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.
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows:
My question is as...
Assume that we have a flat FRW metric expressed in conformal time ##\eta## so that the line element is
$$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)\tag{1}$$
where ##a=1## at the present time ##\eta=0## and the speed of light ##c=1##.
This metric has the following non-zero Christoffel symbols...
Can I say that the Lorentz metric is the specific form ##-c^2dt^2 + dx^2 + dy^2 + dz^2## whereas the Minkowski metric is the metric of Minkowski space which can take the Lorentz form I just gave, but can also, e.g., be written in spherical coordinates?
I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.For the first one I get all my directional derivatives
(0, 0): (0)i + (0)j
(0, 1): (0)i + 2j
(1, 0): 2i + (0)j
(1, 1): 2i + 2j
Then I square them (FOIL):
(0, 0): (0)i...
I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows:
In the above...
I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy.
In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...
Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
In the above...
Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...
In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1).
m^2 = E^2 - \vec{p}^2
This can also be written with the Minkowski metric as:
m^2 = \eta_{\mu\nu} p^\mu p^\nu
More...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help with an aspect of the proof of Theorem 11.1.11...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help with an aspect of the proof of Theorem 11.1.11 ...
I am trying to get an intuition of what a metric is. I understand the metric tensor has many functions and is fundamental to Relativity. I can understand the meaning of the flat space Minkowski metric ημν, but gμν isn't clear to me. The Minkowski metric has a trace -1,1,1,1 with the rest being...
Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
Our 4-D space is ##x^1,x^2,x^3 ,t##.
Our sub-manifold is defined by ##(x^1,x^2,x^3)##
Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold
The metric becomes...
From the geodesic equation
d2xμ/dΓ2+Γμ00(dt/dΓ)2=0,for non-relativistic case ,where Γ is the proper time and vi<<c implying dxi/dΓ<<dt/dΓ.
Now if we assume that the metric tensor doesn't evolve with time (e,g gij≠f(t) ) then Γμ00=-1/2gμs∂g00/∂xs.
If we here assume that the metric components of...
Hello.
I ask for solution help from the integral below, where y and x represent angles in a metric of a spherical, 2-D surface. He was studying how to obtain the geodesic curves on the spherical surface, the sphere of radius r = 1, to simplify. The integral is the end result. It is enough, now...
Since it's possible to choose a coordinate chart where the Schwarzschild metric components are dependent on time, why that's not done? Would'nt there be a scenario where such a choice would be useful?
Hello everyone,
I have been studying perturbation theory in the context of FRW cosmologies, and so far have had a really hard time understanding why the SVT (Scalar, Vector, and Tensor) perturbations associated with the metric tensor "decouple" at first order in perturbation theory.
All...
Homework Statement
I'd like to calculate the form of Liouville operator in a Robertson Walker metric.
Homework Equations
The general form is
$$ \mathbb{L} = \dfrac{\text{d} x^\mu}{\text{d} \lambda} \dfrac{\partial}{\partial x^\mu} - \Gamma^{\mu}_{\nu \rho} p^{\nu} p^{\rho}...
I'm wondering if the galactic rotation curves could be attributed to a deviation of the metric of spacetime from the ideal Schwarzschild metric.
The Schwarzschild-metric is a well tested good approximation for the regions near the central mass - but at the outer space, far away from the...
Does anyone know how to get Maxima to solve the Kerr metric? I enter the terms for that metric that I found on Wikipedia. It tries to print out the Einstein tensor (covariant, leinstein(true)) and the expressions are so long that it literally locks up my computer. And isn’t the Kerr metric a...
Hi, I'm trying to deduce orbit velocity of a particle with mass from Schwarzschild metric. I know for Newtonian gravity it is:
$$v^2=GM\left(\frac{2}{r}-\frac{1}{a}\right)$$
The so called vis-viva equation. Where ##a## is the length of the semi-major axis of the orbit. For Schwarzschild metric...
Let us suppose we have a metric in the form of,
$$ds^2=-c^2dt^2+[(a^2(t)+b(r)e^{-lt})(dr^2+r^2d\Omega^2)]$$
Where scale factor is defined as ##(a^2(t)+b(r)e^{-lt})##
Is this metric describes homogeneity and isotropy or not ? I think it cannot since there's an ##r## dependence, and there are...
Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere.
If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)##
Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...
This is a problem from Tensor Calculus:Barry Spain on # 69
Prove that a space with Schwarzschild's metric is an Einstein space, but not a space of constant curvature.
The metric as given in the book is $$d\sigma^2=-\bigg(1-\frac{2m}{c^2r}\bigg)^{-1}dr^2-r^2d\theta^2-r^2\sin^2 \theta...
I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes:
Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...
I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor.
The given distance element is
$$ ds^2 = e^{2 \lambda} dt^2 -...
Homework Statement
Hi, I am just stuck in why / how we can write minkowski metric where I would usually write delta.
I see that the product rule is used in the first term to cancel the terms in the second term since partials commute for a scalar and so we are left with the d rivative acting...
Homework Statement
Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates.
.
Homework Equations
The Christoffel symbol definition:
Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ)
and the Riemann tensor definition:
Rμνσρ =...
I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa.
Thanks!
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.633&rep=rep1&type=pdf
<Moderator's note: Quotation added to make clear, what the author meant by "Hausdorff metric".>
The article above explores the notion of the Hausdorff metric. In the beginning of the article it describes how, if...
Homework Statement
Identify ##\bar{c}##, ##\bar{c_0}## and ##\bar{c_{00}}## in the metric spaces ##(\ell^\infty,d_\infty)##.
Homework Equations
The ##\ell^\infty## sequence space is
$$
\ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\...
Hello,
I’m new to the cosmological inflation so in this paper: https://arxiv.org/abs/1809.09975
Has some one an idea how to make the Weyl transformation of the metric ## g_{\mu\nu}## Equation (3) , and how to get the potential (4) from the action (3) by this transformation as explained after...
I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe...
In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
Homework Statement
If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##.
Homework Equations
##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
From what I have read the twin paradox can be resolved with the Rindler metric and without the need to bring in general relativity. Special relativity will suffice. But how does the Rindler metric get derived in the context of a constant accelerating reference frame. I haven't seen anything in...
I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates.
A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
Hi PF!
Let's say a boundary condition for an ODE is ##f'(1)+f(1) = 0##. If we solve the ODE numerically, how can I tell if this BC is satisfied "good enough". Let's suppose the numerics generate ##f'(1)+f(1) = 0.134##; is this close enough to zero?
So I have a question regarding the Alcubierre metric and the phenomena of stars on outer edges of galaxies moving at higher velocities than their orbital calculations state they should. When taking the accelerating expansion of space into account due to dark energy, could a sub-luminal...
I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...
Homework Statement
The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong.
Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
Homework Statement
Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y##
Homework Equations
N/A
The Attempt at a Solution
I have already seen various approaches in proving this. Although, I just want to know if...