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. 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...
  2. N

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  3. E

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    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...
  4. P

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  5. S

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  6. J

    Are Metric Space Completions Topologically Equivalent?

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  7. S

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  8. M

    Is this a homeomorphism that does not preserve metric completeness?

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  9. Thinkor

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  10. F

    How to calculate Riemannian Metric by distance function?

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  11. D

    Time Travel - Between two Kerr metric black holes w/detached event horizons

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  12. A

    Metric Connection from Geodesic Equation

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  13. A

    Another christoffel symbols from the metric question

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  14. S

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    So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable...
  15. A

    Equations of motion from Lagrangian and metric

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  16. L

    Equivalence classes and Induced metric

    (X,\rho) is a pseudometric space Define: x~y if and only if ρ(x,y)=0 (It is shown that x~y is an equivalence relation) Ques: If X^{*} is a set of equivalence classes under this relation, then \rho(x,y) depends only on the equivalence classes of x and y and \rho induces a metric on...
  17. A

    Equivalent conditions on a metric space

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  18. L

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  19. M

    Basic topology (differing metric spaces in R^2)

    Got it, thank you
  20. R

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  21. A

    How can we define the induced metric on a brane?

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  22. A

    MHB Show a certain sequence in Q, with p-adict metric is cauchy

    I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...
  23. A

    Show a certain sequence in Q, with p-adict metric is cauchy

    I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...
  24. A

    MHB Surjectivity of an Isometry given the metric space is complete.

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  25. G

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  26. O

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  27. A

    Sequence in Q with p-diatic metric. Show it converges to a rational

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  28. m4r35n357

    Conserved quantities in the Doran Metric?

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  29. 6

    Exploring Closed Sets in Metric Spaces through Infinite Intersections

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  30. D

    Calculating proper time using schwartzchild metric

    I am using the schwartzchild metric given as ds^2 = (1 - \frac{2M}{r})dt^2 - (1 - \frac{2M}{r})^{-1} dr^2 , where I assume the angular coordinates are constant for simplicity. So if a beam of light travels from radius r0 to smaller radius r1, hits a mirror, and travels back to r0, I am...
  31. Fantini

    MHB Should I study metric spaces topology before general topology?

    Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which...
  32. J

    Is the Empty Set Considered a Metric Space?

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  33. W

    Can changing the metric in GR result in different spacetimes?

    Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohio-state.edu/~dhw/A682/notes3.pdf there the FRW Metric is about: "In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3-sphere:"...
  34. W

    Does the expansion of space only occur in unbound systems?

    I'm coming with a good background in Big Bang expansion as the following sci-am article shows (which I've mastered): http://space.mit.edu/~kcooksey/teaching/AY5/MisconceptionsabouttheBigBang_ScientificAmerican.pdf What I'd like to understand is this. Expansion can only be felt in unbound...
  35. michael879

    CPT (M?) symmetries in Kerr-Newman metric

    So the confusion I'm having here really has to do with parity inversion in spherical (or boyer-linquist) coordinates. I've been looking at the discrete symmetries of the Kerr-Newman metric, and I've noticed that depending on how you define parity-inversion, you can get very different results...
  36. 8

    Metric spaces and convergent sequences

    Homework Statement let {xi} be a sequence of distinct elements in a metric space, and suppose that xi→x. Let f be a one-to-one map of the set of xis into itself. prove that f(xi)→x Homework Equations by convergence of xi, i know that for all ε>0, there exists some n0 such that if i≥n0...
  37. D

    Metric Perturbation: Finding Info for Einstein's Field Equation

    Hello guys. I was told to prepare a presentation on perturbed Einstein's field equation by my advisor. I got some of the things I needed to start with in the Weinberg's Cosmology book but it was not enough. Can anyone please tell me a book or anything with information on metric perturbation? Thanks
  38. D

    Vector difference metric that considers the variance of the components

    I am trying to match little square patches in an image. You can imagine that these patches have been "vectorized" in that the values are reordered consistently into a 1D array. At first glance, it seems reasonable to simply do a Euclidean distance style comparison of two of these arrays to get a...
  39. B

    General Relativity - Schwarzschild Metric

    Homework Statement A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c^2 (1) Find the relation between the rate of change of angular position of the spaceship and the proper time and radius of the orbit. Homework...
  40. T

    Connectedness of a subset of a metric space.

    Homework Statement Show that an open (closed) subset of a metric space E is connected if and only if it is not the disjoint union of two nonempty open (closed) subsets of E. Homework Equations The definition of connectedness that we are using is as follows: A metric space E is...
  41. J

    Calculating a metric from a norm and inner product.

    I typed the problem in latex and will add comments below each image. The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I...
  42. D

    Showing that a given metric is Riemannian

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  43. Alesak

    Why do manifolds require a Riemannian metric?

    When reading other threads, following question crept into my mind: When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...
  44. J

    Is U an Open Subset in the Space of Continuous Functions?

    Homework Statement Let a,b \in \mathbb{R} and we define I = [a,b] ([] means closed set). Let \mathcal{C}_{\mathbb{K}}(I) be the space of all continuous functions I \to \mathbb{K} with the norm f \mapsto ||f||_I = \displaystyle \sup _{x \in I} |f(x)|. Let U be the set of all continuous functions...
  45. T

    Triangle Inequality for a Metric

    Homework Statement Prove the triangle inequality for the following metric d d\big((x_1, x_2), (y_1, y_2)\big) = \begin{cases} |x_2| + |y_2| + |x_1 - y_1| & \text{if } x_1 \neq y_1 \\ |x_2 - y_2| & \text{if } x_1 = y_1 \end{cases}, where x_1, x_2, y_1, y_2 \in \mathbb{R}...
  46. N

    [topology] The metric topology is the coarsest that makes the metric continuous

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  47. G

    Tachyonic Energies in a Minkowski Metric

    So I'm working on a problem (Hartle problem 6, chapter 6) dealing with tachyons. So far, I have determined the four-velocity and the four-momentum (up to a sign) of a tachyon. I have, with the four-velocity being a unit spacelike four-vector, u^{\alpha}=\frac{\pm...
  48. 8

    Diameter of a union of metric spaces

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  49. C

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  50. G

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