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. P

    A Metric of the "space" of 3d rotations

    I was recently reading that the space of 3d rotations should have the topology of a real projective space. For confirmation, see wiki, https://en.wikipedia.org/wiki/3D_rotation_group. It seems to me that when we assign coordinates to this space (I was thinking of using the Euler angles, but...
  2. ergospherical

    I Null energy condition constrains the metric

    Another GR question... in the thick of revision season. I would appreciate a sketch of how to approach the problem. You basically are given a metric, involving a positive function ##A(z)##, $$g = A(z)^2(-dt^2 + dx^2 + dy^2) + dz^2$$The game is to figure out somehow that the null-energy...
  3. J

    I How to get metric field using a path dependent parallel propagator

    If a vector ##V(x)## being transported down a path ##l##, The vector field is described with equation: $$\partial_\mu V(x)=\Gamma_\mu V(x)$$ The solution of the equation can be described with parallel propagator ##P(x, x_0)##(in mathematics it is also called product integration): $$V(x)=P(x...
  4. L

    P-adic metric calculate limit

    Hi, I'm not sure if I have calculated the task here correctly Task 4-4b looked like this I have now obtained the following with ##n=-v_p(x-y)## $$\displaystyle{\lim_{n \to \infty}} p^n= \infty$$ $$\sum\limits_{n=0}^{\infty} p^n=\frac{p}{p-1}$$ Is that correct?
  5. D

    I Orbital Period In General Relativity

    What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like T=2pi(GnM/a3). Where a is the semi-major axis, this is for a small body orbiting a larger one. I think I have an idea but I am not 100% sure. I am interested in an...
  6. L

    B Prove that metric tensor is covariant constant

    I'm reading "Problem Book In Relativity and Gravitation". In this book there is a problem 7.5 Show that metric tensor is covariant constant. To prove it, authors suggest to use formulae for covariant derivative: Aαβ;γ=Aαβ,γ−AσαΓβγσ−AσβΓαγσ after that they write this formulae for tensor g and...
  7. jv07cs

    I Do Metric Tensors Always Have Inverses?

    I am reading about musical isomorphisms and for the demonstration of the index raising operation from the sharp isomorphism, we have to multiply the equation by the inverse matrix of the metric. Can we assume that this inverse always exists? If so, how could I prove it?
  8. M

    I Minkowski metric and proper time interpretation

    Using an example of 1 space dimension and 1 time dimension, consider the metric ##d\tau^2 = a dt^2 - dx^2## near a heavy mass (##a>1##). From what I've read a clock ticks slower near a heavy mass, as viewed from an observer far away. A clock tick would be representative of ##d\tau## right...
  9. Euge

    POTW Hölder Continuous Maps from ##R## to a Metric Space

    Let ##\gamma > 1##. If ##(X,d)## is a metric space and ##f : \mathbb{R} \to X## satisfies ##d(f(x),f(y)) \le |x - y|^\gamma## for all ##x,y\in \mathbb{R}##, show that ##f## must be constant.
  10. J

    I A question about metric compatibility equation

    We all know that The gradient of a scalar-valued function ##f(x)## in ##IR^n## is a vector field ##V_\mu(x)=\partial_\mu f(x)##, Such a vector field is said to be conservative. Not all vector fields are conservative. A conservative vector field should meet certain constraints...
  11. S

    I Non-homogeneous and anisotropic metric and laws of physics...?

    In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
  12. E

    A Solving Geodesics with Metric $$ds^2$$

    I have the following question to solve:Use the metric: $$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at $$t=0$$. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...
  13. S

    I Are there non-smooth metrics for spacetime (without singularities)?

    Are there non-smooth metrics for spacetime (that don't involve singularities)? I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics: Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...
  14. O

    Coordinate transformation into a standard flat metric

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  15. S

    I Explore Spacetimes, Metrics & Symmetries in Relativity Theory

    I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970) In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
  16. D

    A Understanding killing vectors and transformations of metric

    Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures. Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations. Q1.Can someone explain how to go from...
  17. K

    B Exploring the Standardization of One Light Year Distance | PF Forum

    Dear PF Forum, It's been a while since I logged in here. And I really do appreciate all the answers that I've been getting here. Now, I wonder. Is there any standardization for 1 light year distance? Is it 10 trillion kilometers, or 299,792,458 * 60 * 60 * 24 * 365.256 = ...
  18. V

    A Metric of a Moving 3D Hypersurface along the 4th Dimension

    Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##. Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
  19. Kostik

    A Static Gravitational Field: Why Must ##g_{m0} = 0##?

    In Dirac's "General Theory of Relativity", he begins Chap 16, with "Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##." It's obvious that...
  20. Onyx

    B Solve General Geodesics in FLRW Metric w/ Conformal Coordinates

    Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
  21. Onyx

    B Calculate Unit Normal Vector for Metric Tensor

    How do I calculate the unit normal vector for any metric tensor?
  22. G

    I Can Spherical Symmetry Be Achieved Without Varying Line Element?

    "Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##" I wonder why: "the line element does not vary when##\theta## and##\phi## are...
  23. M

    B Metric Line Element Use: Do's & Don'ts for Accelerated Dummies?

    From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame: ##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)## ##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...
  24. Sciencemaster

    I Calculating Spacetime Around Multiple Objects

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  25. Sciencemaster

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  26. Onyx

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  27. ergospherical

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  28. Tanujhm

    A Is the FLRW Metric a Valid Approximation for Our Universe on Large Scales?

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  29. Ennio

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  30. JandeWandelaar

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  31. P

    A Curvature & Connection Without Metric

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  32. K

    A Can a conformal flat metric be curved?

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  33. A

    I Schwarzschild Metric & Particle Absorption

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  34. BiGyElLoWhAt

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  35. physicsuniverse02

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  36. RayDonaldPratt

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  37. U

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    Homework Statement:: Please see below. Relevant Equations:: Please see below. I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric: \begin{equation}...
  38. D

    Show that if d is a metric, then d'=sqrt(d) is a metric

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  39. Sciencemaster

    I Adapting Schwarzschild Metric for Nonzero Λ

    So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...
  40. chwala

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  41. George Keeling

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  42. S

    I Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions

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  43. cianfa72

    I Raising/Lowering Indices w/ Metric Tensor

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  44. T

    I Induced Metric Help: Troubleshooting Extrinsic Curvature (12)

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  45. ergospherical

    I Transform Coordinates for Torus Metric in Wald

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  46. H

    A Time Dilation in Reissner-Nordström Metric: Even or Odd?

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  47. yucheng

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  48. D

    Engineering Can you find a surface from a metric?

    if a metric like ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2 ## is given, we know it corresponds to a sphere in spherical coordinates . if you are given an arbitrary metric with two variables for example ##ds^2=\frac{du^2}{u}+dv^2## is ther guarenteed to be a surface embedded in ##R^3##...
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