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

    Proper distance from the FRW metric

    Homework Statement Question: Homework Equations [/B] Dp=R(t)Dc where R(t) is the scale factor, Dc is the comoving distance The Attempt at a Solution [/B] I must have tried solving this starting 10 different ways now, starting with the fact that distance=integral over velocity dt...
  2. S

    Negative scale factor RW metric with scalar field

    Homework Statement The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier. Homework Equations I have this action S=-\frac{1}{2}\int...
  3. D

    Question about Metric Tensor: Learn Differential Geometry

    Hey, I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question. The metric tensor can be written as $$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$ and its also written as...
  4. S. Leger

    Variation of determinant of a metric

    Homework Statement I'm trying to calculate the variation of the following term for the determinant of the metric in the polyakov action: $$h = det(h_{ab}) = \frac{1}{3!}\epsilon^{abc}\epsilon^{xyz}h_{ax}h_{by}h_{cz}$$ I know that there are some other ways to derive the variation of a metric...
  5. evinda

    MHB Showing Completeness of Metric Spaces: Examples

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  6. W

    Writing a general curve on a manifold given a metric

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  7. T

    Metric variation of the covariant derivative

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  8. E

    What is the difference between standard and isotropic metrics?

    The metric $$ds^2=-R_1(r)dt^2+R_2(r)dr^2+R_3(r)r^2(d\theta^2+sin^2d\phi^2)$$ when changed to $$ds^2=-R_1(r)dt^2+R_2(r)(dr^2+r^2d\Omega^2)$$ upon setting ##R_2(r)=R_3(r)##, the later metric holds the name of isotropic metric. My question what is the difference between the first and the second...
  9. I

    Cosmological constant times the metric tensor

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  10. L

    Metric expansion misunderstanding

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  11. B

    Impact parameter of a photon in Schwarzchild metric

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  12. B

    4 velocity in Schwarzchild metric

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  13. L

    How to prove the following defined metric space is separable

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  14. m4r35n357

    Sign of Kretschmann Scalar in Kerr Metric

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  15. darida

    Derive Reissner Nordstrom Eq. (8) Explained

    I want to derive Reissner Nordstrom solution using this paper as a guide: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf, but I get confused by the Eq. (8). Why the Enstein's equation can be rewritten in that form and what is the physical meaning of the Eq. (8)?
  16. RyanH42

    What is the volume of the universe using a spacetime metric approach?

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  17. S

    Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

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  18. lucasLima

    Help proving triangle inequality for metric spaces

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  19. S

    Partial Derivative of x^2 on Manifold (M,g)

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  20. S

    What is the transformation law for tensor components in differential geometry?

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  21. W

    Categorical Counterpart to Relation bet Metric and Measure S

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  22. MattRob

    Validity of Schwarzschild Metric in Real BHs

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  23. jtbell

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  24. T

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  25. S

    Weyl tensor for the Godel metric interpretation

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  26. O

    MHB Proving d(x,A) ≤ d(x,y) + d(y,A) in Metric Spaces

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  27. Tony Stark

    Metric Tensor of a line element

    When we define line element of Minkowski space, we also define the metric tensor of the equation. What actually is the function of the tensor with the line element.
  28. A

    FRW metric, convention misunderstanding?

    So I have been following various derivations of the FRW metric and have a bit of confusion due to varying convention... Would it be correct to say that curvature K can be expressed as both K = \frac{k}{a(t)^2} and K = \frac{k}{R(t)^2} where k is the curvature parameter? If so, is it correct to...
  29. P

    Write Torsion Tensor: Definition, Metric Tensor & Equation

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  30. S

    Clarifying Robertson-Walker Metric Math Objects

    Here is the Robertson Walker metric: ds2= (cdt)2 - R2(t)[dr2/(1- kr2) + r2(dθ2 + sin2(θ)dΦ2)] This metric is seen and discussed in this link: http://burro.cwru.edu/Academics/Astr328/Notes/Metrics/metrics.html Now I am in the process of deriving the general relativistic mathematical objects...
  31. S

    How to prove the bilinearity of a given metric using tensorial product addition?

    How could I proof this ##ds^2=cos^2(v)du^2+dv^2## is bilinear?
  32. S

    What is the mistake in my coordinate transformation for Theorema Egregium?

    I have ##ds^2=\cos^2(v)du^2 + dv^2## , i take a coordinate transformation x=u and cos(v)=##\frac{1}{(cosh(y))}##, I have to find a new metric with this coordinate transformation and proof it is in agreement with Theorema Egregium. ##ds^2=\frac{dx^2}{(cosh^2(y)}) +\frac{dy^2 }{(y^2(1-y^2))}##...
  33. Rasalhague

    Proving that Every Closed Set in Separable Metric Space is Union of Perfect and Countable Set

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  34. P

    Vary Metric w/ Respect to Veirbein: How To?

    Hello! Given a metric in terms of the Veirbein, ##g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}## , how would you go about varying it with respect to ##e^{a}_{\mu}## ? I know that ##{\delta}g_{\mu\nu}={\delta}e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}+e^{a}_{\mu}{\delta}e^{b}_{\nu}{\eta}_{ab}## , with...
  35. P

    Varying Minkowski Metric: Is {\delta}{\eta}_{\mu\nu}=0?

    So I was wondering, is the variation of the Minkowski Metric zero? As in ##{\delta}{\eta}_{\mu\nu}=0## . I would think this is the case because the components of the Minkowski Metric are just numbers (either +1 or -1), so varying it gives you 0. Is this correct?
  36. Tony Stark

    General metric and flat metric

    What is the difference between General metric gαβ and flat metric ηβα in GR? Elaborate answers are appreciated.
  37. P

    Raising and Lowering Indices and metric tensors

    The metric tensor has the property that it can raise and lower indices, but this is on the assumption that it (the metric) is symmetric. If we were to construct a metric tensor that was non-symmetric, would it still raise and lower indices?
  38. B

    Derivative of the mixed metric tensor

    So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...
  39. O

    Is a Metric Feedback Possible in General Relativity?

    Apologies for not doing too much research prior to asking this question; I suppose actually delving into the mathematics would reveal the answer I'm looking for but I haven't taken the time just yet. Considering the concept of GR where matter/energy tells space how to curve and space tells how...
  40. Andre' Quanta

    Metric for a free falling observer

    Is it true that for a free falling observer in a non homogeneuos gravitational field, the metric according to his reference frame is always Minkowski? If it is true, Is it valid only locally?
  41. B

    What is the Induced Metric on the Subspace of Zeros and Ones in ##l^\infty##?

    Homework Statement If ##A## is the subspace of ##l^\infty## consisting of all sequences of zeros and ones, what is the induced metric on ##A##? Homework EquationsThe Attempt at a Solution The metric imposed on ##l^\infty## is ##d(x,y) = \underset{i \in \mathbb{N}}{\sup} |x_i - y_i|##. I...
  42. flaticus

    What are the non-zero Christoffel symbols for 2D polar coordinates?

    Just started self teaching myself differential geometry and tried to find the christoffel symbols of the second kind for 2d polar coordinates. I am checking to see if I did everything correctly. With a line element of: therefore the metric should be: The christoffel symbols of the second kind...
  43. X

    What Is the Induced Metric on a Spacelike Hypersurface t=const?

    Homework Statement Let g_{\mu\nu} be a static metric, \partial_t g_{\mu\nu}=0 where t is coordinate time. Show that the metric induced on a spacelike hypersurface t=\textrm{const} is given by \gamma_{ij} = g_{ij} - \frac{g_{ti} g_{tj}}{g_{tt}} . Homework Equations Let y^i be the coordinates...
  44. Andre' Quanta

    Time Misured by Clock in Non Inertial Sistem?

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  45. E

    Cosmological constant term and metric tensor

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  46. R

    MATLAB Visualizing Robertson-Walker Metric in MATLAB/Maple

    Hi, I was wondering if there's any way to plot/visualize a metric (mostly the spatial part). I want to see how the robertson-walker metric differs from a rotating rw-metric; \begin{align} ds^2=-(1-\omega^2a^2r^2\sin^2\theta)dt^2+a^2[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta...
  47. U

    Frequency of Photon in Schwarzschild Metric?

    Homework Statement The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##. (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...
  48. C

    Tensor Variation with Respect to Metric in First Order Formalism

    Homework Statement I'm just wondering if I'm doing this calculation correct? eta and f are both tensors Homework EquationsThe Attempt at a Solution \frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3}...
  49. ShayanJ

    Forming Spherically Symmetric Metric: Math Analysis & Omitted Steps

    In most GR textbooks, the general form of a spherically symmetric metric is obtained by inspection which is acceptable. But in the textbook I'm reading, the author does that with a mathematical analysis just to illustrate the method. But I can't follow his calculations. In fact he omits much of...
  50. U

    Satellite orbiting around Earth - Spacetime Metric

    Homework Statement The metric near Earth is ##ds^2 = -c^2 \left(1-\frac{2GM}{rc^2} \right)dt^2 + \left(1+\frac{2GM}{rc^2} \right)\left( dx^2+dy^2+dz^2\right)##. (a) Find all non-zero christoffel symbols for this metric. (b) Find satellite's period. (c) Why does ##R^i_{0j0} \simeq \partial_j...
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