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. B

    Inverse of Metric in Topology: Schwarz Inequality

    In Topology: is the multiplicative inverse of a metric, a metric? How do we define the Schwarz inequality then? if ##d(x,z) ≤ d(x,y) + d(y,z)## the inverse ##1/d(x,z)## would give the opposite?
  2. C

    Solving Problems in French Railway Metric

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  3. C

    Find Christoffel symbols from metric

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  4. C

    Understanding Nordstrom Metric & Freely Falling Massive Bodies

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  5. S

    Munkres Topology - Chapter 7 - Complete Metric Spaces and Function Spaces

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  6. P

    Rotating Flat Spacetime in Minkowski Metric

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  7. S

    Killing vectors and Geodesic equations for the Schwarschild metric.

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  8. F

    Does integration require a metric?

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  9. W

    Find tetrad component from metric

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  10. J

    Uniform Convergence and the Uniform Metric

    Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...
  11. J

    Question concerning a topology induced by a particular metric.

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  12. I

    Second order expansion of metric in free-fall

    Hello, I have read that, in a freely-falling frame, the metric/ interval will be of the form: ds2 = -c2dt2(1 + R0i0jxixj) - 2cdtdxi(\frac{2}{3} R0jikxjxk) + (dxidxj(δij - \frac{1}{3} Rikjlxkxl) to second order. Does anyone know where I could find a derivation of this result?
  13. U

    MHB Proving $\text{diam}(A)=\text{diam}(\overline A)$ for Metric Spaces

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  14. popbatman

    Left and right invariant metric on SU(2)

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  15. A

    Deriving the Metric from the Energy-Momentum Tensor

    Say we were given an expression for the energy-momentum tensor (also assuming a perfect fluid), without getting into an expression with multiple derivatives of the metric, are there any cases where it would be possible to deduce the form of the metric?
  16. D

    How do I calculate Lie derivation of a metric?

    Homework Statement I've searched everywhere, and I cannot find an example of calculation of Lie derivation of a metric. If I have some vector field \alpha, and a metric g, a lie derivative is (by definition, if I understood it): \mathcal{L}_\alpha g=\nabla_\mu \alpha_\nu+\nabla_\nu...
  17. stevendaryl

    Why the square-root of the metric

    In a paper about field theory in curved spacetime, an author says that the Lagrangian density for a free scalar particle is L = \sqrt{-g} ((\nabla_\mu \Phi)(\nabla^\mu \Phi) - m^2 \Phi^2) Is there a simple explanation for why this is scaled by \sqrt{-g}?
  18. I

    Variation of the metric tensor determinant

    Homework Statement This is not homework but more like self-study - thought I'd post it here anyway. I'm taking the variation of the determinant of the metric tensor: \delta(det[g\mu\nu]). Homework Equations The answer is \delta(det[g\mu\nu]) =det[g\mu\nu] g\mu\nu...
  19. P

    A discrete subset of a metric space is open and closed

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  20. G

    Alcubierre metric and gravitational waves

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  21. P

    Convergent sequence in compact metric space

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  22. P

    Types of points in metric spaces

    Hi, I'm reading Baby Rudin and have a quick question regarding topology. Given a nonempty subset E of a metric space X, is it true that the only points in E are either isolated points or limit points? (b/c all interior points are by definition limit points, but not all limit points are...
  23. R

    Metric Tensor Division: Is It Proper?

    If you know that {{x}^{a}}{{g}_{ab}}={{x}_{b}} is it proper to say that you also know {{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}
  24. C

    Metric space proof open and closed set

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  25. D

    Every metric space is Hausdorff

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  26. Barnak

    Symmetrizing 3xMetric Tensor: H^{\mu \nu \lambda \kappa \rho \sigma}

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  27. S

    Ricci scalar and curveture of FRW metric

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  28. R

    Understanding the metric tensor

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  29. A

    How to Find Killing Vectors for a Given Metric

    In my general relativity course we recently covered the definition of a killing vector and their importance. However, I am not completely comfortable calculating the killing vectors for a given metric (in a particular case, the 2-sphere), and would like to know if anyone knows of a good...
  30. J

    Minkowski Metric: Timelike vs Spacelike

    hello Whic one of these to metric are Minkowski metric ds^2 =-(cdt)^2+(dX)^2 ds^2 =(cdt)^2-(dX)^2 and what about timelike (ds^2<0) and spacelike (ds^2>0) for each metric? With my appreciation to those who answer
  31. A

    Family of continuous functions defined on complete metric spaces

    Homework Statement Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y. Homework Equations Definition of...
  32. N

    Questions about the Schwarzschild metric

    Hello everybody! I have some questions concerning the structure of the Schwarzschild metric, which is given by $$ ds^2=-(1- \frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+ r^2(d\theta^2+ \sin^2(\theta) d\phi^2) $$ where we set $c=1$. I would like to know the following: \\ \\ 1. Why is it...
  33. Mentz114

    Vaidya metric and Wiki article

    In the Wiki article http://en.wikipedia.org/wiki/Vaidya_metric it states that but it looks to me as if the 'particles' are traveling at the speed of light ( null propagation vector field ) and so must have zero rest mass. Is this a typo or have I misunderstood something ?
  34. mnb96

    Geometric interpretation of metric tensor

    Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.
  35. G

    Double contraction of curvature tensor -> Ricci scalar times metric

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  36. T

    Israel's Formalism: The Metric Junction Method

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  37. F

    How Does a Diagonal Metric Affect the Symmetry and Tensor Equations in Space?

    Hey guys! I am considering a space with a diagonal metric, which is maximally symmetric. It can be proven that in that case of a diagonal metric the following equations for the Christoffel symbols hold: \Gamma^{\gamma}_{\alpha \beta} = 0 \Gamma^{\beta}_{\alpha \alpha} =...
  38. T

    S. Martin's Supersymmetry primer Metric

    Hi all, S. Martin's Supersymmetry primer (http://arxiv.org/abs/hep-ph/9709356) is a wonderful source from which to learn SUSY. But, what really causes me (and others around me) huge consternation is Martin's use of mostly plus metric, when particle physicists use the mostly minus metric...
  39. S

    Apply Gauss's theorem when the metric is unknown

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  40. Orion1

    Kerr-Newman Metric Equation Solution | Verified by Experts

    Kerr–Newman metric: c^{2} d\tau^{2} = - \left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + (c \; dt - \alpha \sin^2 \theta \; d\phi)^2 \frac{\Delta}{\rho^2} - ((r^2 + \alpha^2) d\phi - \alpha c \; dt)^2 \frac{\sin^2 \theta}{\rho^2} I used the Kerr–Newman metric equation form listed on...
  41. N

    How can I obtain the inverse of the Finsler metric in a given geometry?

    Given a Finsler geometry (M,L,F) and $$g_{ab}^L=\frac{1}{2} \frac{\partial^2 L}{\partial y^a \partial y^b}$$ $$g_{ab}^F=\frac{1}{2} \frac{\partial^2 F^2}{\partial y^a \partial y^b}$$ $$F(x,y)=|L(x,y)|^{1/r}$$ I manage to get the following form $$g_{ab}^F=\frac{2|L|^{2/r}}{rL}(...
  42. A

    Unraveling The Minkowski Metric: Intuitive Explanation

    yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively. The quantity: dx2+dy2+dy2-c2dt2 is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I...
  43. M

    How does changing the metric on a manifold affect the shape of the manifold?

    Hi all, I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase. A metric tensor's purpose is to provide a coordinate invariant...
  44. S

    Proving Metric Equivalence for Subset Y in Metric Space X

    Y ⊂ X where X is a metric space with the function d. Prove that (Y,d) is a metric space with the same function d. The metric function d: X x X -> R. I know that the function for Y is: d* : Y x Y -> R How do I show that d is the same as d*.
  45. A

    Is there any relation which holds between energy and metric space ?

    Like any mathematical relativity between them as per General Relativity?
  46. L

    Analysis - Metric space proof (prove max exists)

    http://imageshack.us/a/img12/8381/37753570.jpg I am having trouble with this question, like I do with most analysis questions haha. It seems like I must show that the maximum exists. So E is compact -> E is closed To me having E closed seems like it is clear that a maximum distance...
  47. F

    Change of variable in integral using metric

    What is the formula to evaluate a multi-integral by a change of coordinates using the squareroot of the metric instead of the determinate of the Jacobian? Thanks.
  48. M

    Deriving L-T Metric: Understanding Schwarzschild & Einstein's GR

    Hi, I have posted this question in the PF relativity forum because I am trying to understand the derivation of the Lense-Thirring (L-T) metric. Various sources suggest that L-T produced a solution of Einstein’s field equations of general relativity in 1918, just a few years after Schwarzschild...
  49. G

    Continuity in Metric Spaces: Proving the Convergence of a Sequence

    Homework Statement Show that if (x_{n}) is a sequence in a metric space (E,d) which converges to some x\inE, then (f(x_{n})) is a convergent sequence in the reals (for its usual metric). Homework Equations Since (x_{n}) converges to x, for all ε>0, there exists N such that for all...
  50. E

    Visualization of metric tensor

    Barbour writes: the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components, corresponding to the four values the indices u and v can each take: 0 (for the time direction) and 1; 2; 3 for the three spatial directions. Of the ten components, four merely...
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