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. H

    Is the Schwarzschild metric dimensionless?

    Homework Statement The problem is I am wanting to know if the expression on the right hand side is dimensionless. Homework Equations ds^2 = (1 - \frac{2GM}{c^2 r})c^2 dt^2 The Attempt at a Solution Since the Schwarzschild radius is r = \frac{2GM}{c^2} would I be right in saying that...
  2. N

    Covariant/contravariant transform and metric tensor

    H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G-1 is its inverse. Consider an oblique coordinates system with angle between the axes = α I have G = 1/sin2α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrix I compute H = G*M where M =...
  3. grav-universe

    Problem with Schwarzschild metric derivation

    In this Wiki link for the derivation of the Schwarzschild metric, in the section "simplifying the components", g_22 and g_33 are derived. The problem is that upon deriving them, they first set those local measurements of the components for the metric upon a 2_sphere (on the left side) equal to...
  4. B

    Comparing Open Sets in Metric Spaces

    Homework Statement Let M be a metric space with metric d, and let d_{1} be the metric defined below. Show that the two metric spaces (M,d), (M,d_{1}) have the same open sets. Homework Equations d_1:\frac{d(x,y)}{1+d(x,y)} The Attempt at a Solution I tried to show that the neighborhoods...
  5. Markus Hanke

    Metric of Manifold with Curled up Dimensions

    Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter. Not going anywhere with this, I am just curious as to how...
  6. M

    Anti-de Sitter spacetime metric and its geodesics

    Hello, everybody. I have some doubts I hope you can answer: I have read that the n+1-dimensional Anti-de Sitter (from now on AdS_{n+1}) line element is given, in some coordinates, by: ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2} This can be...
  7. P

    Can Metric Tensors Have Equal Determinants?

    Hello, So, given two points, x and x', in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in x the determinant of the metric is g and in the point x' is g'. How are g and g' related?By any means can g=g'? In what conditions? I'm sorry if this is a dumb...
  8. L

    Form of Lorentz Transformation Using West-Coast Metric

    This is a fairly trivial question I think. I'm only asking it here because after some googling I was unable to find its answer. I was at one point led to believe that the form of the Lorentz-transformation matrix is dependent on the convention used for the Minkowski metric. Specifically it...
  9. grav-universe

    Schwarzschild metric and spherical symmetry

    In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.
  10. M

    Calculating Induced Metric on Vector Bundle E

    hi friends, Suppose we have a vector bundle E equipped with a hermitian metric h, and in a subbundle of E noted SE . I would like tocalculate explicitly the induced metric Sh defined on SE. How to proceed?
  11. P

    Variation of Laplace-Beltrami wrt metric tensor

    I have a very limited knowledge of tensor calculus, and I've never had proper exposure to general relativity, but I hope that the people reading this forum are able to help out. So I'm doing some stat. mech. and a part of a system's free energy is \mathcal{F} = \int V(\rho)\nabla^2\rho dx I'd...
  12. M

    MATLAB Plot unit circle in chebychev metric in MATLAB

    Ok, so I'm trying to plot the unit circle using the chebyvhev metric, which should give me a square. I am trying this in MATLAB, using the 'pdist' and 'cmdscale' functions. My uber-complex code is the following: clc;clf;clear all; boundaryPlot=1.5; % Euclidean unit circle for i=1:360...
  13. E

    Minkowski Metric and the Sign of the Fourth Dimension

    Why is the unit vector for time in Minkowski space i.e. the fourth dimension unit vector always opposite in sign to the three other unit vectors? The standard signature for Minkowski spacetime is either (-,+,+,+) or (+,-,-,-). Is there some particular reason or advantage for making time...
  14. G

    Confused about the metric tensor

    Now let's say I have the metric for some curved two surface ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal) Now my question is, since the metric defines the scalar product of two vectors, let's say (1,0) and (0,1), for...
  15. S

    Taub-Nut or NUT metric, that is the question

    Hello, We know that NUT spacetime is just like a massless rotating black hole, that this consideration introduces a new concept "magnetic mass", and I know just a little about its metric form and the parameters appear in it. While I was searching for NUT spacetime and its metric, I mostly...
  16. S

    Axiom of Choice and Metric Spaces.

    So the axiom of choice is confusing to me, apperently there is a distinction between the exsistence of an element and the actual selection of an element? I'm confused as to how much the axiom of choice is needed in elementary metric space theorems? As an example, is the Axiom of Choice needed...
  17. E

    Solve Schwarzschild Metric: Transformation & Acceleration

    Hi! Given the schwarzschild metric ds^2=-e^{2\phi}dt^2+\frac{dr^2}{1-\frac{b}{r}} I can make this coordinate transformation \hat e_t'=e^{-\phi}\hat e_t \\ \hat e_r'=(1-b/r)^{1/2}\hat e_r and I will get a flat metric. Is this correct? Another thing I'm a lot confused about: if I am at...
  18. G

    Developing Inner Product in Polar Coordinates via metric

    Hey all, I've never taken a formal class on tensor analysis, but I've been trying to learn a few things about it. I was looking at the metric tensor in curvilinear coordinates. This Wikipedia article claims that you can formulate a dot product in curvilinear coordinates through the following...
  19. V

    Metric of a static, spherically symmetric spacetime

    The (0,0) and (r,r) components are: g_{00}= -e^{2\phi},g_{rr}=e^{2\Lambda}. From the first component, combined with the fact that the dot product of the four velocity vector with itself is -1, one can find in the MCRF, U^0=e^{-\phi}. What does this mean? In the MCRF, the rate of the two clocks...
  20. A

    Metric space and absolute value of difference.

    I'm beginning self-study of real analysis based on 'Introductory Real Analysis' by Kolmogorov and Fomin. This is from section 5.2: 'Continuous mappings and homeomorphisms. Isometric Spaces', on page 45, Problem 1. This is my first post to these forums, but I'll try to get the latex right...
  21. R

    Confused by Metric Space Notation: What Does It Mean?

    I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying. In regards to a Metric space A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function {{\bf{d: M \times...
  22. T

    Difference between open sphere and epsilon-neighbourhood - Metric Spaces

    In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows: An open sphere S(x_0,r) in a metric space R (with metric function \rho(x,y)) is the set of all points x\in R satisfying \rho(x,x_0)<r. The fixed point x_0 is called the...
  23. S

    Minkowski metric - to sperical coordinates transformation

    I need to transform cartesian coordinates to spherical ones for Minkowski metric. Taking: (x0, x1, x2, x3) = (t, r, α, β) And than write down all Christoffel symbols for it. I really have no clue, but from other examples I've seen i should use chain rule in first and symmetry of...
  24. G

    Metric tensor in spherical coordinates

    Hi all, In flat space-time the metric is ds^2=-dt^2+dr^2+r^2\Omega^2 The Schwarzschild metric is ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2 Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...
  25. P

    Completion of Metric Space Proof from Intro. to Func. Analysis w/ Applications

    Completion of Metric Space Proof from "Intro. to Func. Analysis w/ Applications" Homework Statement I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof For any metric space X, there is...
  26. G

    Covariant Derivative and metric tensor

    Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.
  27. S

    When will the US officially adopt the metric system?

    Officially, the US adopted metric units as the legal standard in 1866, but never seriously attempted to implement a plan to phase out "customary" units. As a result, the US is the only industrialized nation which still uses non-metric units widely in commerce and law...
  28. G

    Integrating the metric in 3-D Spherical coordinates

    Guys, I read that integrating the ds gives the arc length along the curved manifold. So in this case, I have a unit sphere and its metric is ds^2=dθ^2+sin(θ)^2*dψ^2. So how to integrate it? What is the solution for S? Note, it also is known as ds^2=dΩ^2 Thanks!
  29. S

    Poincaré disk: metric and isometric action

    Hi! I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way: D = \{z \in \mathbb C | |z| < 1\} with metric ds^2 =...
  30. B

    Looking to Prepare for Metric Differential Geometry

    This is the course description: I want to take this class because the professor comes highly recommended, but I'm a little worried that I won't be entirely prepared for it. Normally this class requires Real Analysis as a prerequisite, and even though the professor explicitly states that...
  31. M

    Calculating the Metric on Quotient Space of E

    Hello friends, I would ask if anyone knows the métrqiue on the quotient space? ie, if one has a metric on a vector space, how can we calculate the metric on the quotient space of E?
  32. Orion1

    Alcubierre metric and General Relativity

    Alcubierre metric: ds^2 = \left( v_s(t)^2 f(r_s(t))^2 -1 \right) \; dt^2 - 2v_s(t)f(r_s(t)) \; dx \; dt + dx^2 + dy^2 + dz^2 What formal conditions are required to verify a valid metric solution of the Einstein field equations? How many possible valid metric solutions are there in General...
  33. S

    Second fundamental form of surface with diagonal metric

    Hello everyone, Let r(u_i) be a surface with i=1,2. Suppose that its first fundamental form is given as ds^2 = a^2(du_1)^2 + b^2(du_2)^2 which means that if r_1 = ∂r/∂u_1 and r_2= ∂r/∂u_2 are the tangent vectors they satisfy r_1.r_2 = 0 r_1.r_1 = a^2 r_2.r_2 =...
  34. P

    Minkowski Metric Sign Convention

    Hello, I believe this is a really stupid question but I can't seem to figure it out. So given a Minkowski spacetime one can choose either the convention (-+++) or (+---). Supposedly it's the same. But given the example of the four momentum: Choosing (+---) in a momentarily comoving...
  35. M

    Varying Energy in a Schwartzschild Metric

    This short work will help to calculate the varying energy for a non-rotating spherical distribution of mass. The Energy changing in a Schwartzschild Metric It is not obvious how to integrate an energy in the Schwartzschild metric unless you derive it correctly. The way this following metric...
  36. G

    Questions involving a set under the usual real metric

    Homework Statement Let S={1/k : k=1, 2, 3, ...} and furnish S with the usual real metric. Answer the following questions about this metric space: (a) Which points are isolated in S? (b) Which sets are open and closed in S? (c) Which sets have a nonempty boundary? (d) Which sets...
  37. N

    Complex Metric Tensor: Meaning, Weak Gravitational Fields & Einstein Eqns

    I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, hαβ turn out to be a complex numbers? What is the physical meaning of the complex metric tensor? Can I just take it's real part? Or there is no such thing as complex metric...
  38. G

    Proving Metric Space Reflexivity with Three Conditions

    Homework Statement Show that the following three conditions of a metric space imply that d(x, y)=d(y, x): (1) d(x, y)>=0 for all x, y in R (2) d(x, y)=0 iff x=y (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R (Essentially, we can deduce a reduced-form definition of a metric space...
  39. C

    Need to find the Ricci scalar curvature of this metric

    Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...
  40. C

    Need to find the Ricci scalar curvature of this metric

    Homework Statement Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2 Homework Equations The Attempt at a Solution I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel...
  41. L

    Deriving Connection-Metric Relation from Palatini Formalism

    Hey all, making my way through Landau and Lifgarbagez classical theory of fields and i had a specific question on the Einstein equations. Following the palatini approach, we assume that the connection and metric are independent variables and are not related a priori. In the footnote, they say...
  42. H

    Does the metric tensor only depend on the coordinate system used?

    I have looked at the definition of the metric tensor, and my sources state that to calculate it, one must first calculate the components of the position vector and compute it's Jacobian. The metric tensor is then the transpose of the Jacobian multiplied by the Jacobian. My problem with this...
  43. T

    From Metric Spaces to Linear Spaces

    Hi there, I've been asking in the Quantum Physics Forum about some foundational stuff and I've got stuck in why we assume in QM that the states space is linear. I have not found any deeper answer than "because it fits with observation". So I am now trying the other way around. I believe that...
  44. C

    Need to find the riemann curvature for the following metric

    Homework Statement Calculate the Riemann curvature for the metric: ds2 = -(1+gx)2dt2+dx2+dy2+dz2 showing spacetime is flat Homework Equations Riemann curvature eqn: \Gammaαβγδ=(∂\Gammaαβδ)/∂xγ)-(∂\Gammaαβγ)/∂xδ)+(\Gammaαγε)(Rεβδ)-(\Gammaαδε)(\Gammaεβγ) The Attempt at a Solution...
  45. K

    Riemann Normal Coordinates and the metric

    Homework Statement Consider a 2D spacetime where space is a circle of radius R and time has the usual description as a line. Thus spacetime can be pictured as a cylinder of radius R with time running vertically. Take the metric of this spacetime to be ds^{2}=-dt^{2}+R^{2}d\phi^{2} in the...
  46. pellman

    How do we infer a closed universe from FLRW metric?

    The Friedmann–Lemaître–Robertson–Walker metric is a solution of the field equations of GR. It tells us the local behavior of spacetime, that is, g(x) at a given spacetime point x If the matter density is high enough, the curvature is positive. It is said then that the universe is closed...
  47. N

    How to caculate the inverse metric tensor

    Given a metric tensor gmn, how to calculate the inverse of it, gmn. For example, the metric g_{\mu \nu }= \left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right] From basic understanding, I would think of divided it, that is...
  48. E

    Non discrete metric space on infinite set

    Homework Statement let d be a metric on an infinite set m. Prove that there is an open set u in m such that both u amd its complements are infinite. Homework Equations If d is not a discrete metric, and M is an infinite set (uncountble), then we can always form an denumerable subset...
  49. P

    Prokhorov Metric - Understanding the Definition

    Homework Statement I am wondering if anyone understands why this metric is defined the way it is because i can't seem to make sennse of it. I get that way we use the underlying metric space to define the borel sigma field and then the set of all borel measures, but the actual definition...
  50. S

    Origins of Scale Factor of FRW Metric and Misc Questions of GR Equations

    In the context of Friedmann's time, 1922, how did he know to make the metric scale factor, a, a function of time when Hubble's redshifts were not yet published? I understand that he took the assumption that the universe is homogenous and isotropic, but does that naturally imply that the universe...
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