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. grav-universe

    Possibly solved the metric without field equations

    Error below For a couple of years now, I have been attempting to solve for the values in GR of the time dilation z and the radial and tangent length contractions, L and L_t respectively, which form the metric c^2 dτ^2 = c^2 z^2 dt^2 - dr^2 / L^2 - d_θ^2 r^2 / L_t^2 (along a plane)...
  2. F

    Proving Variation of Metric K^{a b} with Killing Vector

    if we know K^{a b}= (∇^a*ζ^b -∇^b*ζ^a)/2, ζ is a killing vector, under the variation of metric g_{a b}→g_{a b}+δ(g_{a b}) which preserves the Killing vector δ(ζ^a)=0, h_{a b} = δ(g_{a b}) = ∇^a*ζ^b +∇^b*ζ^a, how to prove δ(K^{a b})= ζ_c*∇^a*h^{b c} - h^{c a}*∇_c*ζ^b - (...
  3. C

    How Is the Killing Metric Normalized for Compact Simple Groups?

    The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by ##K_{ab} = k \delta_{ab}## for some...
  4. C

    Necessity of bi-invariant metric for Yang-Mill's theory.

    The action for Yang-Mill's theory is often written as $$ S= \int \frac{1}{4}\text{Tr} (F^{\mu \nu} F_{\mu \nu})d^4 x = \int d^4 x\frac{1}{4} F^{k \mu \nu} F_{k \mu \nu}$$ where latin indices are indicies in the lie algebra, and the trace is taken with respect to the inner product...
  5. Greg Bernhardt

    Understanding the Metric Tensor: Definition, Equations, and Properties

    Definition/Summary The metric tensor g_{\mu\nu} is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime Equations The proper time is given by the equation d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu} using the Einstein summation convention It is a symmetric...
  6. C

    Converting metric to imperial (12in/1ft)^3

    Hello. I am new to engineering and to imperial units, and currently learning by doing some exercises. I'm stuck on the following conversion: 0.04 g / min x m^3 -> lbm / hr x ft^3 I figured it like this: 0.04 g / min x m^3 x (60min/1hr) x (1m/35,314)^3 x (1 lbm / 454g) = 1,49x10^-4 lbm...
  7. C

    Any biinvariant metric proportional to Killing metric

    The killing form on a lie algebra is defined as $$B(X,Y) = \text{Tr}ad_X \circ ad_Y$$ where ##ad_X: \mathfrak{g} \to \mathfrak{g}## is given by ##ad_X(Y) = [X,Y]##, where the latter is the lie bracket between X and Y in ##\mathfrak{g}##. Expressed in terms of components on a basis on...
  8. Mr-R

    Is the Metric in Spherical Coordinates Truly Flat?

    Dear all, As I was reading my book. It said that the line element of a particular coordinate system (spherical) in R^{3} is so and so. Then it said that the metric is flat. I don't get how the metric is flat in spherical coordinate. Could someone shed some light on this please? Thanks
  9. Mr-R

    Understanding Metric Connection and Geodesic Equations in General Relativity"

    Dear all, In my journey through learning General relativity. I have stumbled upon this problem. I have to calculate the geodesic equation for R^{3} in cylindrical polars. I am not sure how to use the metric connection. The indices confuse me. I would appreciate it if someone could shade some...
  10. S

    Minkowski Metric: Exploring Components and Deriving in 4D Coordinates

    I have recently been studying the tensors on the left side of the Einstein field equations, but I have been studying and deriving them in 3-D. I would now like to move on to adding time into the mixture. I have some questions regarding the Minkowski metric \eta\mu\nu. First, I know that...
  11. S

    Metric Tensor in Spherical Coordinates

    I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...
  12. ChrisVer

    Change of Determinant of Metric Under Var Change

    Under a change of variables: x^{\mu} \rightarrow x^{\mu}+ \delta x^{\mu} How can I see how the determinant of the metric changes? \sqrt{|g(x)|}? Is it correct to see it as a function? f(x) \rightarrow f(x+ \delta x) = f(x) + \delta x^{\mu} \partial_{\mu} f(x) ?
  13. ChrisVer

    Understand Degeneracy of Metric at Schwarzschild Singularity

    I am reading this paper http://arxiv.org/pdf/1111.4837.pdf and I came across under eq12 that the new metric is degenerate... How can someone see that from the metric's form? Degeneracy for a metric means that it has at least 2 same eigenvalues (but isn't that the same for the Minkowski metric...
  14. N

    Metric Prefix Help - Find 2mL=200L

    I keep trying to find a certain metric prefix but i can't seem to find it, i need to know what prefix makes 2mL into 200_L i believe it is 10^-5 but i can't find that on any charts. Any help is much appreciated!
  15. M

    General Expression for Round Metric on an N-sphere

    Homework Statement I want to know the expression for the round metric of an n-sphere of radius r Homework Equations I have obtained this for a 3-sphere dS^{2}=dr^{2}+r^{2}(d\theta_{1}^{2}+sin^{2}\theta_{1}d\theta_{2}^{2} +sin^{2}\theta_{1}sin^{2}\theta_{2}d\theta_{3}^{2}) The...
  16. X

    An empty ball in arbitrary metric space

    Is it possible for a ball(with nonzero radius) to be empty in an arbitrary metric space?
  17. C

    What kind of isometry? A metric tensor "respects" the foliation?

    Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...
  18. darida

    Exploring Ansatz Metric of 4D Spacetime

    Ansatz metric of the 4 dimensional spacetime: ds^2=a^2 g_{ij}dx^i dx^j + du^2 (1) where: Signature: - + + + Metric g_{ij} \equiv g_{ij} (x^i) describes 3 dimensional AdS spacetime i,j = 0,1,2 = 3 dimensional curved spacetime indices a(u)= warped factor u = x^D =...
  19. B

    Exploring Bianchi IX Models for Metric Invariance

    In a Bianchi IX universe the metric must be invariant under the SO(3) group acting on the 3-sphere. Hence, the metric must be translation invariant in the spatial parts, where t=constant. This implies that the metric must take the form such that: ds^2 = dt^2 - g_ij(t)(x^i)(x^j), where g is a...
  20. C

    Showing a metric space is complete

    Homework Statement Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)| is a complete metric space. The Attempt at a Solution Spent a few hours just thinking about this question, trying to prove...
  21. B

    Need clarification on the product of the metric and Levi-Civita tensor

    Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...
  22. S

    Metric Spaces - Distance Between sets and it's closures

    I was trying to prove: d(A,B) = d( \overline{A}, \overline{B} ) I "proved" it using the following lemmas: Lemma 1: d(A,B) = \inf \{ d(x,B) \}_{x \in A} = \inf \{ d(A,y) \}_{y \in B} (By definition we have: d(A,B) = \inf \{ d(x,y) \}_{x \in A, y \in B} ) Lemma 2: d(x_{0},A) = d(x_{0}...
  23. H

    Problem about taking measurements in flat metric spaces

    Hello, I am having a problem about the nature of the measurements of the intervals ds's forming out of infinitesimal displacements dx's of the coordinates and the actual meaning of the measurements of the same dx's, in flat metric spaces. I am certain that this must be a trivial problem...
  24. D

    Is there any difference between Metric, Metric Tensor, Distance Func?

    From what I've understood, 1) the metric is a bilinear form on a space 2) the metric tensor is basically the same thing Is this correct? If so, how is the metric related to/different from the distance function in that space? Some other sources state that the metric is defined as the...
  25. G

    Variation with respect to the metric

    Hi everyone! There is something that I would like to ask you. Suppose you have \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} (g^{ab} u_a u_b + 1))}{\delta g^{cd}} The outcome of this would be ##u_{c}u_{d}## or ##-u_{c} u_{d}## ? I am really confused.
  26. H

    Einsteins-Rosen bridge different from schwarzschild black hole metric?

    Hello, if you could help, I will be glad. I am studying the Einsteins-Rosen bridge (a matematically solution of the black hole) and I thought that the Einsteins-Rosen bridge was what we found making the Schwarzschild metric a change in kruskal coordinates. But reading an scientific article it...
  27. L

    How does metric give complete information about its space?

    Hello, I've been struggling with the so often spoken idea that a metric tensor gives you all necessary information about the geometry of a given space. I accept that from the mathematical point of view as every important calculation (speaking as a physicist with respect to GTR rather than...
  28. MattRob

    Alcubierre Metric: York Time Cause and Effect

    So, there's a paper here that I'm a bit confused about. On pages 3 and 4, it talks about energy density magnitude and York time. What I'm a bit confused about, is in the article that linked me to it, the scientist makes mention of eventually generating negative vacuum energy. However, from...
  29. ChrisVer

    Change of the minkowski metric

    If I am not mistaken, the change of the minkowski metric to: n_{\mu\nu} \rightarrow g_{\mu\nu}(x) will violate the Poincare invariance of (example) the Electromagnetism Action. However it allows us to define a wider set of arbitrary transformations (coordinate transformations). The last...
  30. S

    MHB Complete Metric Space: X, d | Analysis/Explanation

    Hi i am confused of the following question. Suppose we have a Metric Space (X,d), where d is the usual metric. Now are the following subsets complete, if so why?? 1.$$X=[0,1]$$ 2.$$X=[0,1)$$ 3.$$X=[0,\infty)$$ 4.$$(-\infty,0)$$
  31. phosgene

    Show that T is a contraction on a metric space

    Homework Statement Consider the metric space (R^{n}, d_{∞}), where if \underline{x}=(x_{1}, x_{2}, x_{3},...,x_{n}) and \underline{y}=(y_{1}, y_{2}, y_{3},...,y_{n}) we define d_{∞}(\underline{x},\underline{y}) = max_{i=1,2,3...,n} |x_{i} - y_{i}| Assume that (R^{n}, d_{∞}) is...
  32. V

    Killing vecotrs of Schwarzschild metric

    Hello, I have a question, whether is possible to looking for Killing Vectors (KV) in this way (I know about general solution): From Schwarzschild metric I can see two KV \frac{\partial}{\partial t} and \frac{\partial}{\partial\phi} . Then I see that other trivial KV arent there. Metric...
  33. C

    Derivatives in a non-trivial metric

    I'm trying to work out: (∇f)^2 (f is just some function, its not really important) While working in curved space with a metric: ds^2 = α dt^2 + dr^2 + 2c√(α+1) dtdr I'm not really sure how to calculate a derivative in curved space, any help would be appreciated thanks
  34. V

    The present epoch in FRW metric

    In an expanding universe that is modeled by the FRW metric we assume that scale factor of the "present epoch" is unity which is equivalent to a zero redshift. Therefore, most observed galaxies with nonzero redshifts are in our past light cone. But it is unclear to me how much back in time or...
  35. shounakbhatta

    Riemann Metric Tensor: Exploring Basics

    Hello All, Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...
  36. C

    Is the composition of a function and a metric a metric?

    Homework Statement Given that f is continuous and strictly increasing, f:[0, ∞)->[0, ∞), f(0)=0, and d(x,y) is the standard metric on the real number line, Is there a function f such that d'(x,y)=f(d(x,y)) is not a metric on the real number line? The Attempt at a Solution The standard...
  37. C

    Show the following is a metric

    Homework Statement Let (X,d) is a metric space. Show that d_1=log(1+d) is a metric space. The Attempt at a Solution (it's not stated what d is so I'm assumed d=|x-y|) I've checked positivity and symmetry but am having trouble with showing the triangle inequality holds. i.e. log(1+|x-y|)...
  38. E

    MHB About open sets in a metric space.

    Let (E=]-1,0]\cup\left\{1\right\},d) metric space with d metric given by d(x,y)=|x-y|, and ||absolute value. How I can find open sets of E explicitly? Thanks in advance.
  39. B

    FLRW Metric and Explaining the Role of Gravity in the Big Bang Theory

    I am trying to self-study FLRW and I hope someone cares to answer a simple question regarding this explanation:http://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#Newtonian_interpretation If I got it right the expanding matter is contrasted by...
  40. R

    Velocity Term in Alcubierre Metric - Exploring its Significance

    Reading over Alcubierre's paper on his "warp" drive (http://arxiv.org/abs/gr-qc/0009013), the metric in equation 3 has a velocity term, v, that doesn't seem to be needed anywhere. Even in the one spot where it seems potentially valuable, equation 12, he just call it =1 and essentially ignores...
  41. ShayanJ

    Metric and existence of parallel lines

    I don't know very much about differential geometry but from the things I know I think that the metric is somehow the quantity which specifies what kind of a geometry we're talking about(Though not sure about this because different coordinate systems on the same manifold can lead to different...
  42. TrickyDicky

    Levi-Civita connection and pseudoRiemannian metric

    One of the properties of the unique Levi-Civita connection is that it preserves the metric tensor at each point's tangent space, allowing the definition of invariant intervals between points in the manifold. I'd be interested in clarifying: when the metric preserved by the L-C connection is a...
  43. W

    Velocity Addition: Solving Minkowski Space with Metric Techniques

    Hi all. I'm taking a course in GR and trying to get my intuition and mathematical techniques up to speed. I've been trying to derive the velocity addition formula in Minkowski space, but for some reason I can't do it. Here's what I have: I'll use the Minkowski metric of signature...
  44. C

    Stress-energy tensor explicitly in terms of the metric tensor

    I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...
  45. T

    What Defines a Ball, Interior, and Limit Point in Metric Spaces?

    Homework Statement For a metric space (X,d) and a subset E of X, de fine each of the terms: (i) the ball B(p,r), where pεX and r > 0 (ii) p is an interior point of E (iii) p is a limit point of E Homework Equations The Attempt at a Solution i) Br(p) = {xεX: d(x.p)≤r}...
  46. W

    Minkowski Metric and Lorentz Metric

    I am currently studying special relativity on my own and I am looking into space time and space time diagrams. While reading through various sources I came across what seemed to be two methods to describe space time. X0, X1, X2, X3 (ct, x,y,z) -> Lorentz Metric X1, X2, X3, X4 (x,y,z,ict)...
  47. G

    Metric for a surface of a cone

    Homework Statement The metric for this surface is ds^2 = dr^2 + r^2\omega^2d\phi^2, where \omega = sin(\theta_0). Solve the Euler-Lagrange equation for phi to show that \dot{\phi} = \frac{k}{\omega^2r^2}. Then sub back into the metric to get \dot{r} Homework Equations L = 1/2 g_{ab}...
  48. C

    Is the metric tensor constant in polar coordinates?

    I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by: g^{\mu \nu} = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix} Since...
  49. C

    Admitting Metric: Meaning & Explanation

    What does one mean when one says that a certain manifold 'admits' a certain metric?
  50. ferst

    Gödel Metric Theses & Dissertations: 3 in Eng/Port, 1 in Portuguese

    Hi, I'm doing the master in science and one of things that I have to study is the Gödel metric. His paper have a high level for me and I'm seeking theses and dissertations about the Gödel universe. At moment I got three theses about the subject in english and portuguese-BR and one dissertation...
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