersects ∆ in codimension one, at x y = 0 .
{ 1 − 1 } e
The parametrization of C in (4) is a map
0
Υ : R5 (Re 0) T∗R3 0 T∗R3 0 .
x1,x2,x3,y1,z1 × θ3 \ −→ \ × \
(cid:0) (cid:1) (cid:0) (cid:1)
One easily sees that Υ is singular at Σ := Σ (Υ) = x y = x z = 0 ,
1 1 1 1 1
{ − − }
where kerdΥ = R ∂ , which is TΣ . Thus, as discussed below, C is
· ∂z1 6⊂ e 1 0
an open umbrella, exhibiting the simplest kind of singularity of a Lagrangian
e
manifold.
2.3 Open umbrellas
For maps between manifolds of the same dimension, a fold is a singularity
of the type S and a cusp is a singularity of type S [33, 27, 10]. From
1,0 1,1,0
larger dimensional spaces to smaller ones, S maps are submersions with
1,0
folds, while in the opposite direction they are referred to variously as cross
caps or Whitney-Cayley umbrellas [10, 9]. In the lowest possible dimensions,
if g : R2 R3 has a cross cap singularity then, in suitable local coordi-
→
nates, (u,v,w) = g(x,y) = (x2,y,xy), and its image is the algebraic surface
w2 = uv2 , the Whitney-Cayley umbrella. This is the simplest type of non-
{ }
immersed surface singularity in three dimensions. It is an immersion away
from the origin; it is actually an embedding on y = 0 ; and along y = 0 ,
{ 6 } { }
it is folded and hence 2-1.
Now, by adding one dimension to the range space, one may simulta-
neously both unfold the closed umbrella, making the parametrization 1-1
away from the origin, and make it a Lagrangian manifold with singularity in
7
R4. The open (or unfolded) umbrella is the map U : R2 R4, U(x,y) =
→
(x2,y,xy,2x3) (and its image). We have U∗ω = 0, where ω is the symplectic
3
form on R4 T∗R2. Hence, the unfolded umbrella is a Lagrangian inclusion,
≃
i.e., a variety which is a smoothly immersed Lagrangian manifold away from
its singular points [9]. To find a similar conic model, one needs to look in
T∗R3 0; for an unknown function f(t,s), consider
\
2
Λ = s2,t,f(t,s);stθ, t3θ,θ T∗R3 0 : θ = 0 .
1
3 ∈ \ 6
(cid:8)(cid:0) (cid:1) (cid:9)
Then, Λ is a Lagrangian if f is chosen so that dξ dx = 0. Since
1 ∧ |Λ1
dξ dx = d(stθ) d(t2)+d(2t3θ) ds+dθ df = dθ (2st2dt+2t3ds)+dθ df,
∧ ∧ 3 ∧ ∧ ∧ 3 ∧
this holds if df = 2st2dt 2t3ds = d(2t3s); using f = 2t3s yields the
− − 3 − 3 −3
conic Lagrangian,
2 2
Λ = t2,s, t3s;stθ, t3θ,θ : t,s R,θ R 0 ,
1
−3 3 ∈ ∈ \
(cid:8)(cid:0) (cid:1) (cid:9)
exhibiting an open umbrella singularity along (0,0,0;0,0,ξ ) : ξ = 0 .
3 3
{ 6 }
Observe that Λ may be parametrized by a degenerate phase function.
1
Letting ϕ(x,θ ,θ ,τ,η) = (x (τ)2)θ +(x + 2(τ)3x )θ , one has
1 2 1 − η 1 3 3 η 2 2
τ 2 τ
d ϕ = x ( )2, d ϕ = x + ( )3x
θ1 1 − η θ2 3 3 η 2
τ θ τ θ τ θ τ θ
d ϕ = ( 2 1 +2 2x ), d ϕ = ( )2( 2 1 +2 2x ),
τ 2 η 2
η − η η η − η − η η η
so that ϕ is degenerate, with both d ϕ and d ϕ having normal crossings:
τ η
d ϕ = 0 iff τ = 0 or 2θ1 + 2τ θ2x = 0, and d ϕ = 0 on same two sets.
τ − η η η 2 η
For the second case, solving for θ = τx θ we see that ϕ parametrizes Λ ,
1 η 2 2 1
while for the first case, ϕ also parametrizes Λ := N∗ x = x = 0 . Λ
0 1 3 0
{ }
and Λ intersect cleanly in codimension one, except at the singular set of
1
Λ , which is contained in the intersection. ϕ simultaneously parametrizes
1
Λ Λ , although it is not a multiphase function in the sense of [26], used to
0 1
∪
parametrize a pair of cleanly intersecting smooth Lagrangians. The class of
generalized FIOs we define in Sec. 5.1 has similar features.
For the general notion of an open umbrella, let (M,ω) be a smooth sym-
plectic manifold of dimension 2n.
8
Definition 2.2. S M is an open umbrella if it is the image of a map
⊂
ψ : Rn M such that dψ drops rank by one simply at Σ , of codim two;
1
→
ker (dψ) TΣ ; and Range (dψ ) is Lagrangian for p Rn Σ.
|Σ1 6⊂ 1 p ∈ \
The class of open umbrellas is structurally stable under smooth pertur-
bations in the class of Lagrangian inclusions [9]. As checked above, the
parametrization Υ in (4) satisfies the conditions in the definition, so that C
0
is an open umbrella.
e
2.4 Composition for the model flat two sided cusp
We now want to show that the composition A∗A for Fourier integral op-
erators, such as R , associated with the model flat two-sided cusp C in
0 0
(3) results in operators with wave front relation in the union of the diago-
nal ∆ and the open umbrella C from (4). Let A Im(R3,R3;C ), so that
0 0
∈
Af(x) = eiφ(x,y,θ)a(x,y,θ)f(y)dθdy withφ(x,y,θ) = (x y (x y )2))θ
e 2− 2− 1− 1 2
+(x yR (x y )4)θ and a Sm+21, the standard H¨ormander class of
3 − 3 − 1 − 1 3 ∈ 1,0
symbols. We consider the normal operator formed by the composition,
A∗Af(x) = ei(φ(z,y,η)−φ(z,x,θ))a(z,y,η)a¯(z,x,θ)f(y)dydzdθdη.
Z
We have
φ(z,y,η) φ(z,x,θ) = (z y (z y )2))η +(z y (z y )4)η
2 2 1 1 2 3 3 1 1 3
− − − − − − −
(z x (z x )2))θ (z x (z x )4)θ .
2 2 1 1 2 3 3 1 1 3
− − − − − − − −
After a stationary phase in (z2,η2) and (z3,η3) in the integral for KA∗A, we
obtain:
φ(z,y,η) φ(z,x,θ) = (x y +(z x )2 (z y )2)θ +(x y +(z x )4 (z y )4)θ
2 2 1 1 1 1 2 3 3 1 1 1 1 3
− − − − − − − − −
= (x y )θ +(x y )θ +(y x )[(2z x y )θ +(2z y x )((z x )2+(z y )2)θ ]
2 2 2 3 3 3 1 1 1 1 1 2 1 1 1 1 1 1 1 3
− − − − − − − − −
Let z = ζ , and then consider the change of variable,
1 θ3
ζ ζ ζ ζ
θ = (2 x y )θ (2 y x )(( x )2 +( y )2)θ ,
1 1 1 2 1 1 1 1 3
− θ − − − θ − − θ − θ −
3 3 3 3
9
for which
∂θ θ ζ ζ ζ
1 = 2 2 2(( x )2 +( y )2)) 2(2 x y )2,
1 1 1 1
∂ζ − θ − θ − θ − − θ − −
3 3 3 3
∂2θ 12 ζ ∂3θ 24
1 1
= (2 x y ), and = .
∂ζ2 −θ θ − 1 − 1 ∂ζ3 −θ
3 3 3
When ∂2θ1 = 0 then 2 ζ = x + y , θ = 0 and θ = (x1−y1)2θ . Let
∂ζ2 θ3 1 1 1 2 − 2 3
T(t ,t ) = (t t + 1t3,t ) be a normal form for a two-dimensional map with
1 2 1 2 2 1 2
a cusp singularity. To understand the push forward T u of a distribution u
∗
under the map T, note that the kernel of T is
∗
K (z,t) = δ(z T(t)) = ei[(z1−t1t2−12t31)σ1+(z2−t2)σ2]1(σ)dσ,
T∗ − Z
so that
T u(z) = ei[z1σ1+z2σ2] e−i[t2σ2+(t1t2+12t31)σ1]u(t)dt dσ.
∗
Z hZ i
Returning now to A∗A, letting z = θ1 and z = θ2 + 1(x y )2, the
1 θ3 2 θ3 2 1 − 1
phase function of A∗A becomes
φ(x,y,θ,σ) = (x y )θ +(x y )θ +(x y )θ
1 1 1 2 2 2 3 3 3
− − −
θ 1 θ 1
e +( 1 t t t3)σ +( 2 + (x y )2 t )σ
θ − 1 2 − 2 1 1 θ 2 1 − 1 − 2 2
3 3
and the amplitude becomes: a 1(σ). Next, we perform stationary phase,
×
first in t ,σ and then in θ ,σ . We have
2 2 1 1
θ 1
d φ = t σ σ , d φ = 2 + (x y )2 t .
t2 − 1 1 − 2 σ2 θ 2 1 − 1 − 2
3
e e
The Hessian is 1, the phase function is φ = (x y)θ + (θ1 1t3 t [θ2 +
− θ3 − 2 1 − 1 θ3
1(x y )2])σ , and
2 1 − 1 1 e
σ θ 1 θ 1
d φ = x y + 1, d φ = 1 t3 t ( 2 + (x y )2).
θ1 1 − 1 θ σ1 θ − 2 1 − 1 θ 2 1 − 1
3 3 3
e e
The Hessian is 1 . Finally, we introduce the variable τ = t θ and obtain
θ32 1 3
τ 1 τ 1 τ
φ = x y + (x y ) θ + x y + (x y )( )3 + (x y )3 θ ,
2 2 1 1 2 3 3 1 1 1 1 3
(cid:18) − θ − (cid:19) (cid:18) − 2 − θ 2θ − (cid:19)
3 3 3
e
10