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57-569: The TIMED (Thermosphere • Ionosphere • Mesosphere • Energetics and Dynamics) mission is dedicated to study the influences that energetics and dynamics of the Sun and humans have on the least explored and understood region of Earth's atmosphere – the Mesosphere and Lower Thermosphere / Ionosphere (MLTI). The mission was launched from Vandenberg Air Force Base in California on 7 December 2001 aboard

114-500: A ≤ x ≤ b ) {\displaystyle (a\leq x\leq b)} on the string be f ( x ) e {\displaystyle f(x)\mathbf {e} } , where e {\displaystyle \mathbf {e} } is a unit vector pointing vertically and f : [ a , b ] → R . {\displaystyle f:[a,b]\to \mathbb {R} .} Let u ( x ) {\displaystyle u(x)} be

171-558: A Delta II rocket launch vehicle . The project is sponsored and managed by NASA , while the spacecraft was designed and assembled by the Applied Physics Laboratory at Johns Hopkins University . The mission has been extended several times, and has now collected data over an entire solar cycle , which helps in its goal to differentiate the Sun's effects on the atmosphere from other effects. It shared its Delta II launch vehicle with

228-426: A light source , with an emission spectrum . Also in general, light which is not transmitted is absorbed ; for intense light, that can cause significant heating of the filter. However, the optical term absorbance refers to the attenuation of the incident light, regardless of the mechanism by which it is attenuated. Some filters, like mirrors , interference filters, or metal meshes, reflect or scatter much of

285-460: A strongly monotone symmetric linear operator , that is, a linear operator satisfying The energetic inner product is defined as and the energetic norm is The set Y {\displaystyle Y} together with the energetic inner product is a pre-Hilbert space . The energetic space X E {\displaystyle X_{E}} is defined as the completion of Y {\displaystyle Y} in

342-437: A " low pass filter ", without qualification, would be understood to be an electronic filter . Band-pass filters only transmit a certain wavelength band, and block others. The width of such a filter is expressed in the wavelength range it lets through and can be anything from much less than an Ångström to a few hundred nanometers. Such a filter can be made by combining an LP- and an SP filter. Examples of band-pass filters are

399-476: A continuous record of water vapor and carbon dioxide levels in the stratosphere and mesosphere. SABER is able to collect 1,500 water vapor measurements per day, a vast improvement from previous satellites and ground-based observations. SABER had a flaw in its optical filter that caused it to overestimate water vapor levels; this error was discovered and the data were corrected. Based on the corrected data, SABER found that between 2002 and 2018, water vapor levels in

456-498: A figure potentially exceeding that of the 2009 satellite collision between Iridium 33 and Kosmos 2251 . This close miss was particularly concerning to NASA, which highlighted the event at the 39th Space Symposium in Colorado Springs in a broader speech on NASA's new space sustainability strategy plan. TIMED has improved scientific understanding of long-term trends in the upper atmosphere. The SABER instrument has collected

513-447: A function in a finite-dimensional subspace of the true solution space. For example, one might let u h {\displaystyle u_{h}} be a continuous piecewise linear function in the energetic space, which gives the finite element method . The approximation u h {\displaystyle u_{h}} can be computed by solving a system of linear equations . The energetic norm turns out to be

570-531: A glass substrate with a series of optical coatings . Dichroic filters usually reflect the unwanted portion of the light and transmit the remainder. Dichroic filters use the principle of interference . Their layers form a sequential series of reflective cavities that resonate with the desired wavelengths. Other wavelengths destructively cancel or reflect as the peaks and troughs of the waves overlap. Dichroic filters are particularly suited for precise scientific work, since their exact colour range can be controlled by

627-416: A multiple of the cavity's resonance frequency. Etalons are another variation: transparent cubes or fibers whose polished ends form mirrors tuned to resonate with specific wavelengths. These are often used to separate channels in telecommunications networks that use wavelength division multiplexing on long-haul optic fibers . Monochromatic filters only allow a narrow range of wavelengths (essentially

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684-496: A particular range of wavelengths , that is, colours , while absorbing the remainder. They can usually pass long wavelengths only (longpass), short wavelengths only (shortpass), or a band of wavelengths, blocking both longer and shorter wavelengths (bandpass). The passband may be narrower or wider; the transition or cutoff between maximal and minimal transmission can be sharp or gradual. There are filters with more complex transmission characteristic, for example with two peaks rather than

741-400: A portion of it. They are specified by the optical density (OD) of the filter, which is the negative of the common logarithm of the transmission coefficient . They are useful for making photographic exposures longer. A practical example is making a waterfall look blurry when it is photographed in bright light. Alternatively, the photographer might want to use a larger aperture (so as to limit

798-504: A real Hilbert space X {\displaystyle X} with the inner product ( ⋅ | ⋅ ) {\displaystyle (\cdot |\cdot )} and the norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} . Let Y {\displaystyle Y} be a linear subspace of X {\displaystyle X} and B : Y → X {\displaystyle B:Y\to X} be

855-446: A sign error in the flight software. The problem was fixed by temporarily disabling the orbiter's magnetic field sensor and uploading a software patch to fix the sign error. In a separate incident, another software update fixed a problem caused by faulty testing of the Sun sensors . After these corrections, the attitude control system functioned as intended. At approximately 06:30 UTC on 28 February 2024, TIMED passed within 10 meters of

912-448: A single band; these are more usually older designs traditionally used for photography; filters with more regular characteristics are used for scientific and technical work. Optical filters are commonly used in photography (where some special effect filters are occasionally used as well as absorptive filters), in many optical instruments, and to colour stage lighting . In astronomy optical filters are used to restrict light passed to

969-720: A single colour) to pass. The term "infrared filter" can be ambiguous, as it may be applied to filters to pass infrared (blocking other wavelengths) or to block infrared (only). Infrared-passing filters are used to block visible light but pass infrared; they are used, for example, in infrared photography . Infrared cut-off filters are designed to block or reflect infrared wavelengths but pass visible light. Mid-infrared filters are often used as heat-absorbing filters in devices with bright incandescent light bulbs (such as slide and overhead projectors ) to prevent unwanted heating due to infrared radiation. There are also filters which are used in solid state video cameras to block IR due to

1026-408: A very sharp slope (referred to as edge filters), are described by the cut-on wavelength at 50 percent of peak transmission. In fluorescence microscopy, longpass filters are frequently utilized in dichroic mirrors and barrier (emission) filters. Use of the older term 'low pass' to describe longpass filters has become uncommon; filters are usually described in terms of wavelength rather than frequency, and

1083-412: Is a polarizer or polarization filter, which blocks or transmits light according to its polarization . They are often made of materials such as Polaroid and are used for sunglasses and photography . Reflections, especially from water and wet road surfaces, are partially polarized, and polarized sunglasses will block some of this reflected light, allowing an angler to better view below the surface of

1140-601: Is a linear subspace of X . {\displaystyle X.} Consider the operator B : Y → X {\displaystyle B:Y\to X} given by the formula so the deflection satisfies the equation B u = f . {\displaystyle Bu=f.} Using integration by parts and the boundary conditions, one can see that for any u {\displaystyle u} and v {\displaystyle v} in Y . {\displaystyle Y.} Therefore, B {\displaystyle B}

1197-573: Is a symmetric linear operator. B {\displaystyle B} is also strongly monotone, since, by the Friedrichs's inequality for some C > 0. {\displaystyle C>0.} The energetic space in respect to the operator B {\displaystyle B} is then the Sobolev space H 0 1 ( a , b ) . {\displaystyle H_{0}^{1}(a,b).} We see that

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1254-452: Is an optical interference or coloured glass filter that attenuates longer wavelengths and transmits (passes) shorter wavelengths over the active range of the target spectrum (usually the ultraviolet and visible region). In fluorescence microscopy, shortpass filters are frequently employed in dichromatic mirrors and excitation filters. A relatively new class of filters introduced around 1990. These filters are normally filters in reflection, that

1311-407: Is extended from Y {\displaystyle Y} to X E {\displaystyle X_{E}} by where ( u n ) {\displaystyle (u_{n})} and ( v n ) {\displaystyle (v_{n})} are sequences in Y that converge to points in X E {\displaystyle X_{E}} in

1368-409: Is more vertical mixing of CO 2 than previously thought. By collecting upper atmosphere data, TIMED assists the modeling of environmental impacts. Water vapor and carbon dioxide are greenhouse gases and their growth in the upper atmosphere must be factored into climate models. Additionally, upper atmosphere water vapor contributes to ozone depletion. Energetic space Formally, consider

1425-431: Is not possible to do perfect filtering. A perfect filter would remove particular wavelengths and leave plenty of light so a worker can see what he/she is working on. A wedge filter is an optical filter so constructed that its thickness varies continuously or in steps in the shape of a wedge. The filter is used to modify the intensity distribution in a radiation beam. It is also known as linearly variable filter (LVF). It

1482-745: Is now available as colored-film filters, originally made from animal gelatin but now usually a thermoplastic such as acetate , acrylic , polycarbonate , or polyester depending upon the application. They were standardized for photographic use by Wratten in the early 20th century, and also by color gel manufacturers for theater use. There are now many absorptive filters made from glass to which various inorganic or organic compounds have been added. Colored glass optical filters, although harder to make to precise transmittance specifications, are more durable and stable once manufactured. Alternately, dichroic filters (also called "reflective" or "thin film" or "interference" filters) can be made by coating

1539-498: Is simply the function extension of B {\displaystyle B} from Y {\displaystyle Y} to X E . {\displaystyle X_{E}.} Consider a string whose endpoints are fixed at two points a < b {\displaystyle a<b} on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point x {\displaystyle x} (

1596-414: Is they are notch filters in transmission. They consist in their most basic form of a substrate waveguide and a subwavelength grating or 2D hole array. Such filters are normally transparent, but when a leaky guided mode of the waveguide is excited they become highly reflective (a record of over 99% experimentally) for a particular polarization , angular orientations, and wavelength range. The parameters of

1653-499: The Jason-1 oceanography mission. The Mesosphere , Lower Thermosphere and Ionosphere (MLTI) region of the atmosphere to be studied by TIMED is located between 60 and 180 kilometres (37 and 112 mi) above the Earth's surface, where energy from solar radiation is first deposited into the atmosphere. This can have profound effects on Earth's upper atmospheric regions, particularly during

1710-545: The Lyot filter and the Fabry–Pérot interferometer . Both of these filters can also be made tunable, such that the central wavelength can be chosen by the user. Band-pass filters are often used in astronomy when one wants to observe a certain process with specific associated spectral lines . The Dutch Open Telescope and Swedish Solar Telescope are examples where Lyot and Fabry–Pérot filters are being used. A shortpass (SP) Filter

1767-471: The deflection of the string at the point x {\displaystyle x} under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is and the total potential energy of the string is The deflection u ( x ) {\displaystyle u(x)} minimizing the potential energy will satisfy the differential equation with boundary conditions To study this equation, consider

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1824-456: The depth of field ); adding an ND filter permits this. ND filters can be reflective (in which case they look like partially reflective mirrors) or absorptive (appearing grey or black). A longpass (LP) Filter is an optical interference or coloured glass filter that attenuates shorter wavelengths and transmits (passes) longer wavelengths over the active range of the target spectrum (ultraviolet, visible, or infrared). Longpass filters, which can have

1881-414: The accurately controlled optical properties and precisely defined transmission curves of filters designed for scientific work, and sell in larger quantities at correspondingly lower prices than many laboratory filters. Some photographic effect filters, such as star effect filters, are not relevant to scientific work. In general, a given optical filter transmits a certain percentage of the incoming light as

1938-606: The bulk or have interference coatings. The optical properties of filters are completely described by their frequency response , which specifies how the magnitude and phase of each frequency component of an incoming signal is modified by the filter. Filters mostly belong to one of two categories. The simplest, physically, is the absorptive filter; then there are interference or dichroic filters . Many optical filters are used for optical imaging and are manufactured to be transparent ; some used for light sources can be translucent . Optical filters selectively transmit light in

1995-435: The defunct Kosmos 2221 satellite. As neither TIMED nor Kosmos 2221 can be maneuvered, the conjunction was unavoidable. LeoLabs, a satellite tracking company, had estimated a satellite collision probability of as high as 8% prior to the encounter. A collision between the two satellites, both traveling at hypervelocity speeds relative to each other, was projected to generate between 2,500 and 7,500 fragments of space debris ,

2052-770: The dual X E ∗ {\displaystyle X_{E}^{*}} (by the strong monotonicity property of B {\displaystyle B} ). Via these identifications, it follows from the above formula that B E u = B u . {\displaystyle B_{E}u=Bu.} In different words, the original operator B : Y → X {\displaystyle B:Y\to X} can be viewed as an operator B : Y → X E ∗ , {\displaystyle B:Y\to X_{E}^{*},} and then B E : X E → X E ∗ {\displaystyle B_{E}:X_{E}\to X_{E}^{*}}

2109-571: The duality bracket between X E ∗ {\displaystyle X_{E}^{*}} and X E , {\displaystyle X_{E},} so ⟨ B E u | v ⟩ E {\displaystyle \langle B_{E}u|v\rangle _{E}} actually denotes ( B E u ) ( v ) . {\displaystyle (B_{E}u)(v).} If u {\displaystyle u} and v {\displaystyle v} are elements in

2166-554: The elastic energy of the string which motivated this study is so it is half of the energetic inner product of u {\displaystyle u} with itself. To calculate the deflection u {\displaystyle u} minimizing the total potential energy F ( u ) {\displaystyle F(u)} of the string, one writes this problem in the form Next, one usually approximates u {\displaystyle u} by some u h {\displaystyle u_{h}} ,

2223-451: The energetic norm. X E {\displaystyle X_{E}} can be considered a subset of the original Hilbert space X , {\displaystyle X,} since any Cauchy sequence in the energetic norm is also Cauchy in the norm of X {\displaystyle X} (this follows from the strong monotonicity property of B {\displaystyle B} ). The energetic inner product

2280-561: The energetic norm. The operator B {\displaystyle B} admits an energetic extension B E {\displaystyle B_{E}} defined on X E {\displaystyle X_{E}} with values in the dual space X E ∗ {\displaystyle X_{E}^{*}} that is given by the formula Here, ⟨ ⋅ | ⋅ ⟩ E {\displaystyle \langle \cdot |\cdot \rangle _{E}} denotes

2337-514: The filters are designed by proper choice of the grating parameters. The advantage of such filters are the few layers needed for ultra-narrow bandwidth filters (in contrast to dichroic filters), and the potential decoupling between spectral bandwidth and angular tolerance when more than 1 mode is excited. Filters for sub-millimeter and near infrared wavelengths in astronomy are metal mesh grids that are stacked together to form LP, BP, and SP filters for these wavelengths. Another kind of optical filter

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2394-402: The following four main instruments: The data collected by the satellite's instruments are made freely available to the public. TIMED experienced minor problems with attitude control when, after launch, the magnetorquers failed to slow the spacecraft's spin as intended. An engineer installing the magnetorquers had mistakenly recorded the reverse of their actual polarities, which generated

2451-533: The high sensitivity of many camera sensors to unwanted near-infrared light. Ultraviolet (UV) filters block ultraviolet radiation, but let visible light through. Because photographic film and digital sensors are sensitive to ultraviolet (which is abundant in skylight) but the human eye is not, such light would, if not filtered out, make photographs look different from the scene visible to people, for example making images of distant mountains appear unnaturally hazy. An ultraviolet-blocking filter renders images closer to

2508-540: The lens. Polarized filters are also used to view certain types of stereograms , so that each eye will see a distinct image from a single source. An arc source puts out visible, infrared and ultraviolet light that may be harmful to human eyes. Therefore, optical filters on welding helmets must meet ANSI Z87:1 (a safety glasses specification) in order to protect human vision. Some examples of filters that would provide this kind of filtering would be earth elements embedded or coated on glass, but practically speaking it

2565-417: The lower stratosphere were increasing at an average rate of 0.25 ppmv (around 5%) per decade, and in the upper stratosphere and mesosphere, water vapor levels were increasing at an average rate of 0.1-0.2 ppmv (around 2-3%) per decade. Growth in methane levels is thought to be partially responsible for the growth in water vapor levels, as methane oxidizes into carbon dioxide and water vapor, but changes driven by

2622-418: The natural norm in which to measure the error between u {\displaystyle u} and u h {\displaystyle u_{h}} , see Céa's lemma . Optical filter An optical filter is a device that selectively transmits light of different wavelengths , usually implemented as a glass plane or plastic device in the optical path , which are either dyed in

2679-473: The non-transmitted light. The ( dimensionless ) Optical Density of a filter at a particular wavelength of light is defined as − log 10 ⁡ T {\displaystyle -\log _{10}T} where T is the (dimensionless) transmittance of the filter at that wavelength. Optical filtering was first done with liquid-filled, glass-walled cells; they are still used for special purposes. The widest range of color-selection

2736-442: The norm being given by Let Y {\displaystyle Y} be the set of all twice continuously differentiable functions u : [ a , b ] → R {\displaystyle u:[a,b]\to \mathbb {R} } with the boundary conditions u ( a ) = u ( b ) = 0. {\displaystyle u(a)=u(b)=0.} Then Y {\displaystyle Y}

2793-534: The original subspace Y , {\displaystyle Y,} then by the definition of the energetic inner product. If one views B u , {\displaystyle Bu,} which is an element in X , {\displaystyle X,} as an element in the dual X ∗ {\displaystyle X^{*}} via the Riesz representation theorem , then B u {\displaystyle Bu} will also be in

2850-522: The peak of the Sun's 11-year solar cycle when the greatest amounts of its energy are being released. Understanding these interactions is also important for our understanding of various subjects in geophysics , meteorology , aeronomy , and atmospheric science , as solar radiation is one of the primary driving forces behind atmospheric tides . Changes in the MLT can also affect modern satellite and radio telecommunications . The spacecraft payload consists of

2907-407: The solar cycle may also be responsible. SABER has also monitored carbon dioxide levels in the upper atmosphere. The instrument found that carbon dioxide levels in the upper atmosphere are increasing: at an altitude of 110 kilometres (68 mi), CO 2 levels were rising at an average rate of 12% per decade. This rate is faster than what has been predicted by climate models, and suggests that there

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2964-494: The space X = L 2 ( a , b ) , {\displaystyle X=L^{2}(a,b),} that is, the Lp space of all square-integrable functions u : [ a , b ] → R {\displaystyle u:[a,b]\to \mathbb {R} } in respect to the Lebesgue measure . This space is Hilbert in respect to the inner product with

3021-427: The spectral band of interest, e.g., to study infrared radiation without visible light which would affect film or sensors and overwhelm the desired infrared. Optical filters are also essential in fluorescence applications such as fluorescence microscopy and fluorescence spectroscopy . Photographic filters are a particular case of optical filters, and much of the material here applies. Photographic filters do not need

3078-422: The thickness and sequence of the coatings. They are usually much more expensive and delicate than absorption filters. They can be used in devices such as the dichroic prism of a camera to separate a beam of light into different coloured components. The basic scientific instrument of this type is a Fabry–Pérot interferometer . It uses two mirrors to establish a resonating cavity. It passes wavelengths that are

3135-424: The visual appearance of the scene. As with infrared filters there is a potential ambiguity between UV-blocking and UV-passing filters; the latter are much less common, and more usually known explicitly as UV pass filters and UV bandpass filters. Neutral density (ND) filters have a constant attenuation across the range of visible wavelengths, and are used to reduce the intensity of light by reflecting or absorbing

3192-430: The water and better vision for a driver. Light from a clear blue sky is also polarized, and adjustable filters are used in colour photography to darken the appearance of the sky without introducing colours to other objects, and in both colour and black-and-white photography to control specular reflections from objects and water. Much older than g.m.r.f (just above) these first (and some still) use fine mesh integrated in

3249-434: The wavelength changes. This is measured by a spectrophotometer . As a linear material, the absorption for each wavelength is independent of the presence of other wavelengths. A very few materials are non-linear , and the transmittance depends on the intensity and the combination of wavelengths of the incident light. Transparent fluorescent materials can work as an optical filter, with an absorption spectrum, and also as

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