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. F

    Change of variable in integral using metric

    What is the formula to evaluate a multi-integral by a change of coordinates using the squareroot of the metric instead of the determinate of the Jacobian? Thanks.
  2. M

    Deriving L-T Metric: Understanding Schwarzschild & Einstein's GR

    Hi, I have posted this question in the PF relativity forum because I am trying to understand the derivation of the Lense-Thirring (L-T) metric. Various sources suggest that L-T produced a solution of Einstein’s field equations of general relativity in 1918, just a few years after Schwarzschild...
  3. G

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  4. E

    Visualization of metric tensor

    Barbour writes: the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components, corresponding to the four values the indices u and v can each take: 0 (for the time direction) and 1; 2; 3 for the three spatial directions. Of the ten components, four merely...
  5. O

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    I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers. Consider the space, L, of all bounded sequences with the metric \rho_1 \displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t| Show that a...
  6. H

    Is the Schwarzschild metric dimensionless?

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  7. N

    Covariant/contravariant transform and metric tensor

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  8. grav-universe

    Problem with Schwarzschild metric derivation

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  9. B

    Comparing Open Sets in Metric Spaces

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  10. Markus Hanke

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  11. M

    Anti-de Sitter spacetime metric and its geodesics

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  12. 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...
  13. 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...
  14. 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.
  15. 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?
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. A

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    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...
  26. R

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  27. 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...
  28. S

    Minkowski metric - to sperical coordinates transformation

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  29. 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...
  30. 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...
  31. 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.
  32. 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...
  33. 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!
  34. 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 =...
  35. B

    Looking to Prepare for Metric Differential Geometry

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  36. M

    Calculating the Metric on Quotient Space of E

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  37. Orion1

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  38. S

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    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 =...
  39. 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...
  40. M

    Varying Energy in a Schwartzschild Metric

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  41. G

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  42. N

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

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  43. G

    Proving Metric Space Reflexivity with Three Conditions

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  44. C

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  45. C

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  46. L

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  47. H

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

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  48. T

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  49. C

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  50. K

    Riemann Normal Coordinates and the metric

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