text:
The view from (and of) Chatham–a conundrum
December 5, 2010 December 5, 2010 / Historian
The view from the terraces at Chatham. Photo by Donald Pfanz.
It is perhaps the most famous, photographed view in the Fredericksburg region: from Chatham's middle terrace, looking over the cannons, across the Rappahannock River to Fredericksburg, with its skyline featuring the three dominant elements that have marked it since 1855–the Baptist Church, St. George's Episcopal Church, and the Circuit Courthouse. But, while it's pleasing and useful, it's not the most important view related to Chatham.
The view from the slopes in front of Chatham, 1863--devoid of trees.
Think for a moment about the great antebellum plantations. Most of them have a signature element that bespeaks the owners' wealth and power. Kenmore has its stunning plaster work inside. Mannsfield was an elaborate, elegant place built of a material (sandstone) almost unheard of it major construction in the Fredericksburg region. Shirley has its exquisite woodwork.
But what about Chatham? What's its signature?
On the inside, Chatham has always been rather plain. It has beautiful paneling, but little ornamentation. Architecturally, it's fairly simple, though elegant. What distinguishes Chatham is its size. The main house is 185 feet long. Include the flanking laundry and kitchen, and the entire complex amounts to a horizontal slab of brick 220 feet long.
Situated as it is on Stafford Heights overlooking Fredericksburg, Chatham was not just meant to be seen; it was meant to awe.
Chatham during the war, as seen from the Fredericksburg side of the river--looking much as it did for its first 150 years. That's the ruins of the Chatham Bridge in the river.
Which brings us to the point of the day: historically by far the most important view or vista related to Chatham is that OF Chatham, not the view FROM Chatham. Clearly when William Fitzhugh's had Chatham built in 1771, he meant for the place to project his wealth, status, and power within the community. If you looked east from Fredericksburg, you saw Chatham. Its visibility on the landscape is surely a major reason it attracted so many prominent visitors; certainly it's why Union generals so often chose it as headquarters, especially during the mid-1862 occupation of Fredericksburg.
Today, Chatham's defining place in the landscape of Fredericksburg and Stafford has been lost, diminished by trees along the formerly clear banks of the Rappahannock, disrupted by trees on the grounds of Chatham itself.
Chatham, today surrounded by trees that have transformed its place on the Fredericksburg area landscape. Photo by Brad Graham.
You may leap to the conclusion that this is the result of neglect–that the vistas have stealthily been reduced to the point of nothing by outsmarting unaware NPS maintenance crews over the years . You'd be wrong. In fact, Chatham's virtual disappearance from Fredericksburg's landscape (and, with all due apologies to Stafford County, Chatham was emphatically part of Fredericksburg's landscape) was a conscious decision that in itself tells us something important about the centrality of the Civil War to American history. After the war–after the end of slavery–Chatham underwent a transformation common to many Southern plantations: from a working, profit-oriented plantation of 1300 acres to a vastly smaller (30 acres) refuge for the rich.
Chatham's later owners sought privacy, not prominence. The imperative for tillable land eventually disappeared, and former fields grew into forests. Massive, purposeful trees grew up on the foreslope of Chatham. And, just as significant, as the importance of the Rappahannock to the prosperity of Fredericksburg diminished in the 20th Century, trees grew up along its banks too. Chatham disappeared. Today there are precious few places in Fredericksburg from which Chatham can be seen, even in winter. Most residents hereabouts don't even know it's there. And that is exactly how the famously private John Lee Pratt (Chatham's final private owner, from 1931 until 1975) wanted it.
While it's easy to recognize that Chatham's literal obscurity is the result of a historical continuum that tells us important things about a changing American economy, society, and landscape, it's also important to recognize that the evolution of Chatham from working plantation to a secluded retreat for wealthy owners came at the expense of its defining physical characteristic: its dominant place on the Fredericksburg landscape. Given modern imperatives for managing watersheds and a societal affection for trees, that's not likely to change.
[It's worth noting that even if the NPS clear-cut the foreslope in front of Chatham, its visibility from Fredericksburg would still be limited. Many of the trees that obscure the view are in the river bottom, on the Fredericksburg bank of the river. Still, it's fun to ponder the effect if Chatham were once again made visible on the Fredericksburg landscape. No doubt it would be as attractive today as it was to the steady flow of visitors that beset Fitzhugh in the 18th century (and in part inspired him to move to a townhouse in Alexandria). Our visitation would probably triple, curious to know the story behind that brick slab that dominates Fredericksburg from Stafford Heights. Alas, it will not be…]
Chatham, Park Development, Stafford County sites, Town and Civilian Sites
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12 thoughts on "The view from (and of) Chatham–a conundrum"
michael snyder
Who was responsible for "Home Sweet Home"?
For those of you who have not been to Chatham, the cast-iron railing on the steps leading down to the middle terrace has musical notes on it–notes, we have determined, that make for a slightly askew version of "Home Sweet Home." The steps and railing date to the ownership of the Devores, in the 1920s. John H.
John Kelley
Is there a date for the image photographed from the west side of the Rappahannock River looking toward the Chatham home?
Was it part of the photographs taken in 1862; or 1863; or May, 1864?
Also, the photograph taken from the grounds of the Chatham residence was photographed in the morning on May 3rd, 1863.
John: I do not have a date at hand. It seems most likely to me that the image was taken in 1864, when other images of the Rappahannock waterfront were taken from the Fredericksburg side of the river. When I get a chance, I will check our files.
For those of you reading this, John has done terrific and important work on the 1863 photographs of Fredericksburg–both in May and June 1863. For an example, check out his article on the images at Franklin's crossing: http://www.civilwarphotography.org/hidden-in-plain-sight
Eric Mik
I found that image while going through the collection of the Western Reserve Historical Society. It is mounted as a stereo on nondescript card stock and lacking any printed backmark. Written, in ink, on the reverse is: "Piers of R.R. Bridge at Fredericksburg May 10, 1864." Refernece to the railroad bridge appears to be an error, but I suspect the date is probably accurate.
I'm amazed at the mind-boggling numbers of old photographs that you are able to find and share. There must be a hum dinger of an archival database maintained somewhere.
Much appreciate the rich diversity of history you and your colleagues provide here.
Larry: Over the years Donald Pfanz of our staff has worked hard to digitize images both from our collection and other sources, and the collection grows almost every week. It's safe to say that we probably have 98% of known wartime images in our digital archives, and there are few pre-1900 images that we know of that haven't been added. We are the beneficiaries of a lot of help in this–from other federal agencies, other archives, and private collectors like Jerry Brent, who has been incredibly generous with his finds. Still, I am amazed that new things still come in, though admittedly at a variable pace. Many thanks for reading. John H.
Thank you all for that image and where it came from as well as the date – Provides an important dimension to Chatham's location.
I know it would not be possible to cut down all the trees, but the trees are hiding the beauty of Fredericksburg and environs as well as the historic importance of the area.
When we talk about the defining characteristics of historic homes, the view of the home is not often mentioned. Thanks for focusing on that very real feature. Another local battlefield landmark currently obscured much like Chatham is Federal Hill. Wouldn't it be great to be able to see that prominent landmark from Willis Hill/Sunken Road area. Perhaps the new owners can be encouraged to replace the too tall evergreens on their foreslope with lower growing trees that obscure the factory while restoring their historic view. Hmmm.
When I was first in the building I thought who would build such a long thin home until I ws on the river side and realized what a power symbol this would be when viewed from the city. Great work.
Robert A. Mosher
As I read this posting, I thought of the following journalist's report. Charles Carleton Coffin, Army Correspondent of The Boston Journal, wrote the following letter to his newspaper from Chatham house on December 9, 1862 (excerpted from "Four Years of Fighting" by Charles Carleton Coffin):
"It is a clear, cold morning. The sky is without a cloud. Standing near General Sumner's quarters, I have a wide sweep of vision. The quarters of the veteran general commanding the right grand division are in a spacious mansion, newly constructed, the property of wealthy planter, whose estate is somewhat shorn of its beauty by the ravages of war. The fences are all gone, the forests are fast disappearing, the fine range of cedars which lined the Belleplain road are no longer to be seen. All around are the white tents of the command, the innumerable camp-fires sending up blue columns of smoke. The air is calm. You hear the rumbling of distant baggage-trains, the clatter of hundreds of axes felling the forests for fuel,–the bugle-call of the cavalrymen, and the rat-a-plan of the drummers, and mingling with all, the steady, constant flow of the falling waters of the winding stream.
Looking far off to the southeast, across the intervals of the river, you see a white cloud of steam moving beneath the fringe of a forest. It is a locomotive from Richmond, dragging its train of cars with supplies for the Rebel camps. The forests and hills beyond are alive with them. Resting my glass against the side of the building to keep it steady, I can count the men grouped around the camp-fires, turning at times to keep themselves warm. Others are bringing in wood. An officer rides along. A train of wagons is winding down the hill toward the town. All along the range of hills are earthworks with sandbag embrasures, and artillery behind,–not quaker guns, I think, but field artillery, so ranged that a movement directly across the river would be marching into the jaws of death,—as hazardous and destructive as the charge of the Light Brigade at Balaklava.
I know that there is a clamor for an onward movement, a desire and expectation for an advance; but I think there are few men in the country who, after taking a look at the Rebel positions, would like to lead in a movement across the stream.
Looking into the town of Fredericksburg we see but few smokes ascending from chimneys, but few people in the streets. It is almost wholly deserted. The women and children have gone to Richmond, or else are shivering in camp. Close upon the river-bank on either side face the pickets, within easy talking distance of each other. There has been no shooting of late. There is constant badinage. The Rebel picket asks the Yankee when he is going to Richmond. The Yankee asks the Rebel if he don't want a pair of boots. I am sorry to say that such conversation is mixed with profane words. Each party seems to think that hard words hit hard.
Last night the southern sky was red with the blaze of Rebel camp-fires. Far off to the southeast I see a hazy cloud, and columns of smoke, indicating the presence of a large army. I do not doubt that if we attempt to cross we shall meet with terrible opposition from a force nearly if not quite as large as our own.
If the President or General Halleck insist upon Burnside's making the movement, it will be made with whatever power, energy, determination, and bravery the army can exhibit. I am as anxious as any one can be to see a great blow given to the Rebellion; but I am not at all anxious to see the attempt made against such disadvantages as are apparent to the most casual observer from this position."
Robert: Thanks very much for sharing Coffin's account…. Coffin, who wrote by the name of "Carleton," was a correspondent for the Boston Journal, and later put his wartime dispatches together into a series of books. Great addition to the post.


text:
Q: UndefVarError: * not defined I'm having a persistent error where Julia tells me that the multiplication operator is undefined when I use multiplication within a package I'm working on. Nowhere in the package do I redefine or expand the Base multiplication operator, and there are some places where multiplication is recognized and others where it is not. For example, in the following code
    Λ = h.metric.Λ # Type Int64
    kSize = 2*Λ+1
    for i = 1:n_steps
        # Tempering
        r = temper(lf, r, (i=i, is_half=true), n_steps)
        # Take a half leapfrog step for momentum variable
        r = r - map(gradi -> ϵ / 2 .* gradi, grad)

the multiplication on line 2 gives me UndefVarError: * not defined wheras the multiplication on the last line compiles and executes without the error.
Multiplication outside the package and in the Julia REPL on Visual Studio Code does not seem to have the same error.
I've tried Using Base: * and import Base.* in both the main module and the individual Julia packages in the file to no avail. Any ideas about what could be going on?

A: This will appear if you accidentally assign to * later in the function or in a branch that didn't get taken.  An easy place where this could happen is in an if expression that intends to ask if a*b == c but mistakenly uses = instead of ==:
julia> function test(a, b, c)
           d = a*b
           if (a*b = c)
               println("got $d")
           end
       end
test (generic function with 1 method)

julia> test(2,3,6)
ERROR: UndefVarError: * not defined
Stacktrace:
 [1] test(a::Int64, b::Int64, c::Int64)
   @ Main ./REPL[4]:2
 [2] top-level scope
   @ REPL[5]:1

A: Without a reproducible example, and full error message it is difficult to help. The only context that I can think of in which * would be undefined is within baremodule
julia> *
* (generic function with 328 methods)

julia> module M1
       *
       end
Main.M1

julia> baremodule M2
       *
       end
ERROR: UndefVarError: * not defined
Stacktrace:
 [1] top-level scope
   @ REPL[2]:2

redpajama_set_name: RedPajamaStackExchange


If you loved Etsy when there were only a few people on it, you'll know my guest! Weird little critters, vibrant patterns, and the positive passwords that helped her along the way. Today is all about the evolution of "ashley g" to "Ashley Goldberg".


text:
\section{Introduction}

Consider the inverse scattering problem of determining an unbounded rough surface in
a two-layered dielectric media from near-field measurements. This kind of problems plays a
fundamental role in diverse scientific areas such as radar and sonar detection, underwater
exploration and non-destructive testing. In such applications, one tries to recover the shape and
location of unknown surfaces by taking the scattered wave-field measurements in certain domain.

Precisely, we consider the scattering of a time-harmonic line source
by an unbounded rough surface. The surface is assumed to
be a local perturbation of a plane which separates the whole space into two unbounded parts.
We are restricted to the two-dimensional case by assuming that the interface and the line source
are invariant in the $x_3$ direction.
Then the wave motion is governed by the two-dimensional Helmholtz equation with the wavenumber
described by a piecewise constant function. Due to the assumption of the interface,
the Sommerfeld radiation condition remains valid to describe the behavior of the scattered wave
away from the rough interface. Given the incident wave (which is a point source in 2D) and the rough
interface, the forward problem is to find the distribution of the scattered wave-field in the whole space.
Many works have been done to study the existence of unique solutions to the forward scattering problem.
We refer the reader to \cite{SE10,MT06,LYZ13,DTS03,QZZ19,ZS03,ZZ13} with either the variational technique
or the boundary integral equation method.

We are mainly interested in the inverse problem of recovering the shape and location of the unknown
interface from the scattered near-field measurements. In \cite{YLZ17}, a global uniqueness theorem
has been proved in a more general case for simultaneously determining the locally rough interface together
with the wavenumber and embedded obstacles in the lower half-space from the near-field measurements
in the upper-half space.
Following this uniqueness result, in this paper we aim to develop an efficient sampling-type method to
solve the associated inverse problem of recovering the unbounded rough interface in the case with no
embedded obstacles.
For the case of unbounded impenetrable surfaces, the above inverse problem has
been extensively studied numerically, most of which focused on iterative optimization algorithms
under some a priori information on the surface. We refer to \cite{CR10} for a Kirsch-Kress approach,
\cite{LB13} for a nonlinear integral equation method, \cite{BGL11,GJ11,GJ13,CP02,QZZ19,RM15,ZZ13} for
Newton-type iterative algorithms, and \cite{GL13} for an inversion algorithm based on the transformed
field expansion in the case when the rough surface is a small and smooth perturbation of a plane.
In addition, several non-iterative sample methods have also been studied
for locating the unbounded rough surface, such as a time domain point source method \cite{LC05},
the factorization method \cite{AL08} and the linear sampling method \cite{DLLY17} for the case
with a Dirichlet condition on the surface.
It is noted that the inversion algorithm in \cite{AL08} was shown to be valid
theoretically only for $\kappa f_+\in (0,\sqrt{2})$, where $\kappa>0$ stands for the wavenumber
and $f_+$ denotes the amplitude of the rough surface. It remains open to show the effectiveness of
the factorization method for the unbounded surface in a general case for $\kappa\in{\mathbb R}_+$.
If the surface is considered to be a penetrable interface in dielectric media, few inversion algorithms
are available for recovering the shape and location of the surface.
A Kirsch-Kress method was proposed in \cite{LZ13} for reconstructing a locally perturbed plane interface,
making use of near-field measurements both above and below the interface. An Newton-type iteration algorithm
was introduced in \cite{CG11} to reconstruct a locally rough interface from far-field measurements.
Further, the inversion algorithm proposed in \cite{GL13} was extended in \cite{GL14} to deal with the case
where the rough surface is a small and smooth perturbation of a plane interface in dielectric media.
Recently, a direct imaging method has been proposed to reconstruct an unbounded rigid
rough surface either from the elastic scattered near-field Cauchy data generated by point sources in \cite{LZZ19}
or from the elastic scattered near-field data generated by elastic plane waves in \cite{HLZZ19} and
to recover an impenetrable or penetrable unbounded rough surface either from the acoustic scattered
near-field Cauchy data generated by point sources in \cite{LZZ18} or from the acoustic scattered near-field
data generated by plane waves in \cite{Zhang20}.
Further, a direct imaging algorithm was developed in \cite{XZZ19} for recovering an unbounded Dirichlet rough
surface from phaseless near-field data generated by plane waves.
However, it seems difficult to develop a linear sampling method for locating an unbounded interface in
dielectric media. As far as we know, no such a result is available so far.

In this paper we will investigate the linear sampling method (LSM) as an analytical as well as a numerical
tool to solve the inverse scattering problem of recovering a locally rough interface in dielectric media
from near-field measurements in a certain domain. It is well-known that the classical LSM was first
introduced \cite{DA96} for inverse acoustic scattering by bounded obstacles and has been extended to
many other inverse problems since then (see, e.g., \cite{CK13}) since the reduced algorithm is fast and
does not need any a priori knowledge on the scatterers.
We remark that a modified version of the LSM was recently proposed in \cite{DLLY17}, where an auxiliary
rough surface is introduced in order to recover a locally rough surface with a Dirichlet boundary condition.
Partially motivated by \cite{DLLY17} and \cite{YLZ17}, we reformulate the interface scattering problem into
an equivalent integral equation formulation in a bounded domain by introducing a class of special rough
surfaces, where the well-posedness easily follows from the classical Riesz-Fredholm alternative.
With this technique, a modified near-field operator equation associated with a special surface
will be constructed, which is proved in a strict way to be valid for recovering the shape and location of
the locally rough interface, leading to a fast imaging algorithm. As two related works, we also
refer the reader to \cite{YZZ1,YZZ2} for a periodical version.

The remaining part of the paper is organized as follows. In Section \ref{sec2}, we introduce the
mathematical model for the forward scattering problem.
In Section \ref{sec3}, we propose a novel version of the classical LSM by constructing a modified
near-field operator equation associated with a special rough surface.
In Section \ref{sec4}, numerical experiments are conducted to demonstrate the effectiveness
of the proposed method.

\section{The mathematical model}\label{sec2}
\setcounter{equation}{0}

We now introduce the mathematical formulation of the scattering problem by an unbounded
rough interface in two dimensions. The unbounded interface is denoted by
the curve $\Gamma=\{(x_1,x_2)\in{\mathbb R}^2: x_2=f(x_1), x_1\in{\mathbb R}^1\}$, where $f\in C^{2,\alpha}({\mathbb R}^1)$,
$\alpha\in (0,1)$, is a smooth function (see Figure \ref{f1}), where
the function $f$ is assumed to satisfy the condition that
there exists a positive constant $M>0$ such that
\begin{eqnarray}\label{Condition-A}
f(x_1)=0\qquad {\rm for}\;\;\;|x_1|\geq M.
\end{eqnarray}
This condition means that the interface $\Gamma$ is only different from the plane
$\Gamma_0=\{(x_1,x_2)\in{\mathbb R}^2: x_2=0\}$ for $x_1$ in a finite interval $(-M,M)$. \
Let $\Omega_1:=\{(x_1,x_2)\in{\mathbb R}^2: x_2>f(x_1), x_1\in{\mathbb R}^1\}$ denote
the unbounded domain above $\Gamma$ which is filled with a homogeneous material characterized
by a constant wavenumber $\kappa_1>0$. Denote by $\Omega_2:={\mathbb R}^2\setminus\overline{\Omega_1}$ the complement of $\Omega_1$
in ${\mathbb R}^2$ which is filled with another homogeneous material characterized by a different
wavenumber $\kappa_2>0$.

\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{pictures/geometry.pdf}
\caption{The physical configuration of the scattering problem}\label{f1}
\end{figure}

Consider an incoming wave induced by the point source in 2D
\begin{eqnarray*}
u^{\rm i}(x) = \Phi_{\kappa_1}(x,y):=\frac{\rm i}{4}H_0^{(1)}(\kappa_1|x-y|)\qquad
{\rm for\;}\;\;y\in\Omega_1,
\end{eqnarray*}
where $H_0^{(1)}(\cdot)$ is the Hunkel function of the first kind of order zero and
$\Phi_{\kappa_1}(\cdot,\cdot)$ denotes the fundamental solution of the two-dimensional Helmholtz equation
satisfying $\Delta \Phi_{\kappa_1}+\kappa_1^2\Phi_{\kappa_1}=-\delta_y$ in ${\mathbb R}^2$
with $\delta_y(\cdot)=\delta(\cdot-y)$. The associated wavelength of the incident wave is then given
by $\lambda=2\pi/\kappa_1$. Thus the scattering problem of $u^{\rm i}$ by $\Gamma$ is modelled
by the Helmholtz equation
\begin{eqnarray}\label{a1}
\Delta u+\kappa^2(x)u=-\delta_y\qquad{\rm in}\;\;\; {\mathbb R}^2,
\end{eqnarray}
where $u:=u^{\rm i}+u^{\rm s}$ denotes the total field in $\Omega_1$ consisting of the incident field
$u^{\rm i}$ and the scattered field $u^{\rm s}$, and $u:=u^{\rm s}$ in $\Omega_2$ denotes
the transmitted field. Moreover, $\kappa(x)$ is defined as $\kappa(x):=\kappa_1$ for $x\in\Omega_1$
and $\kappa(x):=\kappa_2$ for $x\in\Omega_2$.

Due to the condition (\ref{Condition-A}) on the surface $\Gamma$, the Sommerfeld radiation condition
\begin{eqnarray}\label{a2}
\displaystyle\lim_{|x|\rightarrow \infty}\sqrt{|x|}\left(\frac{\partial u^{\rm s}}{\partial |x|}
-{\rm i}\kappa u^{\rm s}\right)=0
\end{eqnarray}
remains valid to describe the asymptotic behavior of the scattered field $u^{\rm s}$ away from $\Gamma$.
It is remarked that the Sommerfeld radiation condition (\ref{a2}) should be replaced by the much weaker
Upward and Downward Propagating Radiation Conditions (cf. \cite{SZ99}) for the globally rough interface case,
which can be shown to be equivalent to (\ref{a2}) for the locally rough interface case. The reader is referred
to \cite{DLLY17,LZ13} for more detailed discussions.

Uniqueness of solutions to the scattering problem (\ref{a1})-(\ref{a2}) can be found in \cite{SZ99}
(see Theorem 5.1 of \cite{SZ99}), where a more complicated scattering problem was dealt with for an infinite
inhomogeneous conducting or dielectric layer at the interface.
The existence of solutions to the problem (\ref{a1})-(\ref{a2}) can be established by the integral equation
method using the Green function of the two-layered medium, that is, the fundamental solution of the
unperturbed problem ($f=0$; cf. \cite{Li2010}).
It can also be proved by either using the boundary integral equation method in \cite{DTS03} or
reducing (\ref{a1})-(\ref{a2}) into an equivalent Lippmann-Schwinger integral equation in a bounded domain
based on the Green function $G_r$ below (see \cite{YLZ17}).

To propose our sampling method in this paper, we follow \cite{YLZ17} to introduce a special class of rough
interfaces $\Gamma_r$ (see Figure \ref{f1}) defined by
\begin{eqnarray*}
\Gamma_r:=\{(x_1,x_2)\in{\mathbb R}^2: x_2=0\quad{\rm for}\;\;|x_1|\geq r,\;\;{\rm and}\;\; x_2=-\sqrt{r^2-x_1^2}\quad{\rm for}\;\;|x_1|< r\}
\end{eqnarray*}
for each fixed $r>0$. It can be seen that $\Gamma_r$ differs from the exact scattering interface $\Gamma$ only in
a finite interval. Let $\Omega_{1,r}$ and $\Omega_{2,r}$ denote the unbounded domains above and below
the interface $\Gamma_r$, respectively. Throughout this paper, we choose a sufficiently large $r>M$ such that
the local perturbation of $\Gamma$ lies totally inside the region $\Omega_{1,r}$, that is,
$\Omega_1\subset\Omega_{1,r}$.

Consider the scattering of the incident point source $\Phi_{\kappa_1}(\cdot, y)$ for
$y\in\Omega_{1,r}$ by $\Gamma_r$, which is to find the solution $G_r(\cdot,y)$ satisfying the problem
\begin{eqnarray}\label{c1}
\left\{\begin{array}{ll}
\Delta G_r(x,y)+\kappa_r^2(x)G_r(x,y)=-\delta_y& {\rm in}\;\;{\mathbb R}^2\\[2mm]
\displaystyle\lim_{|x|\rightarrow\infty}\sqrt{|x|}\left(\frac{\partial G_r(x,y)}{\partial |x|}-{\rm i}\kappa_r G_r(x,y)\right)=0&
\end{array}\right.
\end{eqnarray}
where the wavenumber $\kappa_r(x)$ is defined as $\kappa_r(x):=\kappa_j$ for $x\in\Omega_{j,r}$, $j=1,2$. It follows from \cite{YLZ17} that Problem (\ref{c1}) is uniquely
solvable for each fixed $y\in\Omega_{1,r}$.
It is also well-known that $G_r(\cdot,\cdot)$ corresponds to the
Green's function associated with the special interface $\Gamma_r$. Similar to (\ref{a1})-(\ref{a2}),
$G_r(\cdot,y)$ can be decomposed into the sum of the incident field $\Phi_{\kappa_1}(\cdot, y)$
and its scattered field $G_r^{\rm s}(\cdot,y)$ in $\Omega_{1,r}$ for $y\in\Omega_{1,r}$, i.e.,
$G_r(\cdot,y)=\Phi_{\kappa_1}(\cdot,y)+G_r^{\rm s}(\cdot,y)$ in $\Omega_{1,r}$.

Next, we shall briefly derive an equivalent integral equation of Problem (\ref{a1})-(\ref{a2})
with the help of Problem (\ref{c1}), following the idea in \cite{YLZ17}. To this end,
denote by $D=\Omega_2\cap\Omega_{1,r}$ the intersection between $\Gamma_r$ and $\Gamma$. Applying
the Green's formula for the difference $v_r(x; y):=u(x,y)-G_r(x,y)$, it can be verified that
the solution $u(\cdot,y)$ to Problem (\ref{a1})-(\ref{a2}) satisfies the Lippmann-Swinger integral
equation in $D$:
\begin{eqnarray}\label{c3}
u(\cdot,y)+\eta\int_{D}G_r(\cdot,z)u(z,y){\rm d}z=G_r(\cdot,y)\qquad {\rm for\;\;}
\eta:=\kappa_1^2-\kappa_2^2.
\end{eqnarray}
Define the map $T: L^2(D)\rightarrow L^2(D)$ by
\begin{eqnarray*}
(T\phi)(x):=\int_{D}G_r(x,z)\phi(z){\rm d}z\qquad{\rm for}\;\; x\in D.
\end{eqnarray*}
Then the equation (\ref{c3}) can be rewritten in the operator equation form:
\begin{eqnarray}\label{c4}
(I+\eta T)u(\cdot,y)=G_r(\cdot,y)
\end{eqnarray}

By a similar discussion as in Theorem 8.3 of \cite{CK13}, we have the following equivalence result
between the solvability of Problems (\ref{a1})-(\ref{a2}) and (\ref{c4}).

\begin{theorem}\label{thm2.1}
For $y\in\Omega_1$, if $u(\cdot,y)$ is a solution of (\ref{a1})-(\ref{a2}), then $u(\cdot,y)|_{D}$ is a solution of (\ref{c4}). Conversely, if $u(\cdot,y)$ is a solution of (\ref{c4}), then
\begin{eqnarray*}
u(x,y):=G_r(x,y)-\eta\int_{D}G_r(x,z)u(z,y){\rm d}z\qquad {\rm for}\;\; x\in{\mathbb R}^2,
\end{eqnarray*}
is a solution of (\ref{a1})-(\ref{a2}). Furthermore, the map $I+\eta T$ is an isomorphism on $L^2(D)$.
\end{theorem}

\section{The inverse scattering problem}\label{sec3}
\setcounter{equation}{0}

Based on Theorem \ref{thm2.1}, we study in this section the LSM as an analytical as well as a numerical tool to solve the inverse scattering problem of recovering the shape and location of
the unbounded interface $\Gamma$ by the knowledge of scattered fields $u^{\rm s}(\cdot,y)$ on the
measurement segment $\Gamma_{b,a}:=\{(x_1,x_2)\in{\mathbb R}^2: x_2=b, |x_1|<a\}$. Here, $a>0$ and
$b\in{\mathbb R}$ such that $b>f_+:=\sup_{x_1\in{\mathbb R}}f(x_1)$. Similar to the bounded case \cite{DA96},
our objective is to define an indicator function $I(z)$ by the $L^2$ norm of the solution to
the first kind of integral equation associated with the near-field operator
${\bf N}:L^2(\Gamma_{b,a})\rightarrow L^2(\Gamma_{b,a})$:
\begin{eqnarray}\label{d1}
({\bf N}g)(x): = \int_{\Gamma_{b,a}}u^{\rm s}(x,y)g(y){\rm d}s(y)\qquad \textrm{for}\;\; x\in\Gamma_{b,a},
\end{eqnarray}
or its modified version, which can provide a fast imaging algorithm for $\Gamma$. In (\ref{d1}) $u^{\rm s}(\cdot,y)$ indicates the scattered solution to Problem (\ref{a1})-(\ref{a2}) for the incident point
source $\Phi_{\kappa_1}(\cdot,y)$.

Recalling for the bounded obstacle case, the far-field operator $F$ could be decomposed into
$F=\Lambda H$ with $\Lambda$ a solution operator and $H$ an incidence operator for the obstacle scattering problem. However, it becomes very difficult to directly obtain such decomposition in the unbounded rough surface case especially for a penetrable interface.
To overcome this difficulty, we instead consider a modified near-field operator
 ${\bf N}_{\rm Mod}:L^2(\Gamma_{b,a})\rightarrow L^2(\Gamma_{b,a})$:
 \begin{eqnarray}\label{d22}
({\bf N}_{\rm Mod}g)(x):=\int_{\Gamma_{b,a}}(u^{\rm s}(x,y)-G_r^{\rm s}(x,y))g(y){\rm d}s(y)\qquad \textrm{for}\;\; x\in\Gamma_{b,a}
\end{eqnarray}
by the difference $v_r(x,y):=u(x,y)-G_r(x,y)$ which is easily checked from (\ref{a1})-(\ref{a2}) and (\ref{c1}) to satisfy the boundary value problem
\begin{eqnarray}\label{c2}
\left\{\begin{array}{llllll}
\Delta v_r(\cdot,y)+\kappa_1^2v_r(\cdot,y)=0 & {\rm in}\;\; \Omega_1\\[2mm]
\Delta v_r(\cdot,y)+\kappa_2^2v_r(\cdot,y)=\beta(\cdot) &{\rm in}\;\; D\\[2mm]
\Delta v_r(\cdot,y)+\kappa_2^2v_r(\cdot,y)=0 &{\rm in}\;\; \Omega_{2,r}\\[2mm]
[v_r(\cdot,y)] =[\partial_{n} v_r(\cdot,y)]=0& {\rm on}\;\; \Gamma\cup\Gamma_r\\[2mm]
\displaystyle\lim_{|x|\rightarrow \infty}\sqrt{|x|}\left(\frac{\partial v_r(\cdot,y)}{\partial |x|}-{\rm i}\kappa v_r(\cdot,y)\right)=0&\\[2mm]
\end{array}\right.
\end{eqnarray}
where $\beta(\cdot):=\eta G_r(\cdot,y)$, and $[\cdot]:=\cdot|_{+}-\cdot|_{-}$ with $\cdot|_{\pm}$ indicating  the limits of $\cdot$ from the upward and downward of $\Gamma$ or $\Gamma_{r}$, respectively.
The well-posedness of Problem (\ref{c2}) easily follows from the well-posedness of the original scattering problem (\ref{a1})-(\ref{a2}).
Notice that Problem (\ref{c2}) may be viewed as the scattering by a bounded inhomogeneous
medium $D$ with its boundary containing the local perturbation of the interface $\Gamma$.
Therefore, our sampling method will be based on studying the following integral equation
\begin{eqnarray}\label{d2}
({\bf N}_{\rm Mod}g)(x)=G_r(x,z)|_{\Gamma_{b,a}}\qquad {\rm for}\;\;z\in{\mathbb R}^2.
\end{eqnarray}
It is expected to define an indicator function $I_{\rm Mod}(z)$ by the $L^2$ norm of the density $g$ in  (\ref{d2}), which asymptotic behavior could be used to recover the shape and location of $D$ or $\Gamma$.

To illustrate this point, we introduce the function
\begin{eqnarray}\label{d3}
v_g(x): = \int_{\Gamma_{b,a}}G_r(x,y)g(y){\rm d}s(y)\qquad x\in D,
\end{eqnarray}
for $g\in L^2(\Gamma_{b,a})$ which clearly satisfies the Helmholtz equation $\Delta v_g+\kappa_1^2 v_g=0$ in $D$. Define the space $X:=\{v_g|_{D}: g\in L^2(D)\}$ and
denote by $\overline{X}$ the closure of $X$ in the sense of $L^2(D)$ norm. Moreover,
we introduce the operator $F: L^2(D)\rightarrow L^2(\Gamma_{b,a})$ defined by
\begin{eqnarray*}
(Fh)(x): = -\eta\int_{D}G_r(x,z)h(z){\rm d}z\qquad{\rm for}\;\; x\in\Gamma_{b,a}.
\end{eqnarray*}
By combining equation (\ref{c4}) and Theorem \ref{thm2.1}, we have the following relation
\begin{eqnarray}\nonumber
({\bf N}_{\rm Mod}g)(x)&=&\int_{\Gamma_{b,a}}(Fu(\cdot,y))(x)g(y){\rm d}s(y)\\\label{d4}
&=&\int_{\Gamma_{b,a}}F(I+\eta T)^{-1}G_r(x,y)g(y){\rm d}s(y)\\\nonumber
&=&F(I+\eta T)^{-1}v_g(x)
\end{eqnarray}
from the definition (\ref{d22}). Thus, the near-field operator equation (\ref{d2}) can be
reformulated as
\begin{eqnarray}\label{d5}
F(I+\eta T)^{-1}v_g(\cdot)=G_r(\cdot,z)|_{\Gamma_{b,a}}\qquad {\rm for} \;\;z\in{\mathbb R}^2.
\end{eqnarray}

In order to investigate the solvability of equation (\ref{d5}), similar to the bounded case, we introduce the interior transmission problem in finding a pair of functions $(v,w)\in L^2(D)\times L^2(D)$ such that  $(v,w)$ satisfies the Helmholtz equations
\begin{eqnarray}\label{d6}
\Delta v+\kappa_1^2v=0,\quad \Delta w+\kappa_1^2q w=0\qquad{\rm in}\;\; D,
\end{eqnarray}
and the transmission conditions
\begin{eqnarray}\label{d7}
w-v=G_r(\cdot,z)\quad \frac{\partial w}{\partial n}-\frac{\partial v}{\partial n}=\frac{\partial G_r(\cdot,z)}{\partial n}
\qquad{\rm on }\;\;\partial D,
\end{eqnarray}
for $z\in D$ and $q:=\kappa_2^2/\kappa_1^2$. It is well-known that $\kappa_1^2>0$ is
called a transmission eigenvalue if the homogeneous problem (\ref{d6})-(\ref{d7}) has a pair
of nontrivial solutions $(v,w)\in L^2(D)\times L^2(D)$ such that $v-w\in H_0^2(D)$.
For simplicity, we always assume throughout the paper that {\em $\kappa_1^2>0$ is not a transmission eigenvalue}. Under this assumption, we conclude from \cite{CGH10} that Problem (\ref{d6})-(\ref{d7}) has a unique solution $(v(\cdot,z),w(\cdot,z))\in L^2(D)\times L^2(D)$ for
$z\in D$.

Furthermore, we introduce the space $Y$ defined by
\begin{eqnarray*}
Y:=\{u\in L^2(D): \Delta u\in L^2(D),\;\Delta u+\kappa_1^2 u=0  \;{\rm in}\; D\;{\rm in\;the\;distributional\;sense}\}
\end{eqnarray*}
that is closely related to the set $X$.  It is easily seen that both $\overline{X}$ and $Y$ are closed subspaces of the Hilbert space $L^2_{\Delta}(D):=\{u\in L^2(D): \Delta u\in L^2(D)\}$ equipped with the norm induced by the inner product
$(u,v)_{L^2_{\Delta}(D)}:=(u,v)_{L^2}+(\Delta u,\Delta v)_{L^2}$ for $u,v\in L^2_{\Delta}(D)$.
Hence, both $\overline{X}$ and $Y$ are also Hilbert spaces in the sense of the above norm. For further analysis in next subsections, we need to show that Problem
(\ref{d6})-(\ref{d7}) has a unique solution $(v(\cdot,z),w(\cdot,z))\in \overline{X}\times L^2(D)$ for each
$z\in D$. Thus, it is enough to show the coincidence between $\overline{X}$ and $Y$.
\begin{lemma}\label{lem}
If $\kappa_1^2>0$ is not a Dirichlet eigenvalue of $-\Delta$ in $D$, then $\overline{X}=Y$.
\end{lemma}

\begin{proof}
To prove the assertion, we shall first show that the space $Y$ can be described by the
solution of the Dirichlet problem
\begin{eqnarray}\label{d13}
\left\{\begin{array}{ll}
\Delta \hat{w}+\kappa_1^2\hat{w}=0 &\quad {\rm in}\;\; D,\\[2mm]
\hat{w}=\hat{g}\in H^{-\frac{1}{2}}(\partial D) &\quad {\rm on}\;\; \partial D.
\end{array}\right.
\end{eqnarray}
Hence, we need to prove the uniquely solvability of Problem (\ref{d13}) in $L^2_\Delta(D)$ which
will be done by employing boundary integral equation techniques with seeking a solution
in the form of a combined double and single-layer potential
\begin{eqnarray}\label{d14}
\hat{w}(x) = \int_{\partial D}\left(\frac{\partial \Phi_{\kappa_1}(x,y)}{\partial n(y)}-{\rm i} \Phi_{\kappa_1}(x,y)\right)\psi(y){\rm d}s(y) \qquad {\rm{for}}\;\; x\in {\mathbb R}^2\setminus \partial D,
\end{eqnarray}
where the density $\psi\in H^{-\frac{1}{2}}(\partial{D})$ and $n(y)$ stands for the normal derivative at $y\in\partial D$ directed into the exterior of $D$.
Then we deduce by the boundary condition in (\ref{d13}) that the potential $\hat{w}$ in (\ref{d14}) solve Problem (\ref{d13}), provided the density $\psi$ is a solution of a second-kind of integral equation
\begin{eqnarray}\label{d15}
(I+K-{\rm i}S)\psi = 2\hat{g}
\end{eqnarray}
with the single- and double-layer operators $S$ and $K$, respectively, given by
\begin{eqnarray*}
(S\psi)(x): &=&  2\int_{\partial D}\Phi_{\kappa_1}(x,y)\psi(y)ds(y)\qquad\quad x\in\partial D,\\
(K\psi)(x): &=&  2\int_{\partial D}\frac{\partial \Phi_{\kappa_1}(x,y)}{\partial n(y)}\psi(y)ds(y)\qquad x\in\partial D.
\end{eqnarray*}
It follows from Corollary 3.7 of \cite{CK13} that both $S$ and $K$ are bounded operators from
$H^{-\frac{1}{2}}(\partial D)$ into $H^{\frac{1}{2}}(\partial D)$ due to the piecewise $C^{2,\alpha}$-regularity
of $\partial D$. This, together with the compact imbedding of $H^{\frac{1}{2}}(\partial D)$ into $H^{-\frac{1}{2}}(\partial D)$,
implies that  $I+K-{\rm i}S: H^{-\frac{1}{2}}(\partial D)\rightarrow H^{-\frac{1}{2}}(\partial D)$ is of Fredholm type
with index $0$. The existence of a solution of equation (\ref{d15}) then follows from the Riesz-Fredholm
theory and the uniqueness of Problem (\ref{d13}) for each $\hat{g}\in H^{-1/2}(\partial D)$.
Hence, the space
$\{\hat{w}\in L^2_{\Delta}(D):\hat{w}{\rm\;solves\;Problem\;(\ref{d13})\;for\;}\hat{g}\in H^{-1/2}(\partial D)\}$
coincides with $Y$.

Now, by the trace theorem it is sufficient to prove that $v_g|_{\partial D}$ is dense in $H^{-1/2}(\partial D)$,
which is equivalent to show that the range ${\rm Range}(H)$ of $H: L^2(\Gamma_{b,a})\rightarrow H^{-1/2}(\partial D)$
defined by
\begin{eqnarray*}
(Hg)(x):=\int_{\Gamma_{b,a}}G_r(x,y)g(y){\rm d}s(y)\qquad x\in\partial D
\end{eqnarray*}
is dense in $H^{-1/2}(\partial D)$.
By interchanging the order of integration, the dual operator $H^*:H^{1/2}(\partial D)\rightarrow L^2(\Gamma_{b,a})$
of $H$ is given by
\begin{eqnarray*}
(H^*\psi)(x)&:=& \int_{\partial D}\overline{G_r(y,x)}\psi(y){\rm d}s(y)\\
            &=& \int_{\partial D}\overline{G_r(x,y)}\psi(y){\rm d}s(y)\qquad x\in\Gamma_{b,a}.
\end{eqnarray*}
Here, we have used the reciprocity relation $G_r(x,y)=G_r(y,x)$ which holds for all $x\in\Gamma_{b,a}$
and $y\in\partial D$. Notice that the boundary $\partial D$ is composed of $\Gamma\setminus\Gamma_r$ and $\Gamma_r\setminus \Gamma$.
For the case $y\in\Gamma\setminus\Gamma_r$, it could be similarly proved by the proof of Lemma 3.1 of \cite{DLLY17}
that $G_r(x,y)=G_r(y,x)$ for $x\in\Gamma_{b,a}$. For the case $y\in\Gamma_r\setminus\Gamma$, the function $G_r(\cdot,y)$ is
defined as $G_r(\cdot,y): =\Phi_{\kappa_1}(\cdot,y)+G_r^{1}(\cdot,y)$ in $\Omega_{1,r}$ and
$G_r(\cdot,y):=G_r^{2}(\cdot,y)$ in $\Omega_{2,r}$, where $(G^1_r(\cdot,y),G^2_r(\cdot,y))$ is the solution
to the scattering problem:
\begin{eqnarray}\label{x1}
\left\{\begin{array}{lllll}
\Delta G^1_r(\cdot,y)+\kappa_1^2 G^1_r(\cdot,y)=0&  {\rm in}\;\; \Omega_{1,r}\\[2mm]
\Delta G^2_r(\cdot,y)+\kappa_2^2 G^2_r(\cdot,y)=0& {\rm in}\;\; \Omega_{2,r}\\[2mm]
G^2_r(\cdot,y)-G^1_r(\cdot,y)=\Phi_{\kappa_1}(\cdot,y)&{\rm on}\;\;  \Gamma_r\setminus\{y\}\\[2mm]
\displaystyle\frac{\partial{G^2_r(\cdot,y)}}{\partial n}-\frac{\partial{G^1_r(\cdot,y)}}{\partial n}
=\frac{\partial{\Phi_{\kappa_1}(\cdot,y)}}{\partial n}&{\rm on}\;\;  \Gamma_r\setminus\{y\}\\[2mm]
\displaystyle\lim_{|x|\to \infty}\sqrt{|x|}\left(\frac{\partial G^{j}_r(\cdot,y)}{\partial |x|}
-{\rm i}\kappa_j G^j_r(\cdot,y)\right)=0& {\rm for}\;\; j=1,2.
\end{array}\right.
\end{eqnarray}

To show the existence of a unique solution of Problem (\ref{x1}), define
$E^1_r(\cdot,y):=G^1_r(\cdot,y)$ and $E^2_r(\cdot,y):=G^2_r(\cdot,y)-\Phi_{\kappa_2}(\cdot,y)$
which satisfy Problem (\ref{x1}) with the boundary data replaced with
\begin{eqnarray*}\label{x11}
h_1(z):=\Phi_{\kappa_1}(z,y)-\Phi_{\kappa_2}(z,y)\quad{\rm and}\quad
h_2(z):=\frac{\partial\Phi_{\kappa_1}(z,y)}{\partial n(z)}-\frac{\partial\Phi_{\kappa_2}(z,y)}{\partial n(z)}.
 \end{eqnarray*}
A direct calculation yields that $h_1(\cdot)\in C^1(\Gamma_r)\cap L_{\rm loc}^2(\Gamma_r)$ and
$h_2(\cdot)\in C(\Gamma_r)\cap L_{\rm loc}^2(\Gamma_r)$ for any $y\in \Gamma_r\setminus\Gamma$.
By a similar argument as in \cite{SZ99} it can be proved that Problem (\ref{x1}) is well-posed
in the above setting for $h_1$ and $h_2$, which implies that $E_r^1(\cdot,y)\in H^1_{\rm loc}(\Omega_{1,r})$
and $E_r^2(\cdot,y)\in H^1_{\rm loc}(\Omega_{2,r})$.

We now prove that $G_r(x,y)=G_r(y,x)$ for $y\in \Gamma_r\setminus\Gamma$ and $x\in\Gamma_{b,a}$.
For $s=x,y$, let $B_{\epsilon}(s)$ denote a ball centered at $s$ with sufficiently small radius
$\epsilon>0$ such that $B_{\epsilon}(x)\cap B_{\epsilon}(y)=\emptyset$. Let $B^+_{\epsilon}(y):=B_{\epsilon}(y)\cap\Omega_{1,r}$, $B^-_{\epsilon}(y):=B_{\epsilon}(y)\cap\Omega_{2,r}$, and $B_{r'}:=\{x\in{\mathbb R}^2: |x|=r'>r\}$. Using the Green's theorem in $(\Omega_{1,r}\cap B_{r'})\setminus \overline{(B_{\epsilon}(x)\cup B^+_{\epsilon}(y))}$ gives
\begin{eqnarray*}
0&=&\int_{(\Omega_{1,r}\cap B_{r'})\setminus\overline{(B_{\epsilon}(x)\cup B^+_{\epsilon}(y))}}(\Delta G_r(z,x)G_r(z,y)-\Delta G_r(z,y)G_r(z,x)){\rm d}z\\
&=&\left\{\int_{\partial B^+_{r'}}+\int_{\partial B_{\epsilon}(x)}-\int_{\hat{\Gamma}}+\int_{\partial B_{\epsilon}^+(y)\setminus \Gamma_r}\right\}\left(\frac{\partial G_r(z,x)}{\partial n(z)}G_r(z,y)-\frac{\partial G_r(z,y)}{\partial n(z)}G_r(z,x)\right){\rm d}s(z)
\end{eqnarray*}
where $\hat{\Gamma}:=(B_{r'}\cap\Gamma_r)\setminus B_{\epsilon}(y)$ and $\partial B_{r'}^{+}:=\{x\in{\mathbb R}^2: |x|=r', x_2>0 \}$. Using the Green's theorem again
in $(\Omega_{2,r}\cap B_{r'})\setminus B^-_{\epsilon}(y)$ yields
\begin{eqnarray*}
0&=&\int_{(\Omega_{2,r}\cap B_{r'})\setminus\overline{ B^-_{\epsilon}(y)}}(\Delta G_r(z,x)G_r(z,y)-\Delta G_r(z,y)G_r(z,x)){\rm d}z\\
&=&\left\{\int_{\partial B^-_{r'}}+\int_{\hat{\Gamma}}+\int_{\partial B_{\epsilon}^-(y)\setminus \Gamma_r}\right\}\left(\frac{\partial G_r(z,x)}{\partial n(z)}G_r(z,y)-\frac{\partial G_r(z,y)}{\partial n(z)}G_r(z,x)\right){\rm d}s(z)
\end{eqnarray*}
where $\partial B_{r'}^{-}:=\{x\in{\mathbb R}^2: |x|=r', x_2<0 \}$. Then, combining the above two equalities leads to
\begin{eqnarray}\nonumber
0&=&\left\{\int_{\partial B_{r'}}+\int_{\partial B_{\epsilon}(x)}+\int_{\partial B_{\epsilon}(y)}\right\}\left(\frac{\partial G_r(z,x)}{\partial n(z)}G_r(z,y)-\frac{\partial G_r(z,y)}{\partial n(z)}G_r(z,x)\right){\rm d}s(z)\\\label{x111}
&=:& I_1+I_2+I_3.
\end{eqnarray}
It follows from the Sommerfeld radiation condition that $I_1\rightarrow 0$ as $r'\rightarrow \infty$
and from the mean value theorem that $I_2\rightarrow G_r(x,y)$ as $\epsilon\rightarrow 0$.
For the term $I_3$, by noting $G_r(z,y)=E(z,y)+\Phi(z,y)$ with $E(z,y)=E_r^{j}(z,y)$ and $\Phi(z,y)=\Phi_{\kappa_j}(z,y)$
for $z\in\Omega_{j,r}$ $(j=1,2)$, we can split $I_3$ into two parts, that is, $I_3=I_{31}+I_{32}$, such that
\begin{eqnarray}\label{x222}
I_{31}:=\int_{\partial B_{\epsilon}(y)}\frac{\partial G_r(z,x)}{\partial n(z)}G_r(z,y){\rm d}s(z)\to 0\qquad{\rm as}\;\;\epsilon\to 0
\end{eqnarray}
since $G_r(\cdot,y)\in H^{1,p}(B_{\epsilon}(y))$ for any fixed $p\in (1,2)$.
Further, we have
\begin{eqnarray}\nonumber
I_{32}:&=&-\int_{\partial B_{\epsilon}(y)}\frac{\partial G_r(z,y)}{\partial n(z)}G_r(z,x){\rm d}s(z)\\\nonumber
&=&-\int_{\partial B_{\epsilon}(y)}\frac{\partial E(z,y)}{\partial n(z)}G_r(z,x){\rm d}s(z)
       -\int_{\partial B_{\epsilon}(y)}\frac{\partial \Phi(z,y)}{\partial n(z)}G_r(z,x){\rm d}s(z)\\\label{x333}
&=&: I_{321}+I_{322}.
\end{eqnarray}
By recalling $E_r^j(\cdot,y)\in H^1_{\rm loc}(\Omega_{j,r})$, one has $I_{321}\to 0$ and $I_{322}\to -G_r(y,x)$ as $\epsilon\to 0$, which combines with (\ref{x222}) and (\ref{x333}).
to show $I_3\to -G_r(y,x)$ as  $\epsilon\to 0$. Therefore, we have proved the reciprocity relation $G_r(x,y)=G_r(y,x)$ from equality (\ref{x111}) for all $x\in \Gamma_{b,a}$ and $y\in\partial D$.

We next claim that $H^*$ is injective. To see this, let $H^*\psi=0$ on $\Gamma_{b,a}$ for some $\psi\in H^{\frac{1}{2}}(\partial D)$ and define the function
\begin{eqnarray*}
\tilde{w}(x): = \int_{\partial D}G_r(x,y)\overline{\psi(y)}{\rm d}s(y)\qquad x\in{\mathbb R}^2.
\end{eqnarray*}
Then, $\tilde{w}(x)=0$ on $\Gamma_{b,a}$. The analyticity and the uniqueness of the Dirichlet boundary value problem in $U_b:=\{(x_1,x_2)\in{\mathbb R}^2:x_2>b\}$ implies $\tilde{w}=0$ in $U_b$.
Using the analytic continuation as well as the continuous conditions of $G_r$, $\partial_n G_r$ across $\Gamma_r\cap\Gamma$, we have $\tilde{w}=0$ in ${\mathbb R}^2\setminus\overline{D}$.
Since $G_r(x,y)$ has the same singularity with $\Phi_{\kappa_1}(x,y)$ at $x=y$ for $y\in\partial D$, we deduce that $\tilde{w}$ solves the homogeneous Dirichlet problem
\begin{eqnarray*}
\left\{\begin{array}{lllll}
\Delta \tilde{w}+\kappa_1^2\tilde{w}=0 &\quad {\rm in}\;\; D,\\[2mm]
\tilde{w}=0 &\quad {\rm on}\;\; \partial D.
\end{array}\right.
\end{eqnarray*}
By the assumption on $\kappa^2_1>0$, we conclude $\tilde{w}=0$ in $D$, whence $\psi=0$
follows from the jump relationship of $\partial_n \tilde{w}$ on $\partial D$. Therefore, $H^*$ is injective
which means that ${\rm Range}(H)$ is dense in $H^{-\frac{1}{2}}(\partial D)$. The proof is thus complete.
\end{proof}
\begin{remark}\label{remark1} {\rm
The assumption in Lemma \ref{lem} could be removed by choosing a suitable $r>0$, which may be seen by
the strong monotonicity property and the continuous dependence of the Dirichlet eigenvalues of $-\Delta$
with respect to the domain \cite{L86}. We refer to Remark 3.2 in \cite{DLLY17} for details.
}
\end{remark}

By Lemma \ref{lem}, it is known that there exists a unique solution $(v(\cdot,z),w(\cdot,z))\in \overline{X}\times L^2(D)$ to Problem (\ref{d6})-(\ref{d7}) for each $z\in D$. With this result, we can establish the following two important properties for $F(I+\eta T)^{-1}$ which will play a key role in proposing our sampling method.

\begin{theorem}\label{thm3.1}
If $\kappa_1>0$ is not a transmission eigenvalue of Problem (\ref{d6})-(\ref{d7}), then $F(I+\eta T)^{-1}: \overline{X}\rightarrow L^2(\Gamma_{b,a})$ is injective with dense range.
\end{theorem}

\begin{proof}
Let $F(I+\eta T)^{-1}v=0$ for some $v\in\overline{X}$ and set $w:=(I+\eta T)^{-1}v$.
Define the function
\begin{eqnarray*}
\xi(x):=-\eta \int_{D}G_r(x,z)w(z){\rm d}z\qquad{\rm for}\;\; x\in{\mathbb R}^2.
\end{eqnarray*}
Then, $\xi=0$ on $\Gamma_{b,a}$ follows from $Fw=0$. By the analyticity of $\xi$ in $\Gamma_b:=\{(x_1,x_2)\in{\mathbb R}^2:x_2=b\}$, we conclude $\xi=0$ on $\Gamma_b$. The uniqueness of the Dirichlet boundary value problem in $U_b:=\{(x_1,x_2)\in{\mathbb R}^2:x_2>b\}$ implies that $\xi=0$ in $U_b$. Then the analytic continuation and the continuity of $G_r$ and $\partial_n G_r$ across $\Gamma_r\cap\Gamma$ lead to $\xi=0$ in ${\mathbb R}^2\setminus \overline{D}$. Thus, we conclude that $(v,w)\in\overline{X}\times L^2(D)$  is a solution of Problem (\ref{d6})-(\ref{d7}) with the homogeneous conditions. Since
$\kappa_1>0$ is not a transmission eigenvalue, we derive $v=0$ which means that $F(I+\eta T)^{-1}$ is injective.

To show the denseness we assume that  there exists $g\in L^2(\Gamma_{b,a})$ such that
\begin{eqnarray}\label{x444}
(F(I+\eta T)^{-1}v_h, g)=0\qquad {\rm for\;all\;}v_h\in X.
\end{eqnarray}
Then by (\ref{d4}), we have
\begin{eqnarray*}
\int_{\Gamma_{b,a}}\int_{\Gamma_{b,a}}(u(x,y)-G_r(x,y))h(y){\rm d}s(y)\overline{g(x)}{\rm d}s(x)=0\qquad{\rm for\; all}\;h\in L^2(\Gamma_{b,a}),
\end{eqnarray*}
whence
\begin{eqnarray*}
\int_{\Gamma_{b,a}}(u(y,x)-G_r(y,x))\overline{g(x)}{\rm d}s(x)=0\qquad{\rm for\; all}\;y\in \Gamma_{b,a}.
\end{eqnarray*}
A similar discussion implies
\begin{eqnarray*}
\int_{\Gamma_{b,a}}(u(y,x)-G_r(y,x))\overline{g(x)}{\rm d}s(x)=0\qquad{\rm for\; all}\;y\in{\mathbb R}^2\setminus \overline{D}.
\end{eqnarray*}
Thus, we conclude that $v_{\overline{g}}\in \overline{X}$ and $(\int_{\Gamma_{b,a}}u(y,x)\overline{g(x)}{\rm d}s(x))|_D$,
are the solutions of the homogeneous problem (\ref{d6})-(\ref{d7}). Hence, $v_{\overline{g}}=0$ in $D$
due to the assumption on $\kappa_1$ which further implies $v_{\overline{g}}=0$ in $\Omega_{1,r}\setminus\Gamma_{b,a}$
from the analytic continuation. Using the jump relations of the normal derivative of $v_{\overline{g}}$
gives that $g=0$. This means that, by (\ref{x444}) ${\rm Range}(F(I+\eta T)^{-1}(\overline{X}))$ is dense
in $L^2(\Gamma_{b,a})$.
\end{proof}

\begin{theorem}\label{thm3.2}
If $\kappa_1>0$ is not a transmission eigenvalue Problem (\ref{d6})-(\ref{d7}), then $z\in D$ if and only
if $G_r(\cdot, z)|_{\Gamma_{b,a}}\in {\rm Range}(F(I+\eta T)^{-1})$.
\end{theorem}

\begin{proof}
Since $\kappa_1>0$ is not a transmission eigenvalue, there exists a unique solution
$(v(\cdot,z),w(\cdot,z))\in \overline{X}\times L^2(D)$ to Problem (\ref{d6})-(\ref{d7}) if $z\in D$.
Then we define a function $\widetilde{v}_z$ by
\begin{eqnarray}\label{s2}
\widetilde{v}_z(\cdot):=\left\{\begin{array}{l}
              G_r(\cdot,z) \qquad\qquad\quad\; \textrm{in}\;\; \Omega_1, \\[1mm]
              w(\cdot,z)-v(\cdot,z) \qquad \textrm{in}\;\; D,\\[1mm]
              G_r(\cdot,z) \qquad\qquad\quad\; \textrm{in}\;\; \Omega_{2,r}.
            \end{array}
\right.
\end{eqnarray}
It is easily verified that $\widetilde{v}_z$ solves Problem (\ref{c2}) with
$\beta(\cdot):=\eta v(\cdot,z)$.
It follows from \cite{YLZ17} that $\widetilde{v}_z(\cdot)$ satisfies the Lippmann-Schwinger equation
\begin{eqnarray}\label{s1}
\widetilde{v}_z(x)+\eta\int_D G_r(x,y)\widetilde{v}_z(y){\rm d}y=-\eta\int_DG_r(x,y)v(y,z){\rm d}y
\end{eqnarray}
for $x\in{\mathbb R}^2$, which may be reformulated in the form in $D$:
\begin{eqnarray*}
\widetilde{v}_z(\cdot) = -\eta T(v(\cdot,z)+\widetilde{v}_z(\cdot))\quad{\rm or}\quad
(\widetilde{v}_z(\cdot)+v(\cdot,z)) + \eta T(\widetilde{v}_z(\cdot)+v(\cdot,z)) = v(\cdot,z).
\end{eqnarray*}
Then we have $\widetilde{v}_z(\cdot)+v(\cdot,z)=(I+\eta T)^{-1}v(\cdot,z)$, whence
$F(I+\eta T)^{-1}v(\cdot,z)=F(\widetilde{v}_z(\cdot)+v(\cdot,z))=\widetilde{v}_z(\cdot)|_{\Gamma_{b,a}}=G_r(\cdot,z)|_{\Gamma_{b,a}}$ follows from (\ref{s1}) and (\ref{s2}). Hence, it holds
$G_r(\cdot, z)|_{\Gamma_{b,a}}\in {\rm Range}(F(I+\eta T)^{-1})$ if $z\in D$.

Conversely, assume on the contrary $z\notin D$. Since $G_r(x,z)|_{\Gamma_{b,a}}\in{\rm Range}(F(I+\eta T)^{-1})$,
there exists some $v\in\overline{X}$ such that $F(I+\eta T)^{-1}v=G_r(x,z)|_{\Gamma_{b,a}}$. Set $w:=(I+\eta T)^{-1}v$
and define
\begin{eqnarray*}
\mu(x): = -\eta\int_{D}G_r(x,z)w(z){\rm d}z\qquad x\in{\mathbb R}^2.
\end{eqnarray*}
It is checked that $\mu(x)$ is the solution to Problem (\ref{c2}) with the data $\eta v$. Hence,
we have $F(I+\eta T)^{-1}v=\mu|_{\Gamma_{b,a}}$ and then $\mu(x)=G_r(x,z)$ on $\Gamma_{b,a}$.
Using the analyticity and the uniqueness of the Dirichlet boundary value problem in $U_b$ again,
we conclude that $\mu(\cdot)=G_r(\cdot,z)$ on $U_b$.
We further have $\mu(x)=G_r(x,z)$ for $x\in {\mathbb R}^2\backslash\{D\cup\{z\}\}$ from the analytic continuation.
Note that $G_r(x,z)$ has a singularity at $x=z$ while $\mu(x)$ is smooth at $x=z$, which leads to a
contradiction. The proof is thus completed.
\end{proof}

With the aid of Theorems \ref{thm3.1} and \ref{thm3.2}, we are now able to formulate the main result
of this paper in the following theorem.

\begin{theorem}\label{thm3.3}
Assume that $\kappa_1>0$ is not a transmission eigenvalue of Problem (\ref{d6})-(\ref{d7}). Then we have

1) if $z\in D$, then for each $\epsilon>0$ there exists a solution $g_{z,\epsilon}\in L^2(\Gamma_{b,a})$ of
the inequality
\begin{eqnarray}\label{d8}
\|{\bf N}_{\rm Mod}g_{z,\epsilon}(\cdot)-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}<\epsilon
\end{eqnarray}
such that
\begin{eqnarray*}
\lim_{z\rightarrow \partial D\cap\Gamma}\|g_{z,\epsilon}\|_{L^2(\Gamma_{b,a})}=\infty \quad{\rm and}\quad
\lim_{z\rightarrow \partial D\cap\Gamma}\|v_{g_{z,\epsilon}}\|_{L^2(D)}=\infty
\end{eqnarray*}
where $v_{g_{z,\epsilon}}$ is defined by (\ref{d3}) with the density $g_{z,\epsilon}$;

2) if $z\in {\mathbb R}^2\setminus\overline{D}$, then for each $\epsilon>0$ and $\delta>0$ there exists a solution $g_{z,\epsilon}^{\delta}\in L^2(\Gamma_{b,a})$ of the inequality
\begin{eqnarray}\label{d9}
\|{\bf N}_{\rm Mod}g_{z,\epsilon}^{\delta}(\cdot)-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}<\epsilon+\delta
\end{eqnarray}
such that
\begin{eqnarray*}
\lim_{\delta\rightarrow 0}\|g^{\delta}_{z,\epsilon}\|_{L^2(\Gamma_{b,a})}=\infty \quad{\rm and}\quad
\lim_{\delta\rightarrow 0}\|v_{g^{\delta}_{z,\epsilon}}\|_{L^2(D)}=\infty
\end{eqnarray*}
where $v_{g^{\delta}_{z,\epsilon}}$ is defined by (\ref{d3}) with the density $g^{\delta}_{z,\epsilon}$.
\end{theorem}

\begin{proof}
If $z\in D$, let $(v(\cdot, z),w(\cdot,z))\in\overline{X}\times L^2(D)$ denote the unique solution of
Problem (\ref{d6})-(\ref{d7}). By Theorem \ref{thm3.2}, we have
$F(I+\eta T)^{-1}v(\cdot, z)=G_r(\cdot,z)|_{\Gamma_{b,a}}$. Moreover, for each $\epsilon>0$ there exists
$g_{z,\epsilon}\in L^2(\Gamma_{b,a})$ such that $\|v_{g_{z,\epsilon}}(\cdot)-v(\cdot,z)\|_{L^2(D)}<\epsilon$.
This, combined with the equality $F(I+\eta T)^{-1}={\bf N}_{\rm Mod}$, implies that
\begin{eqnarray*}
\|{\bf N}_{\rm Mod}g_{z,\epsilon}(\cdot)-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}<C_0\epsilon
\end{eqnarray*}
for a fixed constant $C_0>0$.

Assuming that $\|g_{z,\epsilon}\|_{L^2(\Gamma_{b,a})}\leq C$ uniformly as $z\rightarrow \partial D\cap\Gamma$,
we have by (\ref{d3}) that $||v_{g_{z,\epsilon}}||_{L^2(D)}\leq C$ as $z\rightarrow \partial D\cap\Gamma$.
Since $\|v_{g_{z,\epsilon}}(\cdot)-v(\cdot, z)\|_{L^2(D)}<\epsilon$, we conclude that
$\|v(\cdot,z)\|_{L^2(D)}\leq C+C_0\epsilon$. Thus it follows from the equality
$w(\cdot, z)=(I+\eta T)^{-1}v(\cdot, z)$ that $\|w(\cdot,z)\|_{L^2(D)}\leq C_2$ for a fixed constant $C_2>0$
as $z\rightarrow\partial D\cap\Gamma$. Using the boundedness of the operator $T:L^2(D)\to H^1(D)$ as well as
the trace theorem gives that $\|v(\cdot,z)-w(\cdot,z)\|_{H^{1/2}(\partial D)}\leq C_3$ for a fixed constant $C_3>0$.
However, this is a contradiction since, by the boundary condition $v(\cdot,z)-w(\cdot,z)=G_r(\cdot,z)$ on $\partial D$
we have $\|G_r(\cdot,z)\|_{H^{1/2}(\partial D)}\rightarrow\infty$ as $z\rightarrow\partial D\cap\Gamma$.

If $z\in{\mathbb R}^2\setminus\overline{D}$, it is known by theorem \ref{thm3.2} that $G_r(\cdot,z)|_{\Gamma_{b,a}}\notin {\rm Range}(F(I+\eta T)^{-1})$. Moreover, it follows from theorem \ref{thm3.1} that $F(I+\eta T)^{-1}$ is injective with dense range in $L^2(\Gamma_{b,a})$. So, for the operator equation $F(I+\eta T)^{-1}v(\cdot)=G_r(\cdot,z)|_{\Gamma_{b,a}}$, there always exists a regularized solution $v_z^{\alpha}\in\overline{X}$ to its regularized equation
\begin{eqnarray*}
\alpha v_z^{\alpha}+A^*Av_z^{\alpha}=A^*(G_r(\cdot,z)|_{\Gamma_{b,a}})
\end{eqnarray*}
which can be represented as
\begin{eqnarray*}
v_z^{\alpha}=\sum_{n=1}^{\infty}\frac{\mu_n}{\alpha+\mu_n^2}(G_r(\cdot,z)|_{\Gamma_{b,a}}, g_n)\varphi_n
\end{eqnarray*}
with $(\mu_n, \varphi_n, g_n)$ a singular system for $A$,
where $A:=F(I+\eta T)^{-1}$ and $\alpha>0$ is a regularization parameter. By theorem 2.13 of  \cite{CC06} it is also known that $v_z^{\alpha}$ is the minimizer of the Tikhonov functional.
Therefore, for every $\delta>0$, we can deduce by choosing $\alpha>0$ that
\begin{eqnarray}\label{d10}
\|F(I+\eta T)^{-1}v_z^{\alpha}-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}<\delta.
\end{eqnarray}
Since $G_r(\cdot, z)|_{\Gamma_{b,a}}\notin {\rm Range}(F(I+\eta T)^{-1})$, using the Picard's theorem implies that $\|v_z^{\alpha}\|_{L^2(D)}\rightarrow \infty$ as $\alpha\rightarrow 0$. By noticing  $v_z^{\alpha}\in\overline{X}$, for sufficiently small $\epsilon>0$ there then exists $g_{z,\epsilon}^{\delta}\in L^2(\Gamma_{b,a})$ such that
\begin{eqnarray}\label{d11}
\|v_{g_{z.\epsilon}^{\delta}}-v_z^{\alpha}\|_{L^2(D)}<\epsilon
\end{eqnarray}
which means
\begin{eqnarray}\label{d12}
\|F(I+\eta T)^{-1}v_{g_{z.\epsilon}^{\delta}}-F(I+\eta T)^{-1}v_z^{\alpha}\|_{L^2(\Gamma_{b,a})}<\epsilon
\end{eqnarray}
in the sense of omitting a constant. We now combine (\ref{d10}) and (\ref{d12}) to obtain
\begin{eqnarray*}
&&\|{\bf N}_{\rm Mod}g_{z,\epsilon}^{\delta}(\cdot)-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}\\
&=&\|F(I+\eta T)^{-1}v_{g_{z.\epsilon}^{\delta}}(\cdot)-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}\\
&\leq& \|F(I+\eta T)^{-1}v_{g_{z.\epsilon}^{\delta}}-F(I+\eta T)^{-1}v_z^{\alpha}\|_{L^2(\Gamma_{b,a})}
+\|F(I+\eta T)^{-1}v_z^{\alpha}-G_r(\cdot,z)\|_{L^2(\Gamma_{b,a})}\\
&<&\epsilon+\delta.
\end{eqnarray*}
Finally, the inequality (\ref{d11}) and the fact $||v_z^{\alpha}||_{L^2(D)}\rightarrow \infty$ as $\alpha\rightarrow 0$ gives
\begin{eqnarray*}
\lim_{\alpha\rightarrow 0}\|v_{g^{\delta}_{z,\epsilon}}\|_{L^2(D)}=\infty
\end{eqnarray*}
whence
\begin{eqnarray*}
\lim_{\alpha\rightarrow 0}\|g^{\delta}_{z,\epsilon}\|_{L^2(\Gamma_{b,a})}=\infty.
\end{eqnarray*}
This ends the proof due to $\alpha\rightarrow 0$ as $\delta\rightarrow 0$.
\end{proof}

By Theorem \ref{thm3.3}, it is known that the solution of the modified near-field equation (\ref{d2}) in the sense of inequalities (\ref{d8}) and (\ref{d9}) has totally different behaviors when the sampling point $z$ lies inside or outside of $D$, which gives a qualitative way to visualize the support $D$. Based on this observation, we define the indicator function
\begin{eqnarray*}
{\rm Ind}(z):=1/\|g_z\|_{L^2(\Gamma_{b,a})}
\end{eqnarray*}
by the solution of (\ref{d8}) and (\ref{d9}). It is easily seen that the indicator function ${\rm Ind}(z)$ is small when the sampling point $z$ approaches the local perturbation of the interface $\Gamma$ from inside of $D$, which can provide a fast imaging algorithm. The following procedure shows how to recover the interface $\Gamma$ by the indicator function ${\rm Ind}(z)$.
\begin{algorithm}\caption{Reconstruction of locally rough interfaces by linear sampling method}\label{alg1}
\begin{itemize}
\item Select a rectangular grid $S$ containing the local perturbation of the scattering interface $\Gamma$.
\item Choose $r>0$ to be large enough such that $\Omega_1\subset\Omega_{1,r}$, that is, the local perturbation
of the interface $\Gamma$ lies totally in $\Omega_{1,r}$. Then solve the  scattering problem (\ref{c1})
and (\ref{a1})-(\ref{a2}) for each $y\in\Gamma_{b,a}$ to obtain the scattered field data $G_r^s(x,y)$
and $u^s(x,y)$ by the Nystr\"{o}m method (cf.\cite{LYZ13}).
\item For each sampling point $z\in S$, solve the modified near-field equation (\ref{d2}) by the
regularization method to obtain the solution $g_z$ and compute the value of the indicator function ${\rm Ind}(z)$.
\item Choose a cut-off value $C>0$ and so that it is numerically reliable that $z\in D$ if and
only if ${\rm Ind}(z)\leq C$.
\end{itemize}
\end{algorithm}

\section{Numerical experiments}\label{sec4}
\setcounter{equation}{0}

In this section, we carry out several numerical examples to illustrate the effectiveness of the modified LSM proposed in Theorem \ref{thm3.3}. It is known by the analyticity of the kernel
$u^{\rm s}(\cdot,y)-G^{\rm s}(\cdot,y)$ that the operator ${\bf N}_{\rm Mod}$ is compact on $L^2(\Gamma_{b,a})$. Hence, equation (\ref{d2}) is severely ill-posed and is solved by a regularization method with the regularization parameter $\alpha(z)$ chosen by the Morozov's discrepancy principle. However, a number of numerical examples we carried out show that the regularization parameter $\alpha(z)$ can be taken as a fixed parameter. In this paper, we choose
$\alpha(z)=5\times 10^{-5}$. In addition, we take the wavenumber $\kappa(\cdot)$ to be
$\kappa_1=1$ and $\kappa_2=10$, where the wavelength is given as $\lambda=2\pi/\kappa_1=2\pi$.

In numerical experiments, the synthetic scattering data $u^{\rm s}(x,y)$ and $G_r^{\rm s}(x,y)$ are obtained by solving the scattering problem (\ref{a1})-(\ref{a2}) and (\ref{c1}) by the Nystr\"{o}m method \cite{LYZ13}. Thus, we can discretize the modified near-field operator ${\bf N}_{\rm Mod}$ into the following finite dimensional matrix
\begin{eqnarray*}
({\bf N}_{\rm Mod})_{N\times N}=(u^{\rm s}(x_p, y_q)-G_r^{\rm s}(x_p, y_q))_{1\leq p, q\leq N}
\end{eqnarray*}
where $x_p$ is the measuring points equally distributed at $\Gamma_{b,a}$ with $p=1,2,...,N$, and $y_q$ is the incident point sources which is also equally distributed at $\Gamma_{b,a}$ with $q=1,2,...,N$.
Then we discrete the test function $G_r(x,z)$ as $G_r(x_p,z)$ with $1\leq p\leq N$ with the sampling points $z$ belonging to a rectangular grid $S$ containing the local perturbation of $\Gamma$.
Therefore, equation (\ref{d2}) can discretized into the following regularized equation
\begin{eqnarray*}
U_{\rm Mod}(\alpha I+S_{\rm Mod})V_{\rm Mod}^*\hat{g}_z=({\bf N}_{\rm Mod})^*_{N\times N}(G_r(\cdot,z)|_{\Gamma_{b,a}})
\end{eqnarray*}
where $U_{\rm Mod}$, $S_{\rm Mod}$ and $V_{\rm Mod}$ are the related matrices in the SVD algorithm satisfying $U_{\rm Mod}S_{\rm Mod}V_{\rm Mod}^*=({\bf N}_{\rm Mod})_{N\times N}^*({\bf N}_{\rm Mod})_{N\times N}$. Then we can define the following discretized form
\begin{eqnarray*}
{\rm Ind}_N(z) = 1{\left/\left[\sum_{j=1}^{N}|\hat{g}_{z,j}|^2\right]^{\frac{1}{2}}\right.}
\end{eqnarray*}
of the indicator function ${\rm Ind}(z)$, where $\hat{g}_z: =(\hat{g}_{z,1},\cdots,\hat{g}_{z,N})^T\in {\mathbb C}^N$.

If ${\rm Ind}_N(z)$ approximates ${\rm Ind}(z)$, then ${\rm Ind}_N(z)$ should be large in $D$ and very small in $\Omega_1$. To present the results under the same standard, ${\rm Ind}_N(z)$ is normalized to a new indicator function
\begin{eqnarray*}
{\rm NInd(z)}:={\rm Ind}_N(z)/\max_{z\in S}{\rm Ind}_N(z)
\end{eqnarray*}
which will be used to reconstruct the local perturbation of the scattering interface $\Gamma$.

To examine the stability of the sampling method, we also exam our method with noisy data. To this end,
let $(\zeta)_{N\times N}=(\zeta_1)_{N\times N}+{\rm i}(\zeta_2)_{N\times N}$ stand for a complex-valued
matrix with its real part $\zeta_1$ and imaginary part $\zeta_2$ consisting of random numbers which obey
the normal distribution $N(0,1)$. Thus, the scattering data with noisy is given as
\begin{eqnarray*}
(({\bf N}_{\rm Mod})_{N\times N})_{\delta}:=({\bf N}_{\rm Mod})_{N\times N}
+\delta\frac{\zeta}{\|\zeta\|_2}\|({\bf N}_{\rm Mod})_{N\times N}\|_2
\end{eqnarray*}
for relative error $\delta>0$.

As shown in Theorem  \ref{thm3.3}, we should remark that the modified LSM always works for every sufficiently
large $r>M$ as long as $\kappa_1>0$ is not a transmission eigenvalue of  Problem (\ref{d6})-(\ref{d7})
in $D=\Omega_2\cap \Omega_{1,r}$. Moreover, it is also noticed that  the computation of $G_r^{\rm s}(\cdot,z)$
may be time-consuming for sufficiently large $r$, which seems to infect the effectiveness of the modified LSM.
However, a variety of numerical experiments we have carried out imply that $G_r^{\rm s}(\cdot,z)$ decays to $0$
in any fixed bounded domain as $r\rightarrow \infty$. We illustrate this fact in the following table for $r=10^t$
with $t=2,4,6,8,10$, where the location of the incident point source is given at $z=(0,1)$, and the receivers
are given at $x_1=(-2,1)$, $x_2=(0,1)$ and $x_3=(2,1)$.

\begin{table}[tbhp]
\caption{\label{us_num} Numerical solutions of $G_r^{\rm s}(x,z)$
as the radius $r\to\infty$}
\centering
\begin{tabular}{ l     l     l}
\hline
 &$r$        &\qquad\;\;$G_r^{\rm s}(x_1,z)$  \qquad\qquad\qquad\;\;\;\; $G_r^s(x_2,z)$ \qquad\qquad\qquad\;\;\;\;\;\;$G^s(x_3,z)$ \\\hline
 &$10^2$  &\quad\;\;-0.0223-0.0045{\rm i}\qquad\qquad\quad -0.0241-0.0051{\rm i}\qquad\qquad\quad -0.0223-0.0045{\rm i}       \\
 &$10^4$     &1.0e-03$\cdot$(0.8017+2.8967{\rm i})   \quad1.0e-03$\cdot$(0.8008+2.8992{\rm i}) \quad\;1.0e-03$\cdot$(0.8017+2.8967{\rm i})   \\
 &$10^6$     &1.0e-03$\cdot$(-0.0312+0.1383{\rm i})\quad1.0e-03$\cdot$(-0.0312+0.1383{\rm i})\quad\
 1.0e-03$\cdot$(-0.0312+0.1383{\rm i})                  \\
 &$10^8$     &1.0e-05$\cdot$(-1.6947-0.8085{\rm i})  \quad1.0e-05$\cdot$(-1.6947-0.8085{\rm i})\quad\;\;
 1.0e-05$\cdot$(-1.6947-0.8085{\rm i})                \\
 &$10^{10}$ &1.0e-06$\cdot$(0.8272+1.7961{\rm i}) \quad1.0e-06$\cdot$(0.8272+1.7961{\rm i})\quad\;
 1.0e-06$\cdot$(0.8272+1.7961{\rm i})         \\
\hline
\end{tabular}
\end{table}

From Table \ref{us_num}, it could be deduced that for any given positive number $m>0$ we always can choose
sufficiently large $r=r(m)$ such that the estimate $\|{\bf N}-{\bf N}_{\rm Mod}\|_{L^2(\Gamma_{b,a})}<10^{-m}$.
For example, $m>0$ is chosen to be $100$. In this case,
equation (\ref{d2}) could be reduced to solve its approximate equation
\begin{eqnarray}\label{e2}
({\bf N}g)(x)=\Phi_{\kappa_1}(x,z)\qquad{\rm for}\;\; x\in\Gamma_{b,a}.
\end{eqnarray}
Similarly, we have the following discrete regularized equation for (\ref{e2}):
\begin{eqnarray}\label{e4}
U(\alpha I+S)V^*\hat{g}_z=({\bf N})^*_{N\times N}(\Phi_{\kappa_1}(\cdot,z)|_{\Gamma_{b,a}})
\end{eqnarray}
where $U$, $S$ and $V$ are the related matrices in the SVD algorithm satisfying
$USV^*={\bf N}_{N\times N}^*({\bf N})_{N\times N}$.
Then Algorithm \ref{alg1} can be reformulated as follows.
\begin{algorithm}\caption{Reconstruction of locally rough interfaces by linear sampling method}\label{alg2}
\begin{itemize}
\item Select a rectangular grid $S$ containing the local perturbation of the scattering interface $\Gamma$.
\item Solve the scattering problem (\ref{a1})-(\ref{a2}) for each $y\in\Gamma_{b,a}$ by the Nystr\"{o}m
method (cf.\cite{LYZ13}).
\item Solve the discretization regularized equation (\ref{e4}) with the regularization parameter
$\alpha(z)=5\times 10^{-5}$ to obtain its solution $g_z^{\alpha}$.
\item Compute the normalized indicator function
\begin{eqnarray*}\label{e3}
{\rm NInd}(z):={\rm Ind}_N(z)/\max\limits_{z\in S} {\rm Ind}_N(z)
\end{eqnarray*}
and then plot the mapping ${\rm NInd}(z)$ against $z$.
\end{itemize}
\end{algorithm}
\newline
\newline
\newline

\textbf{Example 1}. We first consider the interface $\Gamma$ described by the function
\begin{eqnarray*}
f_1(x_1)=0.6\cdot\Omega_3(x_1)\qquad x_1\in{\mathbb R}^1,
\end{eqnarray*}
where $\Omega_3(\cdot)$ denotes cubic B-spline function given by
\begin{eqnarray*}\nonumber
\Omega_3(x_1)=\left\{\begin{array}{cc}
\displaystyle\qquad\quad\;\;\;\;|x_1|^3/2-x_1^2+2/3& |x_1|\leq1,\\
\displaystyle-|x_1|^3/6+x_1^2-2|x_1|+4/3 &\qquad 1<|x_1|<2,\\
\qquad\qquad\qquad\qquad\qquad\; 0 &|x_1|\geq2.
\end{array}\right.
\end{eqnarray*}
which is twice continuously differentiable and has compactly support.

In this example, the local displacement lies totally above the plane $\Gamma_0$. According to the location of the local displacement, we put the sampling points $z$ in the rectangular grid $(-10,10)\times (-1, 0.5)$ with step size $0.5$ in $x$-axis and $0.1$ in $y$-axis. The measurement was taken at $\Gamma_{b,a}$ with $a=15$ and $b=0.55$, and measurement points are chosen as $N=601$ which are uniformly distributed over $\Gamma_{b,a}$. As shown in Figure \ref{f2}, our inversion algorithm can present a satisfactory reconstruction in this case with exact data, $2\%$ noise and $5\%$ noise.

\begin{figure}
\centering
\subfigure[0\%]{\includegraphics[width=0.31\textwidth]{pictures/Example1/e1-exact.jpg}}
\subfigure[2\%]{\includegraphics[width=0.31\textwidth]{pictures/Example1/e1-2percent.jpg}}
\subfigure[5\%]{\includegraphics[width=0.31\textwidth]{pictures/Example1/e1-5percent.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 1 from data with no noise (a), 2\% noise (b) and 5\% noise (c)}
\label{f2}
\end{figure}

\textbf{Example 2}. In this example, the scattering interface $\Gamma$ is described as
\begin{eqnarray*}
f_2(x_1)=\left[-0.3\exp(-x_1^2)-0.4\exp(-4(x_1-2)^2)-0.2\exp(-3(x_1+2)^2)\right]\cdot\chi(x_1),
\end{eqnarray*}
where $\chi(x_1)\in C_0^{\infty}({\mathbb R}^1)$ is defined by
 \begin{equation}\nonumber
  \chi(x_1)=\left\{\begin{array}{cc}\qquad\qquad\qquad\qquad\;\; 1 &|x_1|\leq 4,\\\left[1+\exp\left(\frac{1}{5-|x_1|}+\frac{1}{4-|x_1|}\right)\right]^{-1} &\qquad
  4<|x_1|<5,\\\qquad\qquad\qquad\qquad\;\; 0 &|x_1|\geq 5.\end{array}\right.
\end{equation}
It is seen that the local perturbation of $f_2$ lies totally below the plane $\Gamma_0$. In this example, we set the rectangular grid for $(-10, 10)\times (-1, 0.2)$ with step size $0.5$ in $x$-axis and $0.1$ in $y$-axis. The scattering data are measured at $N=601$ points which are uniformly distributed over $\Gamma_{b,a}$ with $b=0.25$ and $a=15$. The reconstructions with no noise, $2\%$ noise and  $5\%$ noise were presented in Figure \ref{f3} which shows that our inversion algorithm can also provide a satisfactory reconstruction in this case.
\begin{figure}
\centering
\subfigure[0\%]{\includegraphics[width=0.31\textwidth]{pictures/Example2/e2-exact.jpg}}
\subfigure[2\%]{\includegraphics[width=0.31\textwidth]{pictures/Example2/e2-2percent.jpg}}
\subfigure[5\%]{\includegraphics[width=0.31\textwidth]{pictures/Example2/e2-5percent.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 2 from data with no noise (a),\;2\% noise (b) and 5\% noise (c)}
\label{f3}
\end{figure}

\textbf{Example 3}. In this example, the scattering interface $\Gamma$ is described by
a complex function defined as
\begin{equation}\nonumber
  f_3(x_1)=\left\{\begin{array}{c}
                           \exp(16/(x_1^2-16)) \sin(\pi x_1)\qquad\;\; |x_1|<4, \\[1mm]
                           \qquad\qquad\qquad\qquad\qquad\;\;\;\; 0\;\;\qquad|x_1|\geq 4.
                          \end{array}\right.
\end{equation}
The above definition shows that part of $\Gamma$ lies above $\Gamma_0$ and other part lies below $\Gamma_0$.
We choose the same rectangular grid and the same measurement points as in example 1.
 As seen in Figure \ref{f4}, the inversion algorithm remains valid to give a satisfactory reconstruction for a general locally rough interface, even at the noise level of $5\%$.

\begin{figure}
\centering
\subfigure[0\%]{\includegraphics[width=0.31\textwidth]{pictures/Example3/e3-exact.jpg}}
\subfigure[2\%]{\includegraphics[width=0.31\textwidth]{pictures/Example3/e3-2percent.jpg}}
\subfigure[5\%]{\includegraphics[width=0.31\textwidth]{pictures/Example3/e3-5percent.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 3 from data with no noise (a), 2\% noise (b) and 5\% noise (c)}
\label{f4}
\end{figure}

Notice the inversion algorithm depends on the number $N$ of measurement points, the measurement hight $b$ and the measurement width $a$. In the following experiments, we will focus on the influence of these three parameters on the numerical performance of the inversion algorithm. To show the influence of these parameters more clearly, we work only for the case with $2\%$ noise.

\textbf{Example 4}. In this example, we choose the same interface and the same sampling points with Example 1, and exam the influence of $N$. To this end, we fix $a=15$ and $b=0.55$, and present numerical results with $N=201$, $N=401$ and $N=601$ from left to right. As shown in Figure \ref{f5}, a better reconstruction is obtained as measurement points becomes larger.

\begin{figure}
\centering
\subfigure[N=201]{\includegraphics[width=0.31\textwidth]{pictures/Example4/e4-N-201.jpg}}
\subfigure[N=401]{\includegraphics[width=0.31\textwidth]{pictures/Example4/e4-N-401.jpg}}
\subfigure[N=601]{\includegraphics[width=0.31\textwidth]{pictures/Example4/e4-N-601.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 4 from data with $N=201$ (a), $N=401$ (b) and $N=601$ (c) at the same level of 2\% noise}
\label{f5}
\end{figure}

\textbf{Example 5}. In this example, we focus on the influence of the measurement hight $b$.
We choose the same scattering interface and rectangular grid as in example 2. Then we fix $a=15$ and $N=601$, and set the measurement hight $b=0.25$, $b=0.65$, and $b=1.05$, respectively. We present the reconstruction results in Figure \ref{f6} which shows that the quality of the reconstruction becomes worse when $b$ is large.

\begin{figure}
\centering
\subfigure[b=0.25]{\includegraphics[width=0.31\textwidth]{pictures/Example5/e5-b-025.jpg}}
\subfigure[b=0.65]{\includegraphics[width=0.31\textwidth]{pictures/Example5/e5-b-065.jpg}}
\subfigure[b=1.05]{\includegraphics[width=0.31\textwidth]{pictures/Example5/e5-b-105.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 5 from data with $b=0.25$ (a), $b=0.65$ (b) and $b=1.05$ (c) at the same level of 2\% noise}
\label{f6}
\end{figure}

\textbf{Example 6}. In this example, we test the influence of the measurement width $a$. The scattering interface is given by $f(x_1)=0.6\cdot\Omega_3(x_1)-0.4\cdot\Omega_3(x_1-5)$.
We set the same rectangular grid $S$ as in Example 1 and fix $b=0.55$ and $N=601$. Figure \ref{f7} shows the reconstructions with $a=2$, $a=8$, and $a=14$, respectively, from left to right. As shown in Figure \ref{f7}, we can recover the essential feature of the locally rough interface  very well below the measurement curve $\Gamma_{b,a}$.

\begin{figure}
\centering
\subfigure[a=2]{\includegraphics[width=0.31\textwidth]{pictures/Example6/e6-a-2.jpg}}
\subfigure[a=8]{\includegraphics[width=0.31\textwidth]{pictures/Example6/e6-a-8.jpg}}
\subfigure[a=14]{\includegraphics[width=0.31\textwidth]{pictures/Example6/e6-a-14.jpg}}
\caption{Reconstructions of the locally rough interface given in Example 5 from data with $a=2$ (a), $a=8$ (b) and $a=14$ (c) at the same level of 2\% noise}
\label{f7}
\end{figure}

From the results shown in Figures \ref{f2}-\ref{f4}, it is easily verified that our proposed method in Theorem \ref{thm3.3} can recover the local perturbation of the scattering interface very well at  different noise levels. In addition, it is easily seen that our method could give a better reconstruction as $a$ and $b$ become larger which corresponds to more measurement data to be available. It is also noticed that the quality of the reconstructions become worse when $b$ becomes larger. In this case the scattered field decays when the measurement hight $b$ becomes large which leads to that lesser measurement data are available in this case.

\section{Conclusion}

In this paper, we proposed an extended sampling method to recover the shape and location of
a locally rough interface by near-field measurements above the interface.
The idea is mainly based on constructing a modified near-field operator equation by reducing
the interface scattering problem into an equivalently Lippmann-Schwinger integral equation in a bounded domain,
which generalized our previous work for a locally rough surface with a Dirichlet boundary condition.
Numerical experiments showed that the proposed method can provide a stable and satisfactory reconstruction
for locally rough interfaces with different image features. However, it is not clear how to
extend our method with mathematical justification to recover a non-locally rough interface.
Moreover, it is also challenging to develop a valid sampling method to simultaneously recover an
unbounded rough interface and bounded obstacles embedded in a lower half-space, which is more interesting
in practical applications. We hope to report these works in the future.

\section*{Acknowledgements}

This work was partly supported by the NNSF of China grants 11771349, 91730306 and 11601042.

redpajama_set_name: RedPajamaArXiv


text:
Home Sports Gareth Southgate Says England Need to Play Better
Gareth Southgate Says England Need to Play Better
Despite England's comfortable win over Germany last week, Gareth Southgate has made clear that the players need to play better. And there are concerns that Kane Maguire may not be fit for the upcoming game against Iran after having been withdrawn from the squad due to illness.
Harry Maguire's injury scare affects England's attacking potential
Earlier this month Harry Maguire suffered a suspected concussion during a match against Iran. He was later replaced by Eric Dier.
The Maguire incident has had a serious impact on the attacking potential of England. The team is now playing five games before a break for the World Cup.
The injured Maguire will miss England's Carabao Cup clash against Aston Villa on Thursday evening and their match against the USA on Friday. Maguire's injury could leave the World Cup place in doubt.
Maguire has been an integral part of England's resurgence at international level and is set to play a major role in their hopes of winning the World Cup. But the player has also been the target of a lot of criticism.
He was the central figure in England's successful run to the European Championship final last year, and was also named in the squad of the tournament. But Maguire hasn't quite lived up to his billing in the Premier League this season. He's had a troubled year at Manchester United, finishing sixth in the league.
Southgate's performance against Germany
During two matches in the last week, Harry Maguire has been under scrutiny. The Manchester United player has made two errors that could seal England's fate in the coming weeks. Despite these errors, it's England's fightback against Germany which has won the day.
The two games were played at Wembley, where the game ended in a draw. The result won't deflect attention from the debate over Maguire's future at the World Cup.
During the game, Southgate introduced a handful of substitutes, including Bukayo Saka and Mason Mount. Saka was a part of the move that sparked England's fightback. He laid off a pass for Mount to latch onto. Mount then blasted a sweet first-time shot past Marc-Andre ter Stegen.
It's not easy for a team to come from two goals down, but England made it work. It wasn't just a matter of making the right substitutions. It was also about getting the right system right.
Southgate has said he wants to improve on the performance of his team's last two games in the Nations League. He also praised the players for staying together during the matches. He believes the team's battling spirit will prove useful heading into the World Cup in November.
Read More : yalla shoot
Kane Maguire's illness-related withdrawal from Iran game
Despite a brilliant performance against Iran, England defender Harry Maguire has been forced to pull out of the game. He suffered an apparent concussion and he went off early in the second half.
Maguire was playing well for the Three Lions, who dominated the game. He set up Bukayo Saka for England's second goal, and helped England to a 6-2 win over Iran. He received the man of the match award.
Maguire was taken off in the 70th minute as a precaution. He was replaced by Eric Dier. He had a blurred vision and complained of feeling unwell. It was reported on social media that Maguire was under concussion protocol.
The England manager, Gareth Southgate, has now issued a positive update on Maguire's injury. He said that Maguire was feeling unwell before the goal, and that he felt that he should be fit for the game against the United States on Friday.
Harry Maguire was a major player in England's Euro 2020 semi-final last summer, and he played an important role in the England's 6-0 victory over Belgium in the World Cup semi-final last summer. However, he has struggled for game time at Manchester United.
Southgate could revert to three at the back
Despite a disappointing defeat against Croatia in the semifinals, Gareth Southgate is considering switching to a three-man backline. The head coach believes the formation could offer England more creativity and better chances.
The back three system is ideal for England in major tournaments, but Southgate is still not ready to drop the four at the back that he has been using for the last six years. However, England have had success using three in major tournaments in the past.
The three at the back system helps England build a strong defensive shape. It also allows Southgate to play wing backs. This is particularly useful if England are playing against lower-tier teams.
However, it also means that England lack creativity. They have not scored an open-play goal since March's 3-0 friendly win against Ivory Coast.
Southgate also wants England to shine up front. The goal of the three at the back formation is to give England security when they are playing testing games. It also means that England can be more creative in midfield.
Source : يلا شوت
Gareth Southgate
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redpajama_set_name: RedPajamaCommonCrawl


Preventing spam email messages is an ongoing issue and I'm sure that isn't going to change in the foreseeable future. I am pleased to say though, that we spent most of last week implementing new filtering technologies on our mail server that should provide a noticeable reduction in the amount of junk you receive.
We're also monitoring spam messages and adding filtering rules each week to automatically delete messages that are very obviously spam messages. This includes the various 'bodily enhancement' products and drugs, unsolicited job offers and bogus lotteries.
We invite you to participate in our efforts by submitting any spam messages you receive via our spam report page. Just browse to http://cbl.com.au/spam-message-submission/ and paste the contents of the message. If we can add a filter that will detect the message you have posted then we will do just that.


Vintage 1980s Lauren by Ralph Lauren 1/8 oz Pure Parfum Splash Miniature Perfume.
Bottle stands just over 1.5" tall.
Lauren was launched in 1978. It is classified as a fresh green floral fragrance for women.


text:
Reason # 467 to use a mask when traveling in commuter populations
The Silent Role of Biofilms in Chronic Disease › Forums › Biofilm Community › The DENT Connection › Reason # 467 to use a mask when traveling in commuter populations
April 28, 2013 at 5:03 am #3289 Score: 0
Before this study, educated patients and docs knew that sinus infections were indeed highly polymicrobial. This just validates one of the (many) environmental contributors to complex biofilm infections of the sinus.
Microbes Inhaled On The New York City Subway
The microbial population in the air of the New York City subway system is nearly identical to that of ambient air on the city streets. This research, published ahead of print in the journal Applied and Environmental Microbiology, establishes an important baseline, should it become necessary to monitor the subway's air for dispersal of potentially dangerous microbes. Also, the combination of new methodologies in the study, including fast collection of aerosols and rapid sequencing technology, provide an efficient means for monitoring which was not previously available.
The results "are strong testimony for the efficiency of the train pumping system for ventilation," says principal investigator Norman R. Pace of the University of Colorado, Boulder. The wind one feels while walking across a subway grate as the subway clatters beneath also demonstrates just how effective that system is, he says. The only obvious differences in the subway's microbial population are the somewhat higher proportion of skin microbiota, and the doubled density of the fungal population, which Pace suggests may be due to rotting wood. "I was impressed by the similarity of [subway] and outdoor air," he says.
The researchers used a high tech mechanism to collect air at around 300 liters per minute (L/min), a big jump on the previous state of the art, which swallowed 12 L/min. That enabled collecting sufficient volume of air – a couple of cubic meters – to take the bacterial census within 20 minutes, instead of after "hours," says Pace. And analysis by sequencing is far faster and more thorough then using culture.
Pace notes that until now, the microbial content of subway air was unknown, and that the microbiology of indoor air is an emerging field of scientific inquiry. His research was funded by the Alfred P. Sloan Foundation, through its Microbiology of the Built Environment program, which has made 64 grants totaling $28 million to date.
"While it is difficult to predict what will be discovered on the frontier of scientific inquiry, the opportunity exists to better understand these complex microbial ecosystems and how they affect health and the environment. We expect that someday this knowledge will influence design and construction practices and other industrial processes," says Paula Olsiewski, program director, Alfred P. Sloan Foundation.[HR][/HR]References:
A copy of the manuscript can be found online at http://bit.ly/asmtip0413c. Formal publication is scheduled for the July 2013 issue of Applied and Environmental Microbiology.
(C.E. Robertson, L.K. Baumgartner, J.K. Harris, K.L. Peterson, M.J. Stevens, D.N. Frank, and N.R. Pace, 2013. Culture-independent analysis of aerosol microbiology in a metropolitan subway system. Appl. Environ. Microbiol. Published ahead of print 29 March 2013 ,doi:10.1128/AEM.00331-13)
Source: Microbes Inhaled On The New York City Subway
Viewing 6 - 10 of 111 items
Richard Longland started the topic Monitoring the Formation and Removal of Biofilms in the forum General Article Library 2 years, 8 months ago
Monitoring in Real Time the Formation and Removal of Biofilms From Clinically Relevant Pathogens
SAN DIEGO, CA (PRWEB) NOVEMBER 11, 2016
In addition to living in a free-floating "planktonic" state within aqueous environments, bacteria can also colonize solid surfaces present at liquid-solid and air-solid interfaces. Within these micr…[Read more]
Jeremy Murphree replied to the topic Ozone differences among 2011 options for dental professionals in the forum Xpand Network 2 years, 8 months ago
Jeremy Murphree voted up to the topic Ozone differences among 2011 options for dental professionals by Harrison 2 years, 8 months ago
Jeremy Murphree replied to the topic Compound kills 98% of a drug-resistant bacteria in the forum The Human Ecosystem 2 years, 8 months ago


The DEC MSV11-QC EQUIV is available from Keyways.
The MSV11-QC EQUIV is sometimes known as MSV11QCEQUIV.
We warehouse over 75,000 parts in our 30,000 square foot facilities. We repair or exchange most items. Contact us to see if we repair or exchange the DEC MSV11-QC EQUIV.


text:
A growing village in western Albemarle, Old Trail sees record sales
by Charlottesville Tomorrow January 21, 2011 Updated November 9, 2018
By Brian Wheeler
One real estate project in western Albemarle County is seeing significant residential sales growth. Old Trail Village reported this week that 70 homes were sold or placed under contract during 2010, a record number for the mixed-use development near Crozet.
"We had the best selling year that we have ever had," said Andrea McNeill, director of marketing for Beights Corp. "I think in large part that's due to the housing product that our builders are providing, and it's obviously what consumers want."
Old Trail's success comes in the midst of a housing market that last year saw regional sales drop 1.5 percent below their 2009 levels, the fourth straight year of decline.
The 70 homes that sold in 2010 surpasses the total of 27 sold in 2009 and 21 in 2008. McNeill said 58 of the homes were new construction and 12 were re-sales.
In addition, the first phase of Old Trail's mixed-use commercial town center is virtually all leased.
There are almost 300 occupied homes in Old Trail today and the development has been approved to build another 2,300, for about 2,600 homes in total.
Gaylon Beights, president of Beights Corp., attributed part of the recent success to the maturing of the neighborhood.
"I think Old Trail has matured, the amenities are present, and the architecture is established," Beights said.
When Albemarle County approved the development in 2005, many area residents decried it as being too large, a threat to existing businesses in downtown Crozet, and a population bomb that would undermine the 2004 Crozet Master Plan. That plan has now been revised with a great deal of community input to adjust, in part, for the housing density that ended up in Old Trail.
Jim Duncan, a Realtor in Crozet at Nest Realty, said Beights had responded well to market demand.
"I think they are doing something right and that they have created something that people want," Duncan said. "When I talk about it with clients I say, 'It has stuff — a pool, shopping areas, things to walk to.' People want to have an engaged community."
"Success breeds success," he added. "Old Trail has succeeded faster than other developments like Belvedere because they had the commercial areas ready to go. People view this as the beginning of a successful community."
According to Beights, there are 39 apartments above the first floor commercial area in the existing town center, all of which were occupied throughout last year. Recently five apartments became available, but he said that was because those residents moved into other homes within Old Trail.
Melissa Riley lives in one of the apartments and works in the town center.
"This is a close-knit community and you feel safe," Riley said in an interview at Face Value Studios. "I'm a single mom and I plan to start by renting a townhome and then move on up in the neighborhood. Who wouldn't want to wake up every morning and see those mountains?"
Eating lunch at Anna's Pizza in the town center, Jason and Adrienne Augustino said they had relocated from Baltimore and were initially attracted by the surrounding countryside.
"We just liked the area, the farmland," Jason said. "It sounded like a nice area that would have restaurants and a walkable community. Everything that we like and want and need is here."
McNeill said phase 9 of Old Trail is under construction across from the town center. The Village Commons will include another 126 homes.
"Also, the Lodge at Old Trail will have 126 beds in a new senior assisted living facility," McNeill added. "It's on the rental model and that has a range of care from basic assisted living up to end-of-life care."
Both Riley and the Augustinos said they were excited about the next phase of development for Old Trail Village.
"From an investment perspective, it will really help," Adrienne said. "We enjoy being able to walk to restaurants and we are excited about the neighborhood's growth."
The investment is paying off for Beights, too, who led his business through a difficult market.
"The biggest surprise was 2008-2009 when we faced a market like I have never could have imagined in 40 years of development," Beights said. "Maneuvering a project this massive through the uncertainty of those years, I knew we'd be here, but I didn't know if my bank was here for the long haul. It's refreshing to be on this side of 2010 being able to say sales are continuing."
Virginia has new district maps — and some could spell good news for area Democrats
A developer's plan to build new apartments in Scottsville shows just how unprecedented big projects are for the small town
Affordable mobile home parks are disappearing from Charlottesville — a new law may bring them back
Tagged: Albemarle County, Land Use - Crozet, Old Trail Village
Previous County water board briefed on earthen dam progress
Next Albemarle begins redistricting process


Judson C. Kelly, 81, 1841 Shawnee Dr., Sidney, Ohio, died at Fair Haven Shelby County Home on Monday, July 19, 2004 at 2:10 PM. He was born June 11, 1923 in Hamilton County, Ohio the son of Harry James and Grace (Archer) Kelly and they are deceased. He married Arline M. Emert in Kentucky on September 13, 1944 and she died August 11, 2003. Surviving are one son, Patrick and his friend, Jan Doty; several grandchildren and great-grandchildren and several nieces and nephews. Sons, Larry and Michael Kelly are deceased. He served in the United States Army and was honorably discharged in 1943. He was a life member of the Sidney American Legion, Post 217. He was of the Protestant faith. Mr. Kelly was a former State Highway Department employee for 13 years. He also operated and owned J C Motor Sales, a used car lot and garage, prior to retirement. Rev. Harold McKnight will be in charge of funeral services on Thursday, July 22, 2004 at 1:30 PM at Salm-McGill and Tangeman Funeral Home. Calling hours will be from 12 Noon until the hour of service. Burial will be at Miami Memorial Park Cemetery, Covington, Ohio. Memorial contributions may be made to Fair Haven Shelby County Home, 2901 Fair Rd., Sidney, Ohio 45365. Envelopes will also be available at the funeral home.


text:
Q: Column 'date_start' in where clause is ambiguous I wanna SELECT some date, but i have problem with this
SELECT DISTINCT `clients_agreements`.`date_start` , `buildings`.`id` ,
    `buildings`.`street` , `buildings`.`street_nr` ,
    `clients`.`building_id` , `clients_agreements`.`user_id`, `clients_agrees_conds`.`user_id`, `clients_agrees_perss`.`client_agreement_id`
FROM `clients_agreements`
LEFT JOIN `buildings`
    On `clients_agreements`.`user_id` = `buildings`.`id`
LEFT JOIN `clients`
    ON `clients`.`building_id` = `buildings`.`id`
LEFT JOIN `clients_agrees_conds`
    ON `clients_agreements`.`user_id` = `clients_agrees_conds`.`user_id`
LEFT JOIN `clients_agrees_perss`
   ON `clients_agrees_conds`.`user_id` = `clients_agrees_perss`.`user_id`
WHERE `date_start` = (CURRENT_DATE)

I get this:

error #1052 - Column 'date_start' in where clause is ambiguous

A: Where clause should contain alias for date_start to point to particular table:
        SELECT DISTINCT `clients_agreements`.`date_start` , `buildings`.`id` , `buildings`.`street` , `buildings`.`street_nr` , `clients`.`building_id` , `clients_agreements`.`user_id`, `clients_agrees_conds`.`user_id`, `clients_agrees_perss`.`client_agreement_id`
        FROM `clients_agreements`
        LEFT JOIN `buildings` On `clients_agreements`.`user_id` = `buildings`.`id`
        LEFT JOIN `clients` ON `clients`.`building_id` = `buildings`.`id`
        LEFT JOIN `clients_agrees_conds` ON `clients_agreements`.`user_id` = `clients_agrees_conds`.`user_id`
        LEFT JOIN `clients_agrees_perss` ON `clients_agrees_conds`.`user_id` = `clients_agrees_perss`.`user_id`
        WHERE `clients_agreements`.`date_start` = (CURRENT_DATE)

A: In your where clause add the table name next to it like
clients_agreements.date_start = CURRENT_DATE


Palau's campaign ad on conservation, the Palau Pledge, had won a Black Pencil for branding during the D&AD Awards ceremony in London on April 26.
The D&AD's Black Pencil award is one of the most cherished awards in advertising.
The "Palau Pledge" campaign film, which was produced by Australian communications agency Host/Havas in partnership with the Palau Legacy Project, won a Black Pencil for branding with a campaign that required Palau visitors to sign a passport pledge to act in an ecologically responsible way while staying in the country.
The other two winners for the same award were Saatchi & Saatchi New York for "It's a Tide Ad" and McCann New York's Fearless Girl.
The "Palau Pledge" took home a total of eight pencils including the Black Pencil in Campaign Branding and Identity, one Yellow Pencil (gold) in Brand Expression in Print, three Graphite Pencils (silver) and three Wood Pencils (bronze).
D&AD received entries from 50 countries across the globe for this year.