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. Math Amateur

    MHB Understand Example 3.10 (b) Karl R. Stromberg, Chapter 3: Limits & Continuity

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows: My question is as...
  2. J

    I Does G effectively change in an expanding metric?

    Assume that we have a flat FRW metric expressed in conformal time ##\eta## so that the line element is $$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)\tag{1}$$ where ##a=1## at the present time ##\eta=0## and the speed of light ##c=1##. This metric has the following non-zero Christoffel symbols...
  3. K

    I Metric Nomenclature: Lorentz & Minkowski

    Can I say that the Lorentz metric is the specific form ##-c^2dt^2 + dx^2 + dy^2 + dz^2## whereas the Minkowski metric is the metric of Minkowski space which can take the Lorentz form I just gave, but can also, e.g., be written in spherical coordinates?
  4. T

    Did I Get These Metric Tensors Right?

    I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.For the first one I get all my directional derivatives (0, 0): (0)i + (0)j (0, 1): (0)i + 2j (1, 0): 2i + (0)j (1, 1): 2i + 2j Then I square them (FOIL): (0, 0): (0)i...
  5. D

    I Trying to construct a particular manifold locally using a metric

    I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.
  6. Math Amateur

    MHB Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows: In the above...
  7. snoopies622

    I The vanishing of the covariant derivative of the metric tensor

    I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy. In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...
  8. D

    I What is the Purpose of Calculating the Christoffel Symbols in Curved Spacetime?

    Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...
  9. Math Amateur

    MHB Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows: In the above...
  10. K

    I Radius in Schwarzschild Metric: Definition Explained

    Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...
  11. DuckAmuck

    I Invariant Mass in Gravitational Fields: Special Relativity

    In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1). m^2 = E^2 - \vec{p}^2 This can also be written with the Minkowski metric as: m^2 = \eta_{\mu\nu} p^\mu p^\nu More...
  12. Math Amateur

    MHB Metric Spaces .... the Uniform Metric .... Garling, Proposition 11.1.11 ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with an aspect of the proof of Theorem 11.1.11...
  13. Math Amateur

    I Metric Spaces .... the Uniform Metric .... Garling, Proposition 11.1.11

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with an aspect of the proof of Theorem 11.1.11 ...
  14. D

    I Help Understanding Metric Tensor

    I am trying to get an intuition of what a metric is. I understand the metric tensor has many functions and is fundamental to Relativity. I can understand the meaning of the flat space Minkowski metric ημν, but gμν isn't clear to me. The Minkowski metric has a trace -1,1,1,1 with the rest being...
  15. olgerm

    I Invariant properties of metric tensor

    Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
  16. Zuhaa Naz

    Find the Tetrad for Kerr Metric: Step-by-Step Guide

    how to find tetrad of this metric the tetrad given is this one I m a newly born in General Relativity please help me out how this tetrad is derived
  17. A

    I How Does the Metric of a 4-D Spacetime Define Its Symmetry and Expansion?

    Our 4-D space is ##x^1,x^2,x^3 ,t##. Our sub-manifold is defined by ##(x^1,x^2,x^3)## Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold The metric becomes...
  18. A

    I Perturbation to Flat Space Metric: Geodesic Equation

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  19. V

    B Solving an Integral on a Spherical Surface - Tips

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  20. K

    I Schwarzschild metric not dependent on time

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  21. Z

    A Decoupling of SVT Metric Perturbations

    Hello everyone, I have been studying perturbation theory in the context of FRW cosmologies, and so far have had a really hard time understanding why the SVT (Scalar, Vector, and Tensor) perturbations associated with the metric tensor "decouple" at first order in perturbation theory. All...
  22. A

    Liouville operator in Robertson Walker metric

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  23. A

    A Dark matter and spacetime metric

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  24. E

    B Solving the Kerr metric in the program Maxima

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  25. M

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  26. Arman777

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  27. DaTario

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  28. jk22

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  29. E

    Space With Schwarzschild Metric

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  30. George Keeling

    I What is the canonical form of the metric?

    I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes: Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...
  31. saadhusayn

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  32. binbagsss

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  33. Chromatic_Universe

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  34. M

    I Convert Metric Tensor to Gravity in GR

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  35. Mr Davis 97

    I Hausdorff Metric: Definition 1 vs. Definition 2

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  36. Mr Davis 97

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  37. facenian

    I Metric Homeomorphism: Isometry Equivalence?

    Is every homeomorphism between a metric space X and a topological Y equivalent to an isometry? I think it is, but I need to confirm.
  38. Safinaz

    I Weyl Transformation of Metric: Cosmological Inflation

    Hello, I’m new to the cosmological inflation so in this paper: https://arxiv.org/abs/1809.09975 Has some one an idea how to make the Weyl transformation of the metric ## g_{\mu\nu}## Equation (3) , and how to get the potential (4) from the action (3) by this transformation as explained after...
  39. Mr Davis 97

    I Metric Spaces and Topology in Analysis

    I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe...
  40. K

    I Gradient vector without a metric

    Is it possible to introduce the concept of a gradient vector on a manifold without a metric?
  41. Ibix

    I Coordinates for diagonal metric tensors

    In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
  42. F

    Complex numbers sequences/C is a metric space

    Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
  43. C

    I Derivation of Rindler Metric and How It Resolves the Twin Paradox

    From what I have read the twin paradox can be resolved with the Rindler metric and without the need to bring in general relativity. Special relativity will suffice. But how does the Rindler metric get derived in the context of a constant accelerating reference frame. I haven't seen anything in...
  44. E

    A Vec norm in polar coordinates differs from norm in Cartesian coordinates

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  45. M

    I Metric for knowing when numerical BC is "good"

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  46. J

    I Alcubierre Metric: Explaining Dark Matter Halo Speed

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  47. snoopies622

    I How to keep the components of a metric tensor constant?

    I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...
  48. K

    I Metric Compatibility: Is It Forbidden?

    My question is, is it forbidden to have a connection not compatible with the metric?
  49. T

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  50. T

    Is This Approach Valid for Proving the Discrete Metric in a Metric Space?

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