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

    How to differentiate a term with respect to metric

    Homework Statement for example: ##\frac{\partial(F^{ab}F_{ab})}{ \partial g^{ab}} ## where F_{ab} is electromagnetic tensor. or ##\frac{\partial N_{a}}{\partial g^{ab}}## where ##N_{a}(x^{b}) ## is a vector field. Homework EquationsThe Attempt at a Solution i saw people write ##F^{ab}F_{ab}##...
  2. S

    Godel Metric Cosmological Constant

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  3. T

    Covariant and partial derivative of metric determinant

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  4. T

    Varying the action with respect to metric

    Homework Statement i want to find the variation of this action with respect to ## g^{\mu\nu}## , where ##N_\mu(x^\nu)## is unit time like four velocity and ##\phi## is scalar field. ##I_{total}=I_{BD}+I_{N}## ## I_{BD}=\frac{1}{16\pi}\int dx^4\sqrt{g}\left\{\phi...
  5. binbagsss

    Does Metric Signature Affect Torsion Definition?

    I'm looking at 2 sources. One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}## And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}## ##T## the torsion tensor and ##\Gamma^{c}_{ab}## the connection. Or is it more that different texts use different conventions? Thanks.
  6. U

    General Relativity - FRW Metric

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  7. binbagsss

    Deriving Schwarz Metric Weak Limit: Carroll's Lecture Notes 1997

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  8. S

    Type Godel Metric Line Element - Get Help Here!

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  9. U

    Einstein Equations of this metric

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  10. U

    Solving this space-time Metric

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  11. K

    Metric tensor with diagonal components equal to zero

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  12. L

    Proving Horizon Structure of Complicated Metric

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  13. C

    Sphere of Uniform Density: Exact Solution to Einstein's Field Eqns?

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  14. D

    Twin Paradox in Kerr Metric - Help Needed

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  15. T

    Observables vs. continuum and metric?

    Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...
  16. U

    What is the Geodesic Equation for FRW Metric's Time Component?

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  17. E

    Metric for uniform constant acceleration?

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  18. J

    Equivalent Metrics From Clopen Sets

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  19. binbagsss

    Tod & Hughston GR Intro: FRW Metric Derivation w/ R=6k or R=3k?

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  20. binbagsss

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  21. binbagsss

    Raising Indices with Minkowski Metric: Solving Weak Field Approx

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  22. Y

    Help with the variation of the Ricci tensor to the metric

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  23. C

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  24. binbagsss

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  25. K

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  26. N

    Wave equation given a cosmological inflationary metric

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  27. Muthumanimaran

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  28. W

    MHB Real Analysis Help: Metric Spaces

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  29. K

    Question about Metric Tensor: Can g_{rr} be Functions of Coordinate Variables?

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  30. M

    Questions about FLRW Metric: Finiteness, Radial Coord & More

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  31. E

    Metric matrix for binary star system?

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  32. F

    Angular velocity calculation from Schwarzschild metric

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  33. binbagsss

    FRW Metric: Parameter k & Space/Space-Time Relationships

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  34. D

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  35. T

    Solving Einstein's Field Equation: The Schwarschild Metric & Beyond

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  36. binbagsss

    Understanding Homogeneity & Isotropy in FRW Metric

    So in deriving the metric, the space-time can be foliated by homogenous and isotropic spacelike slices. And the metric must take the form: ##ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}##, where ## \gamma_{ij} ## is the metric of a spacelike slice at a constant t QUESTION: So I've read...
  37. binbagsss

    Deriving FRW Metric: Ricci Vector Algebra Explained

    I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf, deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0## I see that any christoffel symbol with a ##0## vanish and so so do any...
  38. binbagsss

    When can a metric be put in diagonal form?

    I'm looking at deriving the Schwarzschild metric in 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' and the comment under eq. 7.8, where he seeks a diagnoal form of the metric... - Is it always possible to put a metric in diagonal form or are certain symmetries required? - What...
  39. binbagsss

    General Relativity: Manifold/Sub-Manifold Metric Theorem Q-Schwarzschild

    I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are...
  40. U

    Proper distance, Area and Volume given a Metric

    Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
  41. ChrisVer

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    I was wondering if I can expand the FRW metric in d spatial dimensions, like: g_{\mu \nu}^{frw} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & - \frac{a^2(t)}{1-kr^2} & 0 & 0 \\ 0 & 0 & - a^2(t) r^2 & 0 \\ 0 & 0 & 0 & -a^2 (t) r^2 \sin^2 \theta \end{pmatrix} \rightarrow g_{MN} = \begin{pmatrix} g_{\mu...
  42. binbagsss

    Schwarzchild metric spherically symmetric space or s-t?

    This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...
  43. T

    The Schwarzschild Metric: Obtaining Equation M=Gm/c^2 & Newton Law at Infinity

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  44. Math Amateur

    MHB Metric Spaces - Fixed Point Theorem (Apostol, Theorem 4.48)

    I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48) The Fixed Point Theorem and its proof read as follows: In the above proof Apostol writes: " ... ... Using the triangle inequality we find for m \gt n, d(p_m, p_n) \le \sum_{k=n}^{m-1}...
  45. L

    Kerr metric and rotating stars

    I have recently come across the notion that Kerr metric describes the spacetime outside a rotating black hole but not outside a rotating (electrically neutral) star. Unlike Schwarzschild metric, which works both for non-rotating spherically symetric black hole without charge as well as any other...
  46. Math Amateur

    MHB Metric Spaces & Compactness - Apostol Theorem 4.28

    I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition). Theorem 4.28 reads as follows:In the proof of the above theorem, Apostol writes: " ... ... Let m = \text{ inf } f(X). Then m is adherent to f(X) ... ... " Can someone please explain to me...
  47. binbagsss

    Levi-Civita Connection & Riemannian Geometry for GR

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  48. J

    Spacetime Interval & Metric: Equivalent?

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  49. K

    How Do You Convert Gallons to Metric Units and Calculate Paint Thickness?

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  50. G

    Calculate metric tensor in terms of Mass

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