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.

View More On Wikipedia.org
  1. jv07cs

    I Do Metric Tensors Always Have Inverses?

    I am reading about musical isomorphisms and for the demonstration of the index raising operation from the sharp isomorphism, we have to multiply the equation by the inverse matrix of the metric. Can we assume that this inverse always exists? If so, how could I prove it?
  2. M

    I Minkowski metric and proper time interpretation

    Using an example of 1 space dimension and 1 time dimension, consider the metric ##d\tau^2 = a dt^2 - dx^2## near a heavy mass (##a>1##). From what I've read a clock ticks slower near a heavy mass, as viewed from an observer far away. A clock tick would be representative of ##d\tau## right...
  3. Euge

    POTW Hölder Continuous Maps from ##R## to a Metric Space

    Let ##\gamma > 1##. If ##(X,d)## is a metric space and ##f : \mathbb{R} \to X## satisfies ##d(f(x),f(y)) \le |x - y|^\gamma## for all ##x,y\in \mathbb{R}##, show that ##f## must be constant.
  4. J

    I A question about metric compatibility equation

    We all know that The gradient of a scalar-valued function ##f(x)## in ##IR^n## is a vector field ##V_\mu(x)=\partial_\mu f(x)##, Such a vector field is said to be conservative. Not all vector fields are conservative. A conservative vector field should meet certain constraints...
  5. S

    I Non-homogeneous and anisotropic metric and laws of physics...?

    In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
  6. E

    A Solving Geodesics with Metric $$ds^2$$

    I have the following question to solve:Use the metric: $$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at $$t=0$$. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...
  7. S

    I Are there non-smooth metrics for spacetime (without singularities)?

    Are there non-smooth metrics for spacetime (that don't involve singularities)? I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics: Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...
  8. O

    Coordinate transformation into a standard flat metric

    I started by expanding ##dx## and ##dt## using chain rule: $$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$ $$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$ and then expressing ##ds^2## as such: $$ds^2 =...
  9. S

    I Explore Spacetimes, Metrics & Symmetries in Relativity Theory

    I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970) In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
  10. D

    A Understanding killing vectors and transformations of metric

    Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures. Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations. Q1.Can someone explain how to go from...
  11. K

    B Exploring the Standardization of One Light Year Distance | PF Forum

    Dear PF Forum, It's been a while since I logged in here. And I really do appreciate all the answers that I've been getting here. Now, I wonder. Is there any standardization for 1 light year distance? Is it 10 trillion kilometers, or 299,792,458 * 60 * 60 * 24 * 365.256 = ...
  12. V

    A Metric of a Moving 3D Hypersurface along the 4th Dimension

    Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##. Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
  13. Kostik

    A Static Gravitational Field: Why Must ##g_{m0} = 0##?

    In Dirac's "General Theory of Relativity", he begins Chap 16, with "Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##." It's obvious that...
  14. Onyx

    B Solve General Geodesics in FLRW Metric w/ Conformal Coordinates

    Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
  15. Onyx

    B Calculate Unit Normal Vector for Metric Tensor

    How do I calculate the unit normal vector for any metric tensor?
  16. G

    I Can Spherical Symmetry Be Achieved Without Varying Line Element?

    "Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##" I wonder why: "the line element does not vary when##\theta## and##\phi## are...
  17. Lluis Olle

    B Metric Line Element Use: Do's & Don'ts for Accelerated Dummies?

    From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame: ##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)## ##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...
  18. Sciencemaster

    I Calculating Spacetime Around Multiple Objects

    In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship...
  19. Sciencemaster

    I Calculate Gaussian Curvature from 4D Metric Tensor

    I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
  20. Onyx

    A Proper Volume on Constant Hypersurface in Alcubierre Metric

    I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...
  21. ergospherical

    I 4D d'Alembert Green's function for linearised metric

    Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...
  22. Tanujhm

    A Robertson-Walker metric

    The Robertson-Walker-Metric is given by To calculate the Friedmann equations ist is choosed with despite Minkowskis, Schwarzschilds and Kerrs What is the reason for this difference? Tanu
  23. Ennio

    I Exploring Proper Time in FLRW Metric: Meaning and Visualization

    The FLRW metric has been introduced to characterize the homogeneity and isotropy of the Universe and accordingly to obtain "easy" manageable solutions in Friedmann equations. The FLWR metric is where the LHS can be written as where is the proper time (despite we know that time is...
  24. JandeWandelaar

    A What does the metric of a 6D space with 3 compactified dimensions look like?

    I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally. Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended...
  25. P

    A Curvature & Connection Without Metric

    In the absence of a metric, we can not raise and lower indices at will. There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two...
  26. K

    A Can a conformal flat metric be curved?

    5/18/22 I am an MS in physics. I need to find out if the following CONFORMAL METRIC produces zero or nonzero curvature? I suspect the curvature is zero, but others have said it's probably not? MAXIMA sometimes says it is, and other times produces a Ricci scalar that looks like the FRW scalar...
  27. A

    I Schwarzschild Metric & Particle Absorption

    The Schwarzschild metric implies a potential different from that of Newtonian gravity. Is there a relationship between it and the process by which particles can be absorbed by other particles? (I haven't studied QFT yet)
  28. BiGyElLoWhAt

    I Equations of motion for the Schwarzschild metric (nonlinear PDE)

    I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
  29. physicsuniverse02

    Does anyone know which are Ricci and Riemann Tensors of FRW metric?

    I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.
  30. RayDonaldPratt

    I How to convert density ratio of grams/ mm^3 into metric tonnes/ m^3

    For the dimensions of a right cylinder, I am given three significant digits for the diameter (17.4 mm) and the height (50.3 mm). The formula for the volume of a right cylinder is V = Pi x r^2 x h, which would lead here to Pi x (17.4 mm / 2)^2 x 50.3 mm = 11,960.69354 mm^3 before rounding to 3...
  31. U

    Help with identifying a reference for the time-invariant Kaluza-Klein metric

    Homework Statement:: Please see below. Relevant Equations:: Please see below. I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric: \begin{equation}...
  32. docnet

    Show that if d is a metric, then d'=sqrt(d) is a metric

    ##d'## is a metric on ##X## because it satisfies the axioms of metrics: Identity of indiscernibles: ##x=y\Longleftrightarrow d(x,y)=0\Longleftrightarrow \sqrt{d(x,y)}=\sqrt{0}## Symmetry: ##d(x,y)=d(y,x)\Longrightarrow \sqrt{d(x,y)}=\sqrt{d(y,x)}## Triangle inequality: ##d(x,z)\leq...
  33. Sciencemaster

    I Adapting Schwarzschild Metric for Nonzero Λ

    So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...
  34. chwala

    Understanding of the Metric Space axioms - (axiom 2 only)

    Am refreshing on Metric spaces been a while... Consider the axioms below; 1. ##d(x,y)≥0## ∀ ##x, y ∈ X## - distance between two points 2. ## d(x,y) =0## iff ##x=y##, ∀ ##x,y ∈ X## 3.##d(x,y)=d(y,x)## ∀##x, y ∈ X## - symmetry 3. ##d(x,y)≤d(x,z)+d(z,y)## ∀##x, y,z ∈ X## - triangle inequality...
  35. George Keeling

    I Contracted Christoffel symbols in terms determinant(?) of metric

    M. Blennow's book has problem 2.18: Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...
  36. S

    I Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions

    I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar. Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...
  37. cianfa72

    I Raising/Lowering Indices w/ Metric Tensor

    I'm still confused about the notation used for operations involving tensors. Consider the following simple example: $$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$ Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...
  38. T

    I Induced Metric Help: Troubleshooting Extrinsic Curvature (12)

    I am having trouble calculating the extrinsic curvature (12) in the following paper: https://arxiv.org/pdf/gr-qc/0310107.pdf Specifically, I am unsure of what term to plug in for the induced metric h_{ab} in (8). If I am calculating the \sigma term in (12) is h_{ab} all of (4)? Also I would like...
  39. ergospherical

    I Transform Coordinates for Torus Metric in Wald

    I can't figure out how to transform the coordinates to get to the given metric \begin{align*}ds^2 = \cos x (dy^2 - dx^2) + 2\sin x dx dy \end{align*} for a 2-torus. Initially I parameterised it by two angles ##\theta## (around the ##z## axis) and ##\phi## (around the torus axis), to write...
  40. H

    A Time Dilation in Reissner-Nordström Metric: Even or Odd?

    In the Reissner–Nordström metric, the charge ##Q## of the central body enters only as its square ##Q^2##. The same is true for the Kerr-Schild form. This would seem to imply that all effects are even functions of ##Q##. For example, the gravitational time dilation is often written as $$\gamma =...
  41. yucheng

    Derivative of Determinant of Metric Tensor With Respect to Entries

    We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...
  42. docnet

    Engineering Can you find a surface from a metric?

    if a metric like ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2 ## is given, we know it corresponds to a sphere in spherical coordinates . if you are given an arbitrary metric with two variables for example ##ds^2=\frac{du^2}{u}+dv^2## is ther guarenteed to be a surface embedded in ##R^3##...
  43. sophiatev

    A Characterizing GR Traj in Minkowski Space

    In Minkowski space, with line element $$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$ (and ##c = 1##) we take spacelike trajectories to have ##ds^2 > 0##, null trajectories to have ##ds^2 = 0##, and timelike trajectories to have ##ds^2 < 0##. This makes sense given our definition of the line element...
  44. CanoJones

    I Kerr Metric Bibliography: Resources for Timelike Geodesics

    Hi all: As stated in the summary I'm in need for bibliography about timelike geodesics in the Kerr metric. I have tried using the "Mathematical Theory of Black Holes" by S. Chandrasekhar but I find it a bit to complex. Is there any other good books or articles about this that you might know...
  45. Haorong Wu

    A Does quantizing metric fields mean quantum gravity?

    (I am not sure which forum this post belongs to. Hope someone kindly helps me move it to a proper forum.) In papers, for example, here, here, and here, the authors start from the Lagrangian for matters and gravitational fields, then Dirac's constrained canonical quantization is used. They...
  46. W

    Distances in Knn: Are they metric functions in the Mathematical sense?

    Hi, Just curious as to whether distances 'd' , used in Knn ; K nearest neighbors, in Machine Learning, are required to be metrics in the Mathematical Sense, i.e., if they are required to satisfy, in a space A: ##d: A \times A \rightarrow \mathbb R^{+} \cup \{0\} ; d(a,a)=0 ; d(a,b)=d(b,a) ...
  47. Hubble_92

    I Variation of Four-Velocity Vector w/ Respect to Metric Tensor

    Hi everyone! I'm having some difficulty showing that the variation of the four-velocity, Uμ=dxμ/dτ with respect the metric tensor gαβ is δUμ=1/2 UμδgαβUαUβ Does anyone have any suggestion? Cheers, Rafael. PD: Thanks in advances for your answers; this is my first post! I think ill be...
  48. George Keeling

    I Linearized Gravity & Metric Perturbation when Indices Raised

    I have just met linearized gravity where we decompose the metric into a flat Minkowski plus a small perturbation$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\ \ \left|h_{\mu\nu}\ll1\right|$$from which we 'immediately' obtain $$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$$I don't obtain that. In my rule book...
Back
Top