|
|
| Line 1: |
Line 1: |
| '''Isoline retrieval''' is a [[remote sensing]] [[Inverse problem|inverse method]] that retrieves one or more [[isoline]]s of a trace atmospheric constituent or variable. When used to validate another contour, it is the most accurate method possible for the task. When used to retrieve a whole field, it is a general, nonlinear inverse method and a robust estimator.
| | It is a material that is used to build the front fork of the bike. If you keep winning competitions, you will be invited to bigger and bigger races. If you have any inquiries pertaining to where and how to use [http://wallpaper.lazonegeek.com/profile/sapeltier Womens mountain bike sizing.], you can contact us at the web-page. If you go on a regular mountain biking, you will develop a healthier body and disposition. Are you in search of a comfort bike, a mountain bike, or a bike that is designed just for cruising and do you prefer a standard road bike or are you looking for an hybrid. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races. <br><br> |
|
| |
|
| ==For validating advected contours==
| | It'll have you on the edge of your seat and holding your breath for Zink to land this record-breaking backflip. One way to increase exercise fun is to find a friend to run with you. The price you can expect to pay for stems is actually determined by the materials used along with its weight. Sometimes they are internal (drum and coaster brakes), and sometimes they are external (disc brakes). Browse a number of the on-line forums to determine what other riders like and do not like about their bicycles. <br><br>The steeper the angles, the more beneficial it would be for stability and high speed pedaling. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. Apply a generous amount of mountain bike lube to your chain as you move the pedals around backward. Mounting biking is a fun sport for professionals and beginners alike. There is a key point that you should always remember when going on any bicycle ride, especially one of great length. <br><br>With knowledge on the strengths and weakness of the MTB model, they are a great source of help. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. These four websites are important to serious bikers. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. You should be selective in choosing mountain bike tires since mountain bike is already made heavy and if you choose heavy tires you should consider yourself about how you will handle the bike and the terrain at the same time. <br><br>And there are some people who rightfully bring up the issue of the fuel costs and environmental impact of producing the electricty used to power the bike in the first place. Listed below are some of the common terms which you will come across. Something that sounds to me, to be nothing more than hard work. This speed is great for going up a hill that is steep, but it is not a good speed for flat ground or racing downhill. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way". |
| ===Rationale===
| |
| | |
| Suppose we have, as in [[contour advection]], inferred knowledge of a
| |
| single contour or isoline of an atmospheric constituent, ''q''
| |
| and we wish to validate this against satellite remote-sensing data. | |
| Since satellite instruments cannot measure the constituent directly,
| |
| we need to perform some sort of inversion.
| |
| In order to validate the contour, it is not necessary to know,
| |
| at any given point, the exact value of the constituent. We only need to
| |
| know whether it falls inside or outside, that is, is it greater
| |
| than or less than the value of the contour, ''q<sub>0</sub>''.
| |
| | |
| This is a classification problem. Let:
| |
| | |
| : <math>
| |
| j = \begin{cases} 1; & q < q_0 \\
| |
| 2; & q \geq q_0\end{cases}
| |
| </math>
| |
| | |
| be the discretized variable.
| |
| This will be related to the satellite ''measurement vector'', <math>\vec y</math>,
| |
| by some conditional probability, <math>P(\vec y|j)</math>,
| |
| which we approximate by collecting samples, called ''training data'', of both the
| |
| measurement vector and the state variable, ''q''.
| |
| By generating classification results over the region of interest
| |
| and using any contouring algorithm to separate the | |
| two classes, the isoline will have been "retrieved."
| |
| | |
| The accuracy of a retrieval will be given by integrating
| |
| the conditional probability over the area of interest, ''A'':
| |
| | |
| :<math>
| |
| a = \frac {1}{A} \int_A P \left[c(\vec{r}) | \vec{y}(\vec{r}) \right]
| |
| \, d\vec{r}
| |
| </math> | |
| | |
| where ''c'' is the retrieved class at position, <math>\vec r</math>.
| |
| We can maximize this quantity by maximizing the value of the integrand
| |
| at each point:
| |
| | |
| :<math>
| |
| \max(a) = \frac{1}{A} \int_A \left \lbrace \max_j P \left [j |
| |
| \vec{y}(\vec{r}) \right ] \right \rbrace \, d\vec{r}
| |
| </math>
| |
| | |
| Since this is the definition of maximum likelihood,
| |
| a [[statistical classification|classification algorithm]] based on [[maximum likelihood]]
| |
| is the most accurate method possible of validating an advected contour.
| |
| A good method for performing maximum likelihood classification
| |
| from a set of training data is [[variable kernel density estimation]].
| |
| | |
| ===Training data===
| |
| | |
| There are two methods of generating the training data.
| |
| The most obvious is empirically, by simply matching measurements of
| |
| the variable, ''q'', with [[collocation (remote sensing)|collocated]]
| |
| measurements from the satellite instrument. In this case,
| |
| no knowledge of the actual physics that produce the measurement
| |
| is required and the retrieval algorithm is purely statistical.
| |
| The second is with a forward model:
| |
| | |
| :<math>
| |
| \vec y = \vec f(\vec x) \,
| |
| </math> | |
| | |
| where <math>\vec x</math> is the ''state vector'' and
| |
| ''q = x<sub>k</sub>'' is a single component.
| |
| An advantage of this method is that state vectors need not
| |
| reflect actual atmospheric configurations, they need only
| |
| take on a state that could reasonably occur in the real atmosphere.
| |
| There are also none of the errors inherent in
| |
| most [[collocation (remote sensing)|collocation]] procedures,
| |
| e.g. because of offset errors in the locations of the paired samples
| |
| and differences in the footprint sizes of the two instruments.
| |
| Since retrievals will be biased towards more common states,
| |
| however, the statistics ought to reflect those in the real world.
| |
| | |
| ===Error characterization===
| |
| | |
| The conditional probabilities, <math>P(\vec y|j)</math>, provide
| |
| excellent error characterization, therefore the classification
| |
| algorithm ought to return them.
| |
| We define the ''confidence rating'' by rescaling the conditional
| |
| probability:
| |
| | |
| : <math>
| |
| C = \frac{n_c P(c|\vec y) - 1}{n_c - 1}
| |
| </math>
| |
| | |
| where ''n<sub>c</sub>'' is the number of classes (in this case, two).
| |
| If ''C'' is zero, then the classification is little better than
| |
| chance, while if it is one, then it should be perfect.
| |
| To transform the confidence rating to a statistical ''tolerance'',
| |
| the following line integral can be applied to an isoline retrieval | |
| for which the true isoline is known:
| |
| | |
| :<math>
| |
| \delta(C) = \frac{1}{l} \int_0^l h(C - C^\prime(\vec{r})) \, ds
| |
| </math> | |
| | |
| where ''s'' is the path, ''l'' is the length of the isoline
| |
| and <math>C^\prime</math> is the retrieved confidence as a function | |
| of position. | |
| While it appears that the integral must be evaluated separately
| |
| for each value of the confidence rating, ''C'', in fact it may be
| |
| done for all values of ''C'' by sorting the confidence ratings of the
| |
| results, <math>C^\prime</math>.
| |
| The function relates the threshold value of the confidence rating
| |
| for which the tolerance is applicable.
| |
| That is, it defines a region that contains a fraction of the true
| |
| isoline equal to the tolerance.
| |
| | |
| ===Example: water vapour from AMSU===
| |
| | |
| [[Image:tolerance from confidence.png|thumb|right|upright=1.5|alt=Tolerance vs. confidence|Statistical tolerance versus confidence rating for water-vapour isoline retrieval.]]
| |
| | |
| The [[Advanced Microwave Sounding Unit]] (AMSU) series of satellite instruments
| |
| are designed to detect temperature and water vapour. They have a high
| |
| horizontal resolution (as little as 15 km) and because they are
| |
| mounted on more than one satellite, full global coverage can be
| |
| obtained in less than one day.
| |
| Training data was generated using the second method from
| |
| [[European Centre for Medium-Range Weather Forecasts]] (ECMWF) ERA-40
| |
| data fed to a fast [[radiative transfer]] model called
| |
| [[RTTOV (radiative transfer code)|RTTOV]].
| |
| The function, <math>\delta(C)</math> has been generated from
| |
| simulated retrievals and is shown in the figure to the right.
| |
| This is then used to set the 90 percent tolerance in the figure | |
| below by shading all the confidence ratings less than 0.8.
| |
| Thus we expect the true isoline to fall within the shading
| |
| 90 percent of the time.
| |
| | |
| [[Image:ret colour.gif|thumb|center|upright=3|alt=Sample isoline retrieval|Water vapour isoline retrieved from AMSU measurements and compared with ECMWF reanalysis.]]
| |
| | |
| ==For continuum retrievals==
| |
| | |
| [[Image:conditional probability proxy.png|thumb|left|upright=1.5|alt=The conditional probability as proxy for the continuum variable|Specific humidity versus conditional probabilities from water-vapour isoline retrieval.]]
| |
| | |
| Isoline retrieval is also useful for retrieving a continuum variable
| |
| and constitutes a general, [[nonlinear]] [[inverse method]].
| |
| It has the advantage over both a [[neural network]], as well as iterative
| |
| methods such as [[optimal estimation]] that invert the forward model
| |
| directly, in that there is no possibility of getting stuck in a
| |
| [[local minimum]].
| |
| | |
| There are a number of methods of reconstituting the continuum variable
| |
| from the discretized one. Once a sufficient number of contours
| |
| have been retrieved, it is straightforward to [[interpolate]] between
| |
| them. Conditional probabilities make a good [[Proxy (statistics)|proxy]] for
| |
| the continuum value.
| |
| | |
| Consider the transformation from a continuum to a discrete variable:
| |
| | |
| :<math>
| |
| P(1 | \vec{y}) = \int_{-\infty}^{q_0} P(q | \vec{y}) \, dq
| |
| </math>
| |
| | |
| :<math>
| |
| P(2 | \vec{y}) = \int^{\infty}_{q_0} P(q | \vec{y}) \, dq
| |
| </math>
| |
| | |
| Suppose that <math>P(q | \vec y)</math> is given by a Gaussian:
| |
| | |
| :<math>
| |
| P(q | \vec y) = \frac{1}{\sqrt{2 \pi} \sigma_q}
| |
| \exp \left \lbrace - \frac{\left [q - \bar q (\vec y)\right ]^2}{2 \sigma_q} \right \rbrace
| |
| </math>
| |
| | |
| where <math>\bar q</math> is the expectation value and <math>\sigma_q</math>
| |
| is the standard deviation, then the conditional probability is related to the
| |
| continuum variable, ''q'', by the error function:
| |
| | |
| :<math>
| |
| R=P(2 | \vec{y})-P(1 | \vec{y}) = \mathrm{erf} \left [ \frac{q_0 - \bar q (\vec y)}{\sqrt 2 \sigma_q} \right ]
| |
| </math>
| |
| | |
| The figure shows conditional probability versus specific humidity for the example
| |
| retrieval discussed above.
| |
| | |
| ===As a robust estimator===
| |
| | |
| The location of ''q''<sub>0</sub> is found by setting the conditional probabilities
| |
| of the two classes to be equal:
| |
| | |
| : <math>
| |
| \int_{-\infty}^{q_0} P(q | \vec{y}) \, dq =
| |
| \int^\infty_{q_0} P(q | \vec{y}) \, dq
| |
| </math>
| |
| | |
| In other words, equal amounts of the "zeroeth order moment" lie on either side
| |
| of ''q''<sub>0</sub>. This type of formulation is characteristic of a [[robust estimator]].
| |
| | |
| ==References==
| |
| | |
| * {{Cite journal
| |
| | author = Peter Mills
| |
| | title = Isoline retrieval: An optimal method for validation of advected contours
| |
| | journal = Computers & Geosciences
| |
| | volume = 35
| |
| | number = 11
| |
| | pages = 2020–2031
| |
| | year = 2009
| |
| | doi = 10.1016/j.cageo.2008.12.015
| |
| | url = http://peteysoft.users.sourceforge.net/Mills2009.pdf
| |
| }}
| |
| | |
| * {{Cite journal
| |
| | author = Peter Mills
| |
| | title = Efficient statistical classification of satellite measurements
| |
| | journal = International Journal of Remote Sensing
| |
| | doi = 10.1080/01431161.2010.507795
| |
| | year = 2010
| |
| | url = http://peteysoft.users.sourceforge.net/TRES_A_507795.pdf
| |
| }}
| |
| | |
| ==External links==
| |
| * [http://isoret.sourceforge.net Software for isoline retrieval]
| |
| | |
| [[Category:Remote sensing]]
| |
| [[Category:Inverse problems]]
| |
It is a material that is used to build the front fork of the bike. If you keep winning competitions, you will be invited to bigger and bigger races. If you have any inquiries pertaining to where and how to use Womens mountain bike sizing., you can contact us at the web-page. If you go on a regular mountain biking, you will develop a healthier body and disposition. Are you in search of a comfort bike, a mountain bike, or a bike that is designed just for cruising and do you prefer a standard road bike or are you looking for an hybrid. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races.
It'll have you on the edge of your seat and holding your breath for Zink to land this record-breaking backflip. One way to increase exercise fun is to find a friend to run with you. The price you can expect to pay for stems is actually determined by the materials used along with its weight. Sometimes they are internal (drum and coaster brakes), and sometimes they are external (disc brakes). Browse a number of the on-line forums to determine what other riders like and do not like about their bicycles.
The steeper the angles, the more beneficial it would be for stability and high speed pedaling. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. Apply a generous amount of mountain bike lube to your chain as you move the pedals around backward. Mounting biking is a fun sport for professionals and beginners alike. There is a key point that you should always remember when going on any bicycle ride, especially one of great length.
With knowledge on the strengths and weakness of the MTB model, they are a great source of help. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. These four websites are important to serious bikers. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. You should be selective in choosing mountain bike tires since mountain bike is already made heavy and if you choose heavy tires you should consider yourself about how you will handle the bike and the terrain at the same time.
And there are some people who rightfully bring up the issue of the fuel costs and environmental impact of producing the electricty used to power the bike in the first place. Listed below are some of the common terms which you will come across. Something that sounds to me, to be nothing more than hard work. This speed is great for going up a hill that is steep, but it is not a good speed for flat ground or racing downhill. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way".