Metric Definition and 1000 Threads

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

    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...
  2. Stephanus

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

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  3. mertcan

    A How Is the Taylor Expansion Applied to Metric Tensors?

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  4. smodak

    I Stupid question on index lowering and raising using metric

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  5. 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...
  6. C

    Finding the geodesic equation from a given line element

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  7. arpon

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  8. e2m2a

    A Einstein & Light Deviation: Compute w/o Schwarzschild Metric

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  9. T

    I Friedman-Robertson-Walker metric

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  10. G

    I Field equations fully written out

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  11. G

    I GR vs SR: Is a Connection Necessary?

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  12. Rococo

    I Metric Conservation Law in 2D Spacetime

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  13. V

    I Observation of distances w.r.t. metric

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  14. BiGyElLoWhAt

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  15. D

    MHB Relationship between metric and inner product

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  16. C

    A Metric of n-sheeted AdS_3: Constructing BTZ

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  17. D

    I Newtonian limit of Schwarzschild metric

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  18. G

    A Meaning of ds^2 according to Carroll

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  19. A

    I Is a Riemannian Metric Invariant Under Any Coordinate Transformation?

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  20. S

    A Calculate the Weyl and Ricci scalars for a given metric

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  21. L

    Orthonormal basis of 1 forms for the rotating c metric

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  22. F

    Kepler's Law in Schwarzchild metric

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  23. MattRob

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  24. BiGyElLoWhAt

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  25. Kevin McHugh

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  26. S

    Minkowski metric in spherical polar coordinates

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  27. PeroK

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  28. W

    Visualizing the space and structure described by a metric

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  29. C

    Calculating Covariant Riemann Tensor with Diag Metric gab

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  30. W

    Find the coordinate transformation given the metric

    Homework Statement Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##. Homework Equations ##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy## ##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
  31. W

    Curvature at the origin of a space as described by a metric

    Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...
  32. W

    Metric for the construction of Mercator map

    Homework Statement The familiar Mercator map of the world is obtained by transforming spherical coordinates θ , ϕ to coordinates x , y given by ##x = \frac{W}{2π} φ, y = -\frac{W}{2π} log (tan (\frac{Θ}{2}))## Show that ##ds^2 = Ω^2(x,y) (dx^2 + dy^2)## and find ##Ω## Homework Equations...
  33. F

    Wormhole Metric: Solving Difficult Differential Equations

    I'm doing a thesis about wormhole and I would like to put a part in which I conjecture that a shuttle(precisely the Endurance from Interstellar) goes to Kepler 422-b using a Thorne-Miller wormhole. The problem is that I don't know hot to solve a difficult differential equations. Thank you very...
  34. DiracPool

    GR Metric Tensor Rank 2: Quadratic vs Shear Forces

    Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I...
  35. Rlam90

    Two-mass Schwarzschild metric instead of Kerr metric?

    Just a thought... Would there be any implicit differences between (A) a two-body metric where the two central masses are drawn ever further together, with angular momentum included, and (B) the Kerr metric? Angular momentum would still be part of the system, but it would be explained by a more...
  36. D

    Lorentz invariance of the Minkowski metric

    I understand that in order to preserve the inner product of two four vectors under a change of coordinates x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\, \nu}x^{\nu} the Minkowski metric must transform as \eta_{\mu^{'}\nu^{'}}=\Lambda^{\alpha}_{\,\...
  37. V

    A Why pseudo-Riemannian metric cannot define a topology?

    It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let...
  38. Boon

    Grasping the Properties of Minkowski Space

    I'm trying to get an intuitive feel for Minkowski space in the context of Special Relativity. I should mention that I have not studied (but hope to) the mathematics of topology, manifolds, curved spaced etc., but I'm loosely familiar with some of the basic concepts. I understand that spacetime...
  39. tomwilliam2

    Notation difficulties with metric and four vector

    I'm reading an introduction to relativity which uses different notation to the standard indices used in my college course. I came across: L(\nu)gL(\nu)g = 1 Where L is the Lorentz transformations four-vector and g is the metric. Without the indices, I'm a little lost. Is there some convention...
  40. evinda

    MHB Properties of Metric $\sigma(x,y)$

    Hello! (Wave) I want to show that if $\rho(x,y)$ is a metric on $X$, then $\sigma (x,y)= \min \{ 1, \rho(x,y) \}$ is a metric. I have thought the following: $\rho(x,y)$ is a metric on $X$, so: $\rho(x,y) \geq 0, \forall x,y \in X$ $\rho(x,y)=0$ iff $x=y$ $\rho(x,y)=\rho(y,x) \forall x,y...
  41. NihalRi

    How is time incorporated into the robertson walker metric?

    I watched a lecture that derived the robertson walker metric by creating a metric to describe a four dimensional sphere in three dimensions. Then from minkowski's equation-...
  42. V

    Understanding Vaidya Metric & Pure Radiation Stress-Energy

    I am following Vaidya metric and how it is related to pure radiation from Wikipedia. But when it reaches the line where stress-energy tensor is equated to product of two four-vectors, I cannot follow from where they are assumed to be null vectors, and why if the stress-energy tensor is given in...
  43. O

    MHB How does ${X}_{w}\subset {X^*}_{w}$ occur in modular metric space?

    Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$ and The two sets ${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$ and...
  44. grav-universe

    Producing a metric with only a single coordinate function

    Given the metric c^2 d\tau^2 = c^2 B(r) dt^2 - A(r) dr^2 - C(r) r^2 d\phi^2 and solving only for a static, spherically symmetric vacuum spacetime, I want to reduce the number of coordinate functions A, B, and C from three to only one using the EFE's. We can then make a coordinate choice for...
  45. O

    MHB Metric spaces and normed spaces

    What is the relation between metric spaces and normed spaces... What is the meaning of " metric spaces are seen as a nonlinear version of vector spaces endowed with a norm" ? Thank you for your attention...Best wishes...:)
  46. E

    Why do those two terms add here?

    When I was studying complex manifolds in a Freedman's SUGRA book, I ran across this. In a complex manifold, the metric is...
  47. O

    MHB How Do Different Metrics Affect Convergence and Divergence of Sequences?

    Let $X=R$ and ${d}_{1}\left(x,y\right)=\frac{1}{\eta}\left| x-y \right|$ $\eta\in \left(0,\infty\right)$ and ${d}_{2}\left(x,y\right)=\left| x-y \right|$..By using ${d}_{1}$ and ${d}_{2}$ please show that ${x}_{n}=\left(-1\right)^n$ is divergent and ${x}_{n}=\frac{1}{n}$ is convergent...
  48. joneall

    Units if conversion between covariant/contravariant tensors

    I am still at the stage of trying to assimilate contravariant and covariant tensors, so my question probably has a simpler answer than I realize. A covariant tensor is like a gradient, as its units increase when the coordinate units do. A contravariant tensor's components decrease when the...
  49. T

    What are the non-zero elements of the Riemann Tensor using the FLRW metric?

    Homework Statement I've been working on Exercise 14.3 in MTW. This starts with the FLRW metric (see attachment) and asks that you find the connection coefficients and then produce the non-zero elements of the Riemann Tensor. The answer given is that there are only 2 non-zero elements vis...
  50. DiracPool

    Metric in polar coordinate derivation

    At time 1:11:20, Lenny introduces the metric for ordinary flat space in the hyperbolic version of polar coordinates? Is that what he is doing here? d(tau)^2 = ρ^2 dω^2 - dρ^2. He then goes on to say that this metric is the hyperbolic version of the same formula for Cartesian space, i. e...
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