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. T

    Metric outside a weakly gravitating body (MTW Ex 19.1)

    Homework Statement This is Exercise 19.1 in MTW - See attachment Homework Equations See attachment The Attempt at a Solution [/B] I've worked through 19.3a, 19.3b and 19.3c ( see post by zn5252 back in March 2013 replied to by PeterDonis) and proved them for my my own satisfaction and I've...
  2. J

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  3. Spinnor

    I Maximize dτ w/ Schwarzschild Metric: Two Masses, m & M

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  4. T

    I Path of Light in Curved Spacetime with Metric g

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  5. C

    I Where the scale factor a(t) appears in the metric

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  6. binbagsss

    I Raising/Lowering Indices w/ Metric & Tensors: Does Order Matter?

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  7. LarryS

    I Spacetime Metric: Which signature is better?

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  8. F

    A Is the Metric Tensor Invariant under Lorenz Transformations in M4?

    I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...
  9. I

    Complete metric space proof

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  10. I

    Convergence of sequence in metric space proof

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  11. L

    A Interpretation of Derivative of Metric = 0 in GR - Learning from Wald

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  12. F

    A Metric on ℝ^2 Invariant under Matrix Transformations

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  13. K

    I Deriving the FLRW Metric: 4D Euclidean Space Needed?

    Why is it needed to consider a 4D Euclidean space to introduce the FLRW metric? Is it because with a fourth parameter, we can set the radius of the 4D sphere formed with the four parametres as constant?
  14. V

    B Solving Field Equations & Schwarszchild Metric

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  15. arpon

    I Two metric tensors describing same geometry

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  16. K

    I Metric Components: Inner Product of Vector U & V

    The component of a one-form W can be represented using the metric as Wβ = gαβUα, where Uα is the component of a vector U, and one can always multiply Wβ by some Vβ to get the inner product between the two vectors U and V. My question is: since the inner product is defined with the two...
  17. Mihai_B

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  18. K

    I Writing Components of a Metric Tensor

    I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this: g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu}) where g1 and g2 are functions of one variable alone and D is the...
  19. R

    A Lack of uniqueness of the metric in GR

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  20. Jonathan Scott

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  21. F

    I Why does time have to be a complex (Minkowski metric)?

    I am studying special relativity, and I found that you have to work with a four dimensional space, where time is a complex variable. If you do so, you end up with the Minkowski metric, were the time component is negative and space components are positive (or vice versa). My questions are, why do...
  22. J

    How do I determine whether this metric is flat or not?

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  23. P

    A Connection, Metric and Torsion

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  24. J

    I On the invariant speed of light being the upper speed limit

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  25. K

    I Is Symmetry on μ and α Valid for the Derivative of the Metric Tensor?

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  26. D

    Four-Velocity and Schwartzchild Metric

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  27. tommyxu3

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  28. P

    MHB Square metric not satisfying the SAS postulate

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  29. S

    Metric tensor and gradient in spherical polar coordinates

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  30. S

    Scale factor of special conformal transformation

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  31. A

    I Quasars: What Metric Should You Use?

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  32. Stella.Physics

    I Christoffel symbols of Schwarzschild metric with Lagrangian

    So the Schwarzschild metric is given by ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2 and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0## with L = dτ/dσ. So for each α=0,1,2,3 we have ##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime also...
  33. A

    I Geodesic Equation: Lagrange Approximation Solution for Schwarzschild Metric

    Hello so if we have geodesic equation lagrange approximation solution: d/ds(mgμνdxν/ds)=m∂gμν∂xλdxμ/ds dxν/ds. So if we have schwarzschild metric (wich could be used to describe example sun) which is:ds2=(1-rs/r)dt2-(1-rs/r)-1dr2-r2[/SUP]-sin22. But that means that ∂gμν/∂xλ=0. So that means that...
  34. B

    Metric Spaces: Interior of a Set

    Homework Statement Let ##(X,d)## be some metric space, and let ##A## be some subset of the metric space. The interior of the set ##A##, denoted as ##int A##, is defined to be ##\bigcup_{\alpha \in I} G_\alpha##, where ##G_\alpha \subseteq A## is open in ##X## for all ##\alpha \in I##. The...
  35. A

    I Ricci tensor for Schwarzschild metric

    Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...
  36. P

    A Metric for Circular Orbit of Two Bodies

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  37. M

    B Schwarzschild Metric Derivation?

    Hi, I was wondering if anybody could help me understand the derivation of the Schwarzschild metric developed by the author of mathpages website. Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary that also points to the mathpages article and explains my...
  38. N

    I Validating a ds^2 Metric in General Relativity

    Hey all, in every theory that involves GR you see they give their space-time metric, but very few show any other math related to it, how does one know if a metric is valid?
  39. directbydirecct

    A Deriving Amplification of Images in Schwartzschild Metric

    In the paper Strong field limit of black hole gravitational lensing, the amplification of images in the Schwartzschild metric was given by $$ \frac{1}{\beta}\sqrt{\frac{2\,D_{LS}}{D_{OL}D_{OS}}} $$However the authors did not derive this expression or explain its origin. Does anyone know how to...
  40. P

    A Linearized metric for GW emitting orbiting bodies

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  41. redtree

    A Relationship between metric tensor and position vector

    Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...
  42. F

    A A question about coordinate distance & geometrical distance

    As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...
  43. A

    B Schwarzschild Metric: Non-Rotating Black Holes & Examples

    Hello I have been reading about Schwarzschild metric and scources what I read said that Schwarzschild metric is used to describe a non-rotating black holes. And what I can not understand is what can you calculate with it? It would be good if you give some examples where you can use it.
  44. L

    A Question about Holonomy of metric connecton

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  45. S

    I Godel Solution Metric: Shapes & Descriptions

    Here is the Godel solution: ds2 = -dt2/(2ω2) - (exdzdt)/ω2 - (e2xdz2)/(4ω2) + dx2/(2ω2) + dy2/(2ω2) Here is the metric tensor for it: g00 = -1/(2ω2) g03 & g30 = -ex/(2ω2) g11 & g22 = 1/(2ω2) g33 = -e2x/(4ω2) Every other element is 0. Now to my question: What shape is this metric? To clarify...
  46. S

    I Morris-Thorne Wormhole Metric Terms

    Here is the Morris-Thorne Wormhole line element: ds2 = - c2dt2 + dl2 + (b2 + l2)(dθ2 + sin2(θ)d∅2) Now my main question here (even though I've asked this before, but never quite understood) is: What exactly is l? I know that b is the radius of the throat of the wormhole. I know that the rest...
  47. Stephanus

    B Calculating Time to Reach 100,000 Trillion Meters in Hubble Flow | PF Forum

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  48. mertcan

    A Taylor expansion metric tensor

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  49. smodak

    I Stupid question on index lowering and raising using metric

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  50. A

    A Metric with Harmonic Coefficient and General Relativity

    Goodmorning everyone, is there any implies to use in general relativity a metric whose coefficients are harmonic functions? For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function? In (1+1)-dimensions is well-know that the Einstein...
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