What is Metric: Definition and 1000 Discussions

METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model developed by the University of Idaho, that uses Landsat satellite data to compute and map evapotranspiration (ET). METRIC calculates ET as a residual of the surface energy balance, where ET is estimated by keeping account of total net short wave and long wave radiation at the vegetation or soil surface, the amount of heat conducted into soil, and the amount of heat convected into the air above the surface. The difference in these three terms represents the amount of energy absorbed during the conversion of liquid water to vapor, which is ET. METRIC expresses near-surface temperature gradients used in heat convection as indexed functions of radiometric surface temperature, thereby eliminating the need for absolutely accurate surface temperature and the need for air-temperature measurements.

The surface energy balance is internally calibrated using ground-based reference ET that is based on local weather or gridded weather data sets to reduce computational biases inherent to remote sensing-based energy balance. Slope and aspect functions and temperature lapsing are used for application to mountainous terrain. METRIC algorithms are designed for relatively routine application by trained engineers and other technical professionals who possess a familiarity with energy balance and basic radiation physics. The primary inputs for the model are short-wave and long-wave thermal images from a satellite e.g., Landsat and MODIS, a digital elevation model, and ground-based weather data measured within or near the area of interest. ET “maps” i.e., images via METRIC provide the means to quantify ET on a field-by-field basis in terms of both the rate and spatial distribution. The use of surface energy balance can detect reduced ET caused by water shortage.
In the decade since Idaho introduced METRIC, it has been adopted for use in Montana, California, New Mexico, Utah, Wyoming, Texas, Nebraska, Colorado, Nevada, and Oregon. The mapping method has enabled these states to negotiate Native American water rights; assess agriculture to urban water transfers; manage aquifer depletion, monitor water right compliance; and protect endangered species.

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  1. S

    Metric function composed with concave function

    Hi, I have been reading about metric spaces and came across an elementary property that I am having difficulty proving. A quick search on these forums and google has also failed. Given a metric space with distance function d, and an increasing, concave function f:\mathbb{R} \rightarrow...
  2. C

    Interesting discovery about a Metric M6 bolt

    Today while working I found that an M6 x 1.0 Thread bold will easily screw into a hole tapped 1/4"-28. At first when I found this, I thought one of my boxes of bolts was labeled wrong, but I tried another set and it work. Has anyone else come across this similarity before? -CR
  3. D

    Continuity of functions between metric spaces question.

    Was trying to learn differential geometry, had to take time off of it to develop some knowledge of topology, namely compactness and Hausdorff's condition. I'm using Sutherland's book on topology and came across something I didn't understand concerning metric spaces, Sutherland speaks of the so...
  4. T

    Proving Continuity in Metric Spaces | Sequential Characterisation of Continuity

    The sequential characterisation of continuity says that f is continuous at x_0 if and only if for every sequence (x_n)_{n\in\mathbb{N}} in X, f(x_n)\to f(x_0) as x_n \to x_0. f is continuous on X if this is the case for all x_0 \in X. I think I've done all the parts of this question up to the...
  5. O

    How can you drive the metric of kaluza klein in 5 D?

    in many research of kaluza klein theory to unified electromagnetic & gravity fields these research begin with 5D metric how can i drive this metric? please any research drive this metric show me. note russian research's by english is very good if you know it show me . please it is very important
  6. T

    Metric Spaces Homework: Showing Cauchy Sequences

    Homework Statement Homework Equations The Attempt at a Solution I've done the first 3 parts. I've come to the bit on Cauchy sequences at the end. How do I show x_n = n is/isn't a Cauchy sequence in the 2 metrics? (x_n) is a Cauchy sequence in a metric space (X,d) if for any...
  7. A

    Have the definitions of time and metric meter changed over years?

    i remember the French came up with metric meter by measuring the distance between equator and north pole and then divided by an integer to come up meter. it that still the defintion for meter? also, it seems now that a second is defined by the integer number of oscillation of atomic clock...
  8. M

    Metric Space: A Proof of diam(A∪B) ≤ diam(A) + diam(B) | Homework Help

    Homework Statement Consider a metric space (X,d) with subsets A and B of X, where A and B have non-zero intersection. Show that diam(A\bigcupB) \leq diam(A) + diam(B) Homework Equations The Attempt at a Solution A hint would be very much appreciated. :smile:Let x\inA, y\inB, z\inA\bigcupB...
  9. P

    A problem about equivalent metric

    Given X=R∞ and its element be squences let d1(x,y)=sup|xi-yi| let d∞(x,y)=Ʃ|xi-yi| then there exists some some x(k) which convergences to x by d1 but not by d∞ ,for example let x be the constant squence 0, i.e xn=0 ,and let x(k)n=(1/k2)/(1+1/k2)n then d1(x(k),x)=1/k2 and...
  10. J

    What Form Must H Take for a Vacuum Plane Gravitational Wave Metric?

    Hello, I'm having problems solving this problem I got in class. I want to learn the concept and how to approach the solution. Here it is: Consider the metric ds=dx^2+dy^2-dudv+2H(x,y,u)du^2 What form must the function H have for this metric to represent a plane gravitational wave...
  11. P

    A problem about equivalent metric

    Given X=R∞ and its element be squences let d1(x,y)=sup|xi-yi| let d∞(x,y)=Ʃ|xi-yi| then there exists some some x(k) which convergences to x by d1 but not by d∞ ,for example let x be the constant squence 0, i.e xn=0 ,and let x(k)n=(1/k2)/(1+1/k2)n then d1(x(k),x)=1/k2 and...
  12. P

    Deriving the Metric Connection Equation

    Homework Statement Show that \frac{\partial\overline{\mathcal{L}}_G}{{\partial}\mathfrak{g}^{ab}_{,c}}=\Gamma^c_{ab}-\frac{1}{2}\delta^c_a\Gamma^d_{bd}-\frac{1}{2}\delta^c_b\Gamma^d_{ad}.Homework Equations...
  13. A

    Solving Differential Coefficients & Metric Tensor in 2D Plane

    Consider a flat 2-dimensional plane. This can be described by standard Cartesian coordinates (x,y). We establish a oblique set of axes labelled p and q. p coincides with x but q is at an angle θ to the x-axis. At any point A has unambiguous co-ordinates (x,y) in the Cartesian system. In...
  14. H

    Help with continuous functions in metric spaces

    hi guys, I have a question I would like assistance with: let (v,||.||) be a norm space over ℝ, and let f:v→ℝ be a linear functional. if f is continuous on 0 (by the metric induced by the norm), prove that there is k>0 such that for each u in v, |f(u)| ≤ k*||u||. thanks :)
  15. J

    Let f be a continuous real function on a metric space X. Let

    Homework Statement Let f be a continuous real function on a metric space X. Let Z(f) be the set of all p in X at which f(p) = 0. Prove that Z(f) is closed. Homework Equations Definition of continuity on a metric space. The Attempt at a Solution Proof. We'll show that X/Z(f) = {p...
  16. T

    Convergence of a sequence in a metric space

    Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...
  17. T

    Open/closed subsets of metric space

    Homework Statement The Attempt at a Solution I've got through this question up to the last bit. I've got B(0,1) = \{0\} and B(0,2) = \{y\in\mathbb{R} : -1<y<1 \} (i.e. the open interval (-1,1).) How do I show that every subset of \mathbb{R} is open (A \subseteq X is open if it...
  18. G

    Exploring the Confusing Concept of Topology and Metric Space

    Hi! I'm a beginner for a subject "topology". While studying it, I found a confusing concept. It makes me crazy.. I try to explain about it to you. For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function. I've also learned that for a set...
  19. I

    Diagonal Metric and General Relativity: A Fundamental Question

    I have an apparently simple question, which is foundamental for a new approach to General Relativity. Is any diagonal metric with constant determinant a solution of Eintein Equations in vacuum? Does someone have the answer?
  20. T

    Metric of 2 Bodies: Superposition & Resulting Tensor

    Hello there. I would like to find the metric tensor produced by the existence of two massive bodies. Does the principle of superposition work for metrics as well? The first idea I got was to add the two metrics for each separate body in order to obtain the resulting one. Is this approach valid...
  21. P

    Compact Sets of Metric Spaces Which Are Also Open

    Are there any down to Earth examples besides the empty set? Edit: No discrete metric shenanigans either.
  22. A

    Is {X, max(d,r)} or (X, min(d,r)) a Metric Space?

    If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?
  23. P

    Derivative of contravariant metric tensor with respect to covariant metric tensor

    Homework Statement Show that \frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d}) Homework Equations The Attempt at a Solution It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming...
  24. M

    Solutions to the space-time metric

    Can someone direct me to the solution to the space-time metric, ds^2 = -dt^2 + dx^2 + dy^2 + dt^2? Thanks.
  25. D

    Compactness of (0,1) when that is the whole metric space

    Hello. In my analysis book, it says that "Any closed bounded subset of E^n is compact" where E is an arbitrary metric space. I looked over the proof and it used that fact that E^n was complete, but it does not say that in the original condition so I was wondering if the book made a mistake in...
  26. Z

    Defining an Empty Set Metric Space: Understanding Properties of d

    Can we define a metric space (\emptyset, d)? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure. Thank you!
  27. jfy4

    Can Structure Constants Define a Metric in a 10D Lie Algebra?

    Hi, Let's say I have a 10 dimensional Lie algebra over some field of functions, something along the lines of at least twice differentiable with twice differentiable inverses. The structure constants have inputs from this field. Is it possible to build a metric from these structure...
  28. A

    On the Reissner-Nordstrom Metric

    The Reissner Nordstrom metric considers charge apart from mass in its composition. Both charge and mass appear in the temporal as well as the spatial components of the metric. By considering a large amount of charge against a small amount of mass we can have an estimate the individual...
  29. B

    Metric Space and Lindelof Theorem

    Homework Statement Assume some metric space (K,d) obeys Lindelof, take (X,d) a metric subspace of (K,d) and show it too must obey Lindelof. The Attempt at a Solution I'm assuming since I know that (K,d) obeys Lindelof then there is some open cover that has a countable subcover say {Ji | i is a...
  30. V

    Metric component confusion

    I am reading a paper and the authors read the tt component of the metric from the line element ds^2=f(r)[g(r) dt^2+h(r) dr^2] as g_{tt}=g(r) instead of (what I expect to be) g_{tt}=f(r) g(r) Could somebody please explain to me why? Thank you.
  31. J

    Regard Q, the set of all rational numbers, as a metric space,

    Homework Statement Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = |p − q|. Let E be the set of all p ∈ Q such that 2 < p2 < 3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q? Homework Equations Definition of interior...
  32. T

    Proving d_N is a Metric with Discrete Metric d_X

    Homework Statement [PLAIN]http://img833.imageshack.us/img833/6932/metric2.jpg The Attempt at a Solution I've shown d_{X\times Y} is a metric by using the fact that d_X and d_Y are metrics. What is a simpler description of d_N with d_X the discrete metric? Is it just: d_N(x,y) =...
  33. C

    Simple proof of continuity of a metric space

    Homework Statement Let X and Y be metric spaces, f a function from X to Y: a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X. b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous...
  34. L

    Confusion over Einstein summation convention and metric tensors.

    My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...
  35. I

    Metric spaces - Clopen property

    Why is it that a metric space (X,d) always has two clopen subsets; namely {0}, and X itself? Rudin calls it trivial, and so do about 15 other resources I've perused. What confuses me is that if we define some metric space to be the circle in ℝ2: x2+y2 ≤ r2, then points on the boundary of...
  36. G

    Showing Openness of U: Let X be a Metric Space & p in X with r>0

    " Let X be a metric space and p be a point in X and be a positive real number. Use the definition of openness to show that the set U(subset of X) given by: U = {x∈X|d(x,p)>r} is open. " I have tried: U is open if every point of U be an interior point of U. x is an interior point of U if there...
  37. Q

    How does GR handle metric transition for a spherical mass shell?

    This is really a continuation from another thread but will start here from scratch. Consider the case of a static thin spherical mass shell - outer radius rb, inner radius ra, and (rb-ra)/ra<< 1, and with gravitational radius rs<< r(shell). According to majority opinion at least, in GR the...
  38. S

    How Can I Better Understand Metric Space Problems?

    Hey All, I have been working on some Metric Space problems for roughly 20hrs now and I cannot seem to grasp some of these concepts so I was hoping someone here could clear a few things up for me. My first problem is detailed below... I have the following metric... d(x,y) = d(x,y)/(1 +...
  39. A

    Schwarzschild Metric - Rindler coordinates

    Hello, well I just read a paper by Atish Dabholkar and Ashoke Sen, titled "Quantum Black Holes", pp. 4-5 as shown below and I tried to find d\xi^{2}\frac{2GM}{\xi}=d\rho^{2} like this which is different from the eq. in the paper. So, could somebody please help me to find my...
  40. F

    Help with a metric tensor derivative

    Hello, Can anyone give me the answer of the following derivative? \frac{\partial{g}}{\partial{g^{\mu \nu}}} Thank you in advance !
  41. Matterwave

    How can a metric connection have torsion?

    Hi, I'm reading this wikipedia article on the metric connection, and it says that the Levi-Civita connection is a metric connection without torsion. If the metric connection is defined so that the covariant derivative of the metric is 0, how can there be torsion? Doesn't this condition force the...
  42. C

    What is the Limit of Max in a Metric Space?

    Homework Statement Prove that \rho_{0}(x,y)=max_{1 \leq k \leq n}|x_{k} - y_{k}|=lim_{p\rightarrow\infty}(\sum^{n}_{k=1}|x_{k}-y_{k}|^{p})^{\frac{1}{p}} Homework Equations The Attempt at a Solution My approach was to define a_{m}=max_{1 \leq k \leq n}|x_{k} - y_{k}| and...
  43. C

    Can somebody give me both, an intuitive and a formal definition of a metric?

    I'm having my first differential geometry course and I can't get the concept.
  44. M

    Show that a metric space is complete

    Homework Statement Given (R+, d), R-Real # d= | ln(x/y) | Show that this metric space is complete Homework Equations The Attempt at a Solution Firstly, I know that to show it is complete I need to have that all Cauchy sequences in that space converge... So I'm not 100%...
  45. J

    Variation of the auxiliary worldsheet metric

    Can somebody clarify how the formula for variation of the auxilliary worldsheet metric is obtained due to reparametrization of the worldsheet in string theory??
  46. D

    Angle on abstract metric space, has sense?

    Hello, I was wondering if if has any sense of talking about angles on an arbitrary http://en.wikipedia.org/wiki/Metric_space" (where only a distance function with some properties is defined) At first sight it seems to not has any sense, only some metric spaces has angles, namely does that...
  47. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...
  48. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...
  49. M

    Proving Convergence of Real Number Sequences with Metric Equations

    Homework Statement Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0 Homework Equations The Attempt at a Solution I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <...
  50. Fredrik

    Generalizations (from metric to topological spaces)

    This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...
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