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Yet another weird SF fan
 I'm a mathematician, a libertarian, and a science-fiction fan. Common sense? What's that?Go to first entry

## Archives

<< current

 Yet another weird SF fan

### Are Leftists in Favor of Open Borders?

Juan Cole isn't:

The first original sin was the contradictory and feckless pledge by the British to sponsor Jewish immigration into their Mandate in Palestine, which they wickedly and fantastically promised would never inconvenience the Palestinians in any way. It was the same kind of original sin as the French policy of sponsoring a million colons in French Algeria, or the French attempt to create a Christian-dominated Lebanon where the Christians would be privileged by French policy.
One of the more plausible reasons to be dubious of Zionism is the fact that open borders are restricted to Jews. If the first complaint about Israel is that it is based on immigration, then the quarrel is no longer open society vs. closed society but tribe vs. tribe and I may as well stick with my own and, for that matter, born-again American Christians may as well stick with Israelites.

this Tea Party.

### How Should I Celebrate?

How should I celebrate the news that I have low cholesterol? (It's rather surprising for somebody as fat as I am.) Maybe a three-egg omelet is called for …

### The Trouble with High-Level Understanding

According to SilasBarta22 on LessWrong, at the highest-level of understanding:

At this stage, not only do you have good, well-connected models of reality, but they are so well-grounded, that they "regenerate" when "damaged".  That is, you weren't merely fed these wonderful models outright by some other Really Smart Being (though initially you might have been), but rather, you also consistently use a reliable method for gaining knowledge, and this method would eventually stumble upon the same model you have now, no matter how much knowledge is stripped away from it.
The trouble with this is you might be fooling yourself into thinking you have such a deep understanding and then derive conclusions based on your erroneous model that you don't think you have to check.

For example, I've read that part of the initial reaction to Starship Troopers by Robert Heinlein came from people who noticed that the book was very pro-military and then deduced from that that Heinlein must have been pro-conscription. That turned out not to be the case. I suspect that the deductions were done by people who thought they had a well-grounded understanding of “militarists.” As a result, they were confident about regenerating Heinlein's opinion of conscription. (I have earlier mentioned other examples of the same phenomenon.)

### Real Estate with a Pre-existing Condition

The following ad appeared at Dr. Boli's Celebrated Magazine a couple of years ago:

WAREHOUSE FOR SALE. 6 floors, 23,000 sq ft. Unheated. Ample parking in separate lot. 6 loading docks. 3 offices and small retail space on 1st floor. Currently on fire. May be said to be heated in that sense. Liberal incentives for early offers. Apply in person, with own hose, at Dr. Boli’s Celebrated Real Estate Agency, Seldom Seen.
There's an analogy here for pre-existing health conditions.

### Paranoia in the Wrong Direction

I doubt if leftists are planning to get more votes via immigration amnesty. For one thing, they can get many of same political advantages by the easier tactic of counting aliens in the census. For another thing, the “problem” of illegal immigration is marvelous excuse for a power grab. It's no wonder the Democrats are dragging their feet.

On the other hand, if the Democrats claim to be pro-immigration, they can get the votes of the second generation—especially if Republicans believe it.

The Foreign Minister of Germany once said to me your country does not dare do anything against Germany, because we have in your country five hundred thousand Germans reservists [emigrants] who will rise in arms against your government if you dare to make a move against Germany. Well, I told him that that might be so, but that we had five hundred thousand and one lamp posts in this country, and that that was where the reservists would be hanging the day after they tried to rise.
The German threat, of course, went nowhere. That just might be an indication that immigrants to the U.S. are less dangerous than feared.

ObSF: “The Other Likeness” by James H. Schmitz. (I've mentioned this before.)

### Leftists and Memory

I noticed a common assumption on the left side during the health-insurance debates: that if a behavior is mandated or prohibited nearly everybody will, of course, obey the law. They don't believe people will respond to incentives to have or drop health insurance but will respond to a legal requirement to do so.

I have heard that people who smoked enough dope in college don't recall it (or, apparently, that anybody else their age had done so) but I didn't believe that until now…

### Is There a Great Filter?

According to Katja Grace (seen via Overcoming Bias):

The great filter, as described by Robin Hanson:

Humanity seems to have a bright future, i.e., a non-trivial chance of expanding to fill the universe with lasting life. But the fact that space near us seems dead now tells us that any given piece of dead matter faces an astronomically low chance of begating such a future. There thus exists a great filter between death and expanding lasting life, and humanity faces the ominous question: how far along this filter are we?

I will argue that we are not far along at all. Even if the steps of the filter we have already passed look about as hard as those ahead of us, most of the filter is probably ahead. Our bright future is an illusion; we await filtering. This is the implication of applying the self indication assumption (SIA) to the great filter scenario, so before I explain the argument, let me briefly explain SIA.

SIA says that if you are wondering which world you are in, rather than just wondering which world exists, you should update on your own existence by weighting possible worlds as more likely the more observers they contain. For instance if you were born of an experiment where the flip of a fair coin determined whether one (tails) or two (heads) people were created, and all you know is that and that you exist, SIA says heads was twice as likely as tails. This is contentious; many people think in such a situation you should think heads and tails equally likely. A popular result of SIA is that it perfectly protects us from the doomsday argument. So now I’ll show you we are doomed anyway with SIA.

She then points out that the SIA indicates we are almost certainly not in a universe in which there was much of a filter in the past. If there is a great filter that means it is in the future.

On the other hand, the theory that there is a great filter rests on the apparent absence of extraterrestrial civilizations. A theory that I saw on Usenet (yes, there are or were occasional glimpses of rationality there) some years ago from Kaehler, Oh, and Krummenacker gives us good reason to think there are extraterrestrial civilizations in the observable universe:

From: kr@cup.portal.com (Markus XXX Krummenacker)
Newsgroups: alt.alien.visitors,sci.astro
Subject: Re: Drake Equation & Fermi Paradox / KOK hypothesis
Date: 10 Jul 1995 03:40:15 -0700
Organization: The Portal System (TM)
Lines: 97
Sender: pccop@unix.portal.com
Distribution: world
Message-ID: <141949@cup.portal.com>
NNTP-Posting-Host: news1.unix.portal.com

………

First I need to admit that I am probably going out on a limb with what follows, and that I do not really know much about cosmology and astrophysics.

However, I would like to ask: How can we know that the universe is indeed not yet infested with Dyson spheres ? Maybe the universe is already 90% full, instead of still empty.

In a very crowded universe, there is a desire to squeeze out as much utility and energy efficiency from scarce starlight as possible, so one would expect Dyson spheres to be constructed which emit "waste radiation" at the lowest photon-energy that is physically possible, "milking" the higher-quality photons for useful work to the fullest extent. Such Dyson spheres would be designed to emit "waste-photons" at a temperature which is just above the thermal background radiation of the universe, i.e. essentially in the region of 3K. It would likely be extremely difficult to detect them, because they will be black objects before a black background.

In general, one can probably assume that even if the universe is populated by many advanced civilizations, we would not be able to recognize and/or comprehend them at all anyway. The current SETI efforts look more like a joke, because I tend to think that such civilizations would not have much interest in sending beacons, which are designed to be easily understood, to us little human critters (much as we do not care to send signals to ant-colonies, to make them recognize that we exist). However, if we live in a densely populated universe, we probably are staring at their artifacts and/or signals already, and are just too stupid to recognize them as such. Advanced civilizations will leave a strong mark on the universe, which can not possibly be overlooked. Presumably, if such evidence can be found in the sky, humans have photographed and recorded it already many times. In some sense, money for SETI projects is not really needed, just quite a bit more imagination for how to re-interpret what we already have in our archives. How about a thorough data-mining session ? :-)

In the above picture of an already densely populated universe, the question would not be whether we can find other civilizations in our galaxy. Very likely, entire galaxies will be taken over completely by civilizations on a short time scale, so finding "partially consumed" galaxies in the sky will be extremely rare. Either entire galaxies are completely black, or still totally pristine. Maybe 90% of all galaxies in the universe are already fully populated, which would be an interesting explanation for the "missing mass" phenomenon. This mass is black, which is why we do not see it. If the universe is so crowded by advanced civilizations, it also becomes plausible why maybe about 10% of the universe was left untouched: they agreed to leave a "natural reservation". In this view, we happen to live in this reservation. Once we will start to expand outwards, we will at some point be made aware of the rules that the other civilizations around us have established regarding resource consumption. Isn't the large scale structure of the visible universe reminiscent of multiple intersecting bubbles ? Would it not make sense for advanced civilizations to establish these reservations at the interface boundaries between large empires, as a buffer zone, so to speak ? (It would be clear to them that expanding further and consuming the tiny buffer zone would not do them much good, because they would not gain all that much more real estate, and they might also risk an intergalactic war by intruding on their neighbor's territory.)

BTW, the hypothesis above was cooked up sometime in fall '93 after a nanotechnology discussion meeting called "The Assembler Multitude", in Palo Alto, CA, and has been tentatively labelled "KOK Hypothesis". KOK stands for Kaehler-Oh-Krummenacker, the names of the three hackers who have conceived this theory, which will force all cosmology text books to be rewritten. :-) :-) (To be taken with a grain of salt, at least until the aliens have been contacted for confirmation of the hypothesis.)

Now I would like to have all the real physicists and cosmologists out there jump on it, and tear apart this hypothesis by pointing out all the fatal flaws. :-)

Have fun...

Many Greetings
Markus Krummenacker

(I was originally planning to give a link to Google groups, but the above message doesn't seem to be there.)

Come to think of it, the SIA indicates we are more likely to be in a universe with lots of extraterrestrials.

### Alms for the Poor?

The proposed health-insurance regulation bill has something for everyone:

Much of the money in the bill would be devoted to subsidies to help families at incomes of up to $88,000 a year pay their premiums. I finally know how much New York Times reporters make. Addendum: I just recalled the following essential quote: Blimey, this redistribution of wealth is trickier than I thought. ### Saturday, March 20, 2010 ### Puberty vs. Learning There's evidence that puberty can interfere with learning. I suppose that means, in addition to “up-tight” parents who don't want their kids fooling around, yuppie parents who want their kids in Ivy-League schools will also try to raise the age of puberty (once this becomes possible). After a generation of that, behavior that's commoner among people with earlier puberty may seem low class. We can expect “liberated” behavior to be as disreputable as smoking is today. ### Thursday, March 18, 2010 ### What We're Complaining About Jorge Chaim (of PhD Comics) finds out what us anti-elitist wingnuts are complaining about. ### Monday, March 15, 2010 ### Do Increasing Subsidies Lower Costs? Evidence they don't. ### Are Politicians Bonkers? Non-fiction answer: Yes. Science-fiction answer: Yes. ### Sunday, March 14, 2010 ### It's 3.14 Do you want your π apple, cherry, or pumpkin? ### Scenery for a Science-Fiction Movie When they film a movie based on A Gift from Earth by Larry Niven, it will be filmed at Mt. Roraima, the real-life Mt. Lookitthat. Addendum: I'm not the first to notice the resemblance. ### Saturday, March 13, 2010 ### Attention: Charles Johnson It's harder to criticize the anti-Darwin right when you're allied with that part of the left able to win Darwin Awards. Darwin Awards are explained here and why part of the left might win them explained in the tag line here. (Please note that this attitude is not limited to Catholics.) ### Wednesday, March 10, 2010 ### Two Quotes on Testing From Hard Facts (quoted at Overcoming Bias): The evidence strongly suggests that students learn better when they are not graded and certainly not when they are graded on a curve. From Homeopathy vs Science - a Metaphor: When it's not being tested, it works, fact. I was reminded somehow. ### Monday, March 08, 2010 ### What Lenin Should Have Said I'm sure you've heard of the Leninist slogan “You can't make an omelet without breaking a few eggs.” Following a recent mishap in my kitchen, I thought of a variant “You can't make an utterly disgusting mess on the floor without breaking a few eggs.” ### Sunday, March 07, 2010 ### If Free Will Is an Illusion If free will is an illusion, according to Anthony Cashmore: Perhaps the most obvious impact of this paradigm shift will be on our judicial system, in which the notions of free will and responsibility form an integral component. Currently, in order to be found guilty, a criminal must be considered responsible for his actions; otherwise, he can be found not guilty by reason of insanity. Cashmore disagrees with these rules, noting that psychiatric research is finding its way more and more into the courts and causing time-wasting debates. (For example, is alcoholism a disease? Are sex crimes an addiction?) Okay. That means we can't hold judges responsible for ignoring psychiatrists. On the other hand, we also can't hold Anthony Cashmore responsible for ignoring Bryan Caplan's defense of free will (which shows that Dr. Cashmore has no empirical proof of his beliefs). We also can't hold him responsible for ignoring the implications of the fact that there are real numbers that are neither recursive nor random (which shows that there is at least one system in which not everything is either definable or random which means Dr. Cashmore has no logical proof of his beliefs). ### Another Narnia While looking at reactions to the essay asking why there is no Jewish Narnia (earlier discussed here, I noticed that may of the commenters seemed to think the question was about the supposed lack of Jewish fantasy writers. If we look at the best-known work by the best-known Jewish sf/fantasy writer (the Foundation series by Isaac Asimov), we see it's about someone who predicts and produces a movement to set up a revived analog of the Roman Empire with some partial success when last heard from. In other words, it's Islam with the serial numbers filed off just as Narnia is Christianity with the serial numbers filed off. (I have not forgotten that Al-Qaeda is Arabic for Foundation.) ### Saturday, March 06, 2010 ### This Was Predictable According to The New York Times, Creationists are using the growing criticism of the anthropogenic global warming hypothesis to push their theories: Critics of the teaching of evolution in the nation’s classrooms are gaining ground in some states by linking the issue to global warming, arguing that dissenting views on both scientific subjects should be taught in public schools. ……… Yet they are also capitalizing on rising public resistance in some quarters to accepting the science of global warming, particularly among political conservatives who oppose efforts to rein in emissions of greenhouse gases. It looks like they had the same reaction as the people at the next table I blogged about a few years ago: “I believe science teaches us that human-caused global warming is an urgent crisis.” “You mean if it's either not a problem or can be fixed easily, it proves science is false?” When the evidence is inadequate, maybe it makes sense to admit it. #### On the other hand … If anthropogenic global warming does turn out to be important, my fellow wingnuts should be prepared for a reaction in other direction. We should try to not tie ourselves to an uncertain dogma. ### Friday, March 05, 2010 ### The Jewish Narnia? At The Jewish Review of Books, Michael Weingrad asks why there is no Jewish Narnia. Let's see … Narnia was basically the story of Jesus Christ with the serial numbers filed off. I suppose the Jewish equivalent would be the story of Moses with the serial numbers filed off. The story is about a great leader who led his people from slavery to freedom but was unable to enter the Promised Land. The first example that comes to mind is They Shall Have Stars by James Blish (part of the Cities in Flight series). There are two problems: 1) James Blish wasn't Jewish. 2) There were other analogues to Jews (the Hamiltonians) later in the series. If we don't insist on the “unable to enter the Promised Land” part, there are other examples, such as Watership Down by Richard Adams or Falling Free by Lois McMaster Bujold. ### Wednesday, March 03, 2010 ### Parody and Reality This is a parody from Woody Allen (back when he was funny): Venal & Sons has at last published the long-awaited first volume of Metterling’s laundry lists (The Collected Laundry Lists of Hans Metterling, Vol. I, 437 pp., plus xxxii-page introduction; indexed;$18.75), with an erudite commentary by the noted Metterling scholar Gunther Eisenbud. The decision to publish this work separately, before the completion of the immense four-volume oeuvre, is both welcome and intelligent, for this obdurate and sparkling book will instantly lay to rest the unpleasant rumors that Venal & Sons, having reaped rich rewards from the Metterling novels, play, and notebooks, diaries, and letters, was merely in search of continued profits from the same lode.
On the other hand, this is apparently intended seriously:

In the words of the American abstract artist, Charles Green Shaw: "Real happiness consists in not what we actually accomplish, but what we think we accomplish."

He is one of the artists featured in an exhibition currently running at the Archives of American Art in Washington.

It is a taxonomist's dream: hundreds of lists drawn up by some of the world's greatest modern artists. Some are scribbled on scraps of paper, while others are elaborately illustrated.

There are lists of ideas, lists of instructions, lists of ambitions, of biographical details, of paintings, of things to do, of colours.

A few years ago, I theorized that reality had been replaced by a very extended episode of Monty Python. I'll have to add Woody Allen to the theory.

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# Current density

1. Jun 30, 2015

### Shreyas Samudra

What is current density

2. Jul 1, 2015

### meBigGuy

Maybe you could google the term first and then maybe ask real questions.

3. Jul 1, 2015

### EverGreen1231

Basically, it represents the amount of charge flowing through a given area. Generally denoted as J with units of A/m2

Last edited: Jul 1, 2015

format: plain


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## FANDOM

167 Pages

In Asteroseismology, the oscillations of a star are observed to study its interior. In this entry we present two example for such inferences. The oscillation spectrum of stars is often harmonic, which means that peaks in the spectrum occur at regular intervals in the frequency $\Delta \nu$. The only relevant time scale in the problem is the time it takes a sound wave to travel from the center of the star to the surface, and from dimensional analysis

$\Delta \nu \propto \left( \int_0^R \frac{dr}{c_s} \right)^{-1} \propto \sqrt{G \bar{\rho}}$

where $\bar{\rho} \propto M/R^3$ is the mean density.

The series of harmonics will continue in higher and higher frequencies, until the wavelength becomes comparable with the scale height

$\nu_{\max} \propto \frac{c_s}{\lambda} \propto \frac{c_s}{H} \propto \frac{c_s}{c_s^2/g} \propto \frac{g}{c_s}$

The speed of sound at the surface can usually be estimated from the temperature and the composition of the atmosphere. So from frequency we can infer the gravitational acceleration on the stellar surface $g = G M / R^2$. When the wavelength becomes smaller than the scale height, the waves will "leak" out of the star rather then being reflected and amplified by constructive interference.


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Now showing items 1-1 of 1

• #### $\phi$ meson production in Pb-Pb collisions at $\sqrt s_{NN}$=5.02 TeV with ALICE at the LHC

(Elsevier, 2017-11)
Strangeness production is a key tool to understand the properties of the medium formed in heavy-ion collisions: an enhanced production of strange particles was early proposed as one of the signatures of the Quark-Gluon ...
Journal article


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# Prime Number of Set Bits in Binary Representation in C++

C++Server Side ProgrammingProgramming

In this problem, we are given two integers L and R. Our task to print the total numbers that have set bits counting to a prime number that is in between L to R.

Let’s take an example to understand the problem

Input: L = 7, R = 12
Output: 6
Explanation:
7 -> 111 , set bits = 2, prime number.
8 -> 1000 , set bits = 1, not prime number.
9 -> 1001 , set bits = 2, prime number
10 -> 1010 , set bits = 2, prime number
11 -> 1011, set bits = 3, prime number
12 -> 1100, set bits = 2, prime number

To solve this problem, we will traverse each and every element within the range. And check the total number of set bits in the number, for this we will use a predefined function in CPP _builtin_popcount(). Then, we will check the set bits of the number for prime. If yes increase count otherwise not.

Program to show the implementation of our solution

## Example

Live Demo

#include <iostream>
using namespace std;
if (n <= 1) return false;
if (n <= 3) return true;
if (n%2 == 0 || n%3 == 0) return false;
for (int i=5; i*i<=n; i=i+6)
if (n%i == 0 || n%(i+2) == 0)
return false;
return true;
}
void printPrimeSetBits(int l, int r) {
int tot_bit, count = 0;
for (int i = l; i <= r; i++) {
tot_bit = __builtin_popcount(i);
count++;
}
cout<<count;
}
int main() {
int L = 7, R = 13;
cout<<"Total numbers with prime set bits between "<<L<<" and "<<R<<" are : ";
printPrimeSetBits(L, R);
return 0;
}

## Output

Total numbers with prime set bits between 7 and 13 are : 6
Published on 03-Feb-2020 10:54:47


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Inscribed Ellipse in a Parallelogram

Geometry Level 3

Parallelogram $$ABCD$$ with $$AB = 10, BC =16, \angle ABC = 60^{\circ}$$ has an inscribed ellipse such that the tangent point $$E$$ satisfies $$CE = \frac{3}{4} CD$$.

Find the length of the semi-major axis of this ellipse.

×


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# Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.

## Definition

In general, if is a multiplicative function, then the Dirichlet series

is equal to

where the product is taken over prime numbers , and is the sum

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes .

An important special case is that in which is totally multiplicative, so that is a geometric series. Then

as is the case for the Riemann zeta-function, where , and more generally for Dirichlet characters.

## Convergence

In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region

Re(s) > C

that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.

In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here. The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GLm.

## Examples

The Euler product attached to the Riemann zeta function , using also the sum of the geometric series, is

.

while for the Liouville function , it is,

Using their reciprocals, two Euler products for the Möbius function are,

and,

and taking the ratio of these two gives,

Since for even s the Riemann zeta function has an analytic expression in terms of a rational multiple of , then for even exponents, this infinite product evaluates to a rational number. For example, since , , and , then,

and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to,

where counts the number of distinct prime factors of n and the number of square-free divisors.

If is a Dirichlet character of conductor , so that is totally multiplicative and only depends on n modulo N, and if n is not coprime to N then,

.

Here it is convenient to omit the primes p dividing the conductor N from the product. Ramanujan in his notebooks tried to generalize the Euler product for Zeta function in the form:

for where is the polylogarithm. For the product above is just

## Notable constants

Many well known constants have Euler product expansions.

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios

where each numerator is a prime number and each denominator is the nearest multiple of four.[1]

Other Euler products for known constants include:

Murata's constant (sequence A065485 in the OEIS):

Strongly carefree constant :

Landau's totient constant :

Carefree constant :

(with reciprocal) :

Feller-Tornier constant :

Quadratic class number constant :

Totient summatory constant :

Sarnak's constant :

Carefree constant :

Strongly carefree constant :

Stephens' constant :

Barban's constant :

Taniguchi's constant :

## Notes

1. Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.

## References

• G. Polya, Induction and Analogy in Mathematics Volume 1 Princeton University Press (1954) L.C. Card 53-6388 (A very accessible English translation of Euler's memoir regarding this "Most Extraordinary Law of the Numbers" appears starting on page 91)
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 (Provides an introductory discussion of the Euler product in the context of classical number theory.)
• G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th ed., Oxford (1979) ISBN 0-19-853171-0 (Chapter 17 gives further examples.)
• George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part I, Springer (2005), ISBN 0-387-25529-X
• G. Niklasch, Some number theoretical constants: 1000-digit values"


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Journal cover Journal topic
Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
Journal topic
Hydrol. Earth Syst. Sci., 22, 4921-4934, 2018
https://doi.org/10.5194/hess-22-4921-2018
Hydrol. Earth Syst. Sci., 22, 4921-4934, 2018
https://doi.org/10.5194/hess-22-4921-2018

Research article 21 Sep 2018

Research article | 21 Sep 2018

# Inflation method for ensemble Kalman filter in soil hydrology

Inflation method for the ensemble Kalman filter in soil hydrology
Hannes H. Bauser1,2, Daniel Berg1,2, Ole Klein3, and Kurt Roth1,3 Hannes H. Bauser et al.
• 1Institute of Environmental Physics (IUP), Heidelberg University, Heidelberg, Germany
• 2HGS MathComp, Heidelberg University, Heidelberg, Germany
• 3Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany
Abstract

The ensemble Kalman filter (EnKF) is a popular data assimilation method in soil hydrology. In this context, it is used to estimate states and parameters simultaneously. Due to unrepresented model errors and a limited ensemble size, state and parameter uncertainties can become too small during assimilation. Inflation methods are capable of increasing state uncertainties, but typically struggle with soil hydrologic applications. We propose a multiplicative inflation method specifically designed for the needs in soil hydrology. It employs a Kalman filter within the EnKF to estimate inflation factors based on the difference between measurements and mean forecast state within the EnKF. We demonstrate its capabilities on a small soil hydrologic test case. The method is capable of adjusting inflation factors to spatiotemporally varying model errors. It successfully transfers the inflation to parameters in the augmented state, which leads to an improved estimation.

1 Introduction

Data assimilation combines information from models and measurements into an optimal estimate of a geophysical field of interest (Reichle2008). It has applications in all branches of the geosciences, with weather forecasting as the driving force behind many recent advances . The advantage of data assimilation methods (in contrast to, e.g., inverse modeling) is the possibility of considering model errors, which are characteristic of geophysical systems.

The ensemble Kalman filter (EnKF) is a popular data assimilation method due to its simple conceptional formulation and ease of implementation (Evensen2003). It is an extension of the Kalman filter (Kalman1960) for nonlinear models.

In hydrology, the EnKF was used for soil moisture estimation from satellite data (e.g., Reichle et al.2002; Crow and Van Loon2006) or from local measurements (e.g., De Lannoy et al.2007, 2009; Camporese et al.2009). However, the largest uncertainties in hydrology are associated with soil hydraulic material properties. They can neither be measured directly nor can they be transferred from the lab to the field, and are typically parameterized. Thus, including material properties in the estimation can be crucial in hydrology. called for an integrated assimilation framework including not only states, but also parameters and even model structure.

The joint estimation of states and parameters in data assimilation might be one possibility to reduce the influence of model errors on parameter estimation . Such a joint estimation in the EnKF with an augmented state was already demonstrated by for an atmospheric model. In hydrology combined an EnKF and the shuffled complex evolution Metropolis algorithm, while used a dual EnKF approach to estimate states and parameters for a rainfall–runoff model. The joint assimilation of states and parameters in an augmented state was successfully performed for example in groundwater research (e.g., Chen and Zhang2006; Hendricks Franssen and Kinzelbach2008; Kurtz et al.2012, 2014; Erdal and Cirpka2016), but also in soil hydrology for land surface models (e.g., Bateni and Entekhabi2012; Han et al.2014; Zhang et al.2017) and on smaller scales based on the Richards equation (e.g., Li and Ren2011; Wu and Margulis2011, 2013; Song et al.2014; Erdal et al.2014, 2015; Shi et al.2015; Bauser et al.2016; Brandhorst et al.2017; Botto et al.2018).

Due to unrepresented model errors and due to a limited ensemble size, the EnKF underestimates model errors, which can lead to filter inbreeding. Systematic model errors are common for example in land surface models . Additionally, in soil hydrology spatially and temporally varying model errors occur due to un- or ill-represented processes like preferential flow or hysteresis. Underestimated errors cause an insufficient ensemble spread in the augmented state. This is especially severe for parameters, which are typically not changed through a forward propagation and consequently cannot increase their uncertainty again. Due to the convergent dynamics in soil hydrology, the uncertainty in the state depends strongly on the parameter spread and becomes too small as well.

Covariance inflation can counteract filter inbreeding. Different methods have been proposed: (i) additive inflation, which adds a model error after the forward propagation. This method is especially useful if prior knowledge about the model error exists. In atmospheric sciences additive inflation has been successfully applied by, e.g., using reanalysis of historical weather prediction errors . (ii) Relaxation methods, which relax the analysis back to a prior perturbation or spread, have been proposed with tuning parameters or based on deviations to measurements . (iii) Multiplicative covariance inflation, which inflates the complete state with a scalar factor, where the inflation factor is either chosen manually or is estimated based on deviations from measurements (e.g., Wang and Bishop2003; Anderson2007; Li et al.2009). This method has been further extended to inflate each state component individually (Anderson2009).

All these inflation methods are developed in an atmospheric sciences context. Their transfer to soil hydrology is limited, due to the spatiotemporally varying model errors and the typically employed augmented state. For groundwater research, reported improved results by employing the inflation method by , and used the constant inflation by . In soil hydrology, however, adjusted methods have been used: for example, and apply a special case of the inflation method by and keep the parameter spread constant to ensure a sufficient ensemble spread. used the method by , but adjusted the inflation of parameters.

Alternatively, no inflation method is reported (e.g., Li and Ren2011; Shi et al.2015), but instead a damping factor , which can alleviate the issue, is employed. This is done by, e.g., , , , , and , where and combined this method with additive inflation.

In this paper, we propose a novel multiplicative inflation method, specifically designed for the needs in the soil hydrology community. The inflation method can vary rapidly in space and time to cope with the typically varying model errors and it is capable of a transfer of the inflation in the state to the parameters in the augmented state. The remainder of this paper is organized as follows: Sect. 2 describes (i) the EnKF, (ii) our proposed inflation method and (iii) a soil hydrologic test case. Section 3 shows the results of our method applied to the test case, followed by discussion and conclusion in Sects. 4 and 5.

2 Method

## 2.1 Ensemble Kalman filter

The EnKF is the Monte Carlo extension of the Kalman filter (Kalman1960) for nonlinear models and assumes unbiased Gaussian error distributions to combine model and measurement information. The filter is a sequential method and alternates between a forecast step and an analysis step. The forecast propagates a state including its uncertainty forward in time. The analysis combines uncertain model information with uncertain measurements at this time into an optimal estimate of the state. These two steps are now explained in more detail.

The forecast propagates an ensemble of states φn forward from time k−1 to time k with a model M,

$\begin{array}{}\text{(1)}& {\mathbit{\phi }}_{k}^{\mathrm{f},n}=M\left({\mathbit{\phi }}_{k-\mathrm{1}}^{\mathrm{a},n}\right)+{\mathbit{\beta }}^{n},\end{array}$

where the superscripts “f” and “a” denote forecast and analysis respectively, while n denotes the ensemble members with n=1, …, N. The uncertainty in the state is directly represented through the ensemble ${\mathbit{\phi }}_{k-\mathrm{1}}^{\mathrm{a},n}$ and then propagated nonlinearly with the model. Unrepresented model errors can be added through the unbiased Gaussian process noise β. This is also called additive inflation. However, the details of the model error are typically unknown and thus not represented adequately. The propagated uncertainties are directly represented through the new forecast ensemble ${\mathbit{\phi }}_{k}^{\mathrm{f},n}$.

The state can be extended by, e.g., model parameters ϕ to an augmented state u=[φ, ϕ]. This requires a forecast for each augmented state component. Parameters are typically assumed to be constant in time:

$\begin{array}{}\text{(2)}& {\mathbit{\varphi }}_{k}^{\mathrm{f},n}={\mathbit{\varphi }}_{k-\mathrm{1}}^{\mathrm{a},n}.\end{array}$

The forecast of the state ${\mathbit{\phi }}_{k}^{\mathrm{f},n}$ now also depends on the corresponding parameter set ${\mathbit{\varphi }}_{k-\mathrm{1}}^{\mathrm{a},n}$. This way, uncertainties in the parameters are propagated as well and can be reduced jointly in the analysis.

Assuming unbiased Gaussian distributions, the ensemble of augmented states is characterized through the forecast error covariance matrix Pf,

$\begin{array}{}\text{(3)}& {\mathbf{P}}_{k}^{\mathrm{f}}=\stackrel{\mathrm{‾}}{\left[{\mathbit{u}}_{k}^{\mathrm{f},n}-\stackrel{\mathrm{‾}}{{\mathbit{u}}_{k}^{\mathrm{f}}}\right]{\left[{\mathbit{u}}_{k}^{\mathrm{f},n}-\stackrel{\mathrm{‾}}{{\mathbit{u}}_{k}^{\mathrm{f}}}\right]}^{T}},\end{array}$

where the overline denotes the expectation value and $\stackrel{\mathrm{‾}}{{\mathbit{u}}_{k}^{\mathrm{f}}}$ is the ensemble mean.

The analysis combines model and measurement information based on the Gaussian error assumption. The measurement error covariance matrix R of the measurements d is defined analogously as

$\begin{array}{}\text{(4)}& {\mathbf{R}}_{k}=\stackrel{\mathrm{‾}}{\left[{\mathbit{ϵ}}_{k}^{n}\right]{\left[{\mathbit{ϵ}}_{k}^{n}\right]}^{T}},\end{array}$

where ϵ is the measurement error. The measurements are linked to the state through the linear measurement operator H, which maps from the state space to the measurement space:

$\begin{array}{}\text{(5)}& {\mathbit{d}}_{k}={\mathbf{H}}_{k}{\mathbit{u}}_{\mathrm{true},k}+{\mathbit{ϵ}}_{k}.\end{array}$

The Kalman gain K weighs the forecast error covariance matrix with the measurement error covariance matrix and maps from the measurement space back to the state space, based on the covariances in the forecast error covariance matrix:

$\begin{array}{}\text{(6)}& {\mathbf{K}}_{k}={\mathbf{P}}_{k}^{\mathrm{f}}{\mathbf{H}}_{k}^{T}{\left[{\mathbf{H}}_{k}{\mathbf{P}}_{k}^{\mathrm{f}}{\mathbf{H}}_{k}^{T}+{\mathbf{R}}_{k}\right]}^{-\mathrm{1}}.\end{array}$

Based on the measurements, the Kalman gain updates the forecast ensemble to the analysis ensemble:

$\begin{array}{}\text{(7)}& {\mathbit{u}}_{k}^{\mathrm{a},n}={\mathbit{u}}_{k}^{\mathrm{f},n}+{\mathbf{K}}_{k}\left[{\mathbit{d}}_{k}+{\mathbit{ϵ}}_{k}^{n}-{\mathbf{H}}_{k}{\mathbit{u}}_{k}^{\mathrm{f},n}\right].\end{array}$

This update to the ensemble ${\mathbit{u}}_{k}^{\mathrm{a},n}$ minimizes the analysis error covariance ${\mathbf{P}}_{k}^{\mathrm{a}}$, which fulfills

$\begin{array}{}\text{(8)}& {\mathbf{P}}_{k}^{\mathrm{a}}=\left[\mathbf{I}-{\mathbf{K}}_{k}{\mathbf{H}}_{k}\right]{\mathbf{P}}_{k}^{\mathrm{f}},\end{array}$

for infinite ensemble sizes.

Through spurious correlations and non-Gaussian distributions, ${\mathbf{P}}_{k}^{\mathrm{a}}$ will become too small, which can lead to filter inbreeding and ultimately filter divergence (e.g., Hamill et al.2001). This is intensified if the model error required in Eq. (1) is unknown.

A common way to alleviate this issue in hydrology is the use of a damping factor $\mathit{\gamma }\in \left[\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\right]$ , which is multiplied to the correction vector in Eq. (7) and consequently lessens the uncertainty reduction. The damping factor can be extended to a vector γ (and an entrywise multiplication) to treat augmented state components differently . Typically, parameters are multiplied by a smaller factor than the state. However, the damping factor can only alleviate and not completely prevent the inbreeding problem.

## 2.2 Multiplicative inflation for soil hydrology

Multiplicative inflation is another heuristic way to avoid filter inbreeding. proposed to increase the distance of each ensemble member to the ensemble mean by multiplying this distance by $\sqrt{\mathit{\lambda }}$ for the inflation factor λ≥1:

$\begin{array}{}\text{(9)}& {\mathbit{u}}_{\mathrm{inf}}^{\mathrm{f},n}=\sqrt{\mathit{\lambda }}\left({\mathbit{u}}^{\mathrm{f},n}-\stackrel{\mathrm{‾}}{{\mathbit{u}}^{\mathrm{f}}}\right)+\stackrel{\mathrm{‾}}{{\mathbit{u}}^{\mathrm{f}}}.\end{array}$

This inflation factor is applied to the complete augmented state and has to be adjusted to the specific problem. By construction, it does not alter the mean value: $\stackrel{\mathrm{‾}}{{\mathbit{u}}_{\mathrm{inf}}^{\mathrm{f}}}=\stackrel{\mathrm{‾}}{{\mathbit{u}}^{\mathrm{f}}}$. A temporally varying inflation factor can be estimated by comparing uncertainties with the distance of measurement and forecast (e.g., Wang and Bishop2003; Anderson2007; Li et al.2009). A spatiotemporally adaptive inflation has been achieved by estimating a vector λ for the complete augmented state (Anderson2009). The author uses the correlation between measurements and augmented state dimensions and asks this question: How much inflation is required in each dimension to explain the observed differences to the measurements? showed that this works excellently for the actual state. However, we experienced possible over-inflation in parameters (which do not have any dynamics to compensate for this), which can lead to filter collapses.

We propose a more conservative inflation method and ask this question: How much of the required change in the inflation are we allowed to transfer to the state dimensions based on the correlation information? This can be achieved by applying a Kalman filter for the inflation within the EnKF.

In this Kalman filter, the inflation vector is treated as the state variable. As for parameters, we choose a constant model for the forecast in time:

$\begin{array}{}\text{(10)}& {\mathbit{\lambda }}_{k}^{\mathrm{f}}={\mathbit{\lambda }}_{k-\mathrm{1}}^{\mathrm{a}}.\end{array}$

For convenience we will drop the time subscript k in the following. Furthermore, we will use the same symbols as for the EnKF, but denote them with the subscript λ. We approximate the forecast error covariance matrix for lambda, ${\mathbf{P}}_{\mathit{\lambda }}^{\mathrm{f}}$, based on the covariance matrix of the augmented state in the EnKF, Pf, as the normalized absolute correlation matrix of the augmented state ensemble. The matrix component ij is determined as

$\begin{array}{}\text{(11)}& {\left({\mathbf{P}}_{\mathit{\lambda }}^{\mathrm{f}}\right)}_{ij}={\mathit{\sigma }}_{\mathit{\lambda }}^{\mathrm{2}}\left|{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{ij}\right|{\left[{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{ii}{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{jj}\right]}^{-\frac{\mathrm{1}}{\mathrm{2}}},\end{array}$

where σλ denotes the uncertainty of the inflation factors. It is a tuning parameter that is kept constant over time and is assigned to all state dimensions. It influences how fast the inflation factors are adjusted. This follows the idea by to avoid a closure problem, where the inflation estimation would require its own inflation. Instead, the uncertainty is kept constant. Furthermore, only the absolute value of the correlation is considered, since the inflation is based on differences between measurement and model, but ignores their direction. Note, that this presumes that the correlations of the model state can be transferred to the inflation. In the presence of unknown model errors this assumption may or may not be correct. However, the estimation at measurement locations will remain meaningful in any case.

For the analysis, the distance dλ between mean forecast and measurement is used as measurement for λ:

$\begin{array}{}\text{(12)}& {\mathbit{d}}_{\mathit{\lambda }}=\left|\mathbit{d}-\mathbf{H}\stackrel{\mathrm{‾}}{{\mathbit{u}}_{\mathrm{inf}}^{\mathrm{f}}}\right|.\end{array}$

The measurement error covariance matrix Rλ of dλ can be calculated based on the error covariance matrices of d and $\mathbf{H}\stackrel{\mathrm{‾}}{{\mathbit{u}}_{\mathrm{inf}}^{\mathrm{f}}}$,

$\begin{array}{}\text{(13)}& {\left({\mathbf{R}}_{\mathit{\lambda }}\right)}_{ij}=\left|\left(\mathbf{R}{\right)}_{ij}+{\left({\mathbf{HP}}_{\mathrm{inf}}^{\mathrm{f}}{\mathbf{H}}^{T}\right)}_{ij}\right|,\end{array}$

where the inflated forecast error covariance matrix ${\mathbf{P}}_{\mathrm{inf}}^{\mathrm{f}}$ can be calculated from the inflation vector and the forecast error covariance matrix by combining Eq. (9) (with vector λ and entrywise multiplication) and Eq. (3): ${\mathbf{P}}_{\mathrm{inf}}^{\mathrm{f}}={\mathbf{P}}^{\mathrm{f}}\circ \left[\sqrt{{\mathbit{\lambda }}^{\mathrm{f}}}{\sqrt{{\mathbit{\lambda }}^{\mathrm{f}}}}^{T}\right]$. The entrywise product is denoted by  and the entrywise square root of λ by $\sqrt{\mathbit{\lambda }}$.

The expected distance between measurement and mean forecast based on the current inflation is

$\begin{array}{}\text{(14)}& {\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}={\left[{\left({\mathbf{R}}_{\mathit{\lambda }}\right)}_{ii}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}},\end{array}$

which combines the uncertainties of d and $\mathbf{H}\stackrel{\mathrm{‾}}{{\mathbit{u}}_{\mathrm{inf}}^{\mathrm{f}}}$. To be able to determine the Kalman gain, we first calculate the Jacobian matrix Hλ of partial derivatives of hλ with respect to λ:

$\begin{array}{ll}{\left({\mathbf{H}}_{\mathit{\lambda }}\right)}_{ij}& =\frac{\partial }{\partial {\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i},\\ & ={\left[\mathrm{2}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}\right]}^{-\mathrm{1}}\\ \text{(15)}& & \sum _{m}\left(\mathbf{H}{\right)}_{ij}\left(\mathbf{H}{\right)}_{im}{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{jm}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}.\end{array}$

With this approximated measurement operator Hλ, the Kalman gain Kλ and the analysis state λa are obtained as

$\begin{array}{}\text{(16)}& & {\mathbf{K}}_{\mathit{\lambda }}={\mathbf{P}}_{\mathit{\lambda }}^{\mathrm{f}}{\mathbf{H}}_{\mathit{\lambda }}^{T}{\left[{\mathbf{H}}_{\mathit{\lambda }}{\mathbf{P}}_{\mathit{\lambda }}^{\mathrm{f}}{\mathbf{H}}_{\mathit{\lambda }}^{T}+{\mathbf{R}}_{\mathit{\lambda }}\right]}^{-\mathrm{1}},\text{(17)}& & {\mathbit{\lambda }}^{\mathrm{a}}={\mathbit{\lambda }}^{\mathrm{f}}+{\mathbf{K}}_{\mathit{\lambda }}\left[{\mathbit{d}}_{\mathit{\lambda }}-{\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right].\end{array}$

Note, that the matrices ${\mathbf{P}}_{\mathit{\lambda }}^{\mathrm{f}}$ and Rλ can possibly become indefinite, due to the absolute value in Eqs. (11) and (13). Consequently, the inverse in Eq. (16) could become unfeasible. However, we never encountered such a case. In a situation with uncorrelated measurements, the issue can be resolved by reducing σλ just for that single time step.

With this Kalman filter, the inflation vector is updated at each time step based on the difference of the mean forecast to the measurements. Following , we additionally prohibit a deflation by constraining the inflation values to (λ)i≥1.

## 2.3 Model

We test the proposed inflation method on a small hydrologic test case. We constructed it specifically to require a strong inflation. This makes it possible to explore features of the inflation in detail on a rather short timescale. Due to a small ensemble size, the results vary depending on the seed of the random numbers. This however, is related to different performance of the EnKF itself. In simulations (results are not shown), we found that the behavior of the inflation remains consistent. We have also tested the inflation method with real-world data by reanalyzing the application by with the main result shown in Appendix B.

The Richards equation describes the change of volumetric soil water content θ (–) in a continuous porous medium,

$\begin{array}{}\text{(18)}& \frac{\partial \mathit{\theta }}{\partial t}-\mathrm{\nabla }\cdot \left[K\left(\mathit{\theta }\right)\left[\mathrm{\nabla }{h}_{\mathrm{m}}\left(\mathit{\theta }\right)-\mathrm{1}\right]\right]=\mathrm{0},\end{array}$

where K (L T−1) is the isotropic conductivity and hm (L) is the matric head. Both are related to the water content. This relation is typically described through parameterized material properties. We choose the Mualem–van Genuchten parameterization ,

$\begin{array}{}\text{(19)}& & K\left(\mathrm{\Theta }\right)={K}_{\mathrm{0}}{\mathrm{\Theta }}^{\mathit{\tau }}{\left[\mathrm{1}-{\left[\mathrm{1}-{\mathrm{\Theta }}^{n/\left[n-\mathrm{1}\right]}\right]}^{\mathrm{1}-\mathrm{1}/n}\right]}^{\mathrm{2}},\text{(20)}& & {h}_{\mathrm{m}}\left(\mathrm{\Theta }\right)=\frac{\mathrm{1}}{\mathit{\alpha }}{\left[{\mathrm{\Theta }}^{-n/\left[n-\mathrm{1}\right]}-\mathrm{1}\right]}^{\mathrm{1}/n},\end{array}$

with the saturation Θ (–),

$\begin{array}{}\text{(21)}& \mathrm{\Theta }:=\frac{\mathit{\theta }-{\mathit{\theta }}_{\mathrm{r}}}{{\mathit{\theta }}_{\mathrm{s}}-{\mathit{\theta }}_{\mathrm{r}}}.\end{array}$

The parameterization is described by a set of six parameters: θs (–), θr (–), α (L−1), n (–), K0 (L T−1), and τ (–).

We additionally consider small-scale heterogeneity through Miller scaling. It assumes geometrical similarity. With this the microscopic geometry of the pore space at a macroscopic position is parameterized by a single length scale ξ and the macroscopic heterogeneity field can be generated with a single scalar field of this length scale. showed that the hydraulic functions scale with this parameter according to

$\begin{array}{}\text{(22)}& & K\left(\mathit{\theta }\right)={K}^{*}\left(\mathit{\theta }\right)\cdot {\mathit{\xi }}^{\mathrm{2}},\text{(23)}& & {h}_{\mathrm{m}}\left(\mathit{\theta }\right)={h}_{\mathrm{m}}^{*}\left(\mathit{\theta }\right)\cdot \frac{\mathrm{1}}{\mathit{\xi }},\end{array}$

where the functions K(θ) and hm(θ) are defined at a reference point * with Miller scaling parameter ξ=1 and from there are projected to all locations.

For the test, we choose a one-dimensional case with a depth of 50 cm for a time of 6 days. We set a groundwater table as the lower boundary condition throughout the whole time and start from equilibrium conditions. The upper boundary condition is no flux, except for a rain event with $\mathrm{2.0}×{\mathrm{10}}^{-\mathrm{7}}$ (m s−1) during the fourth day. As observations we choose two water content measurements at depths of 9.5 and 19.5 cm as they would be available from time domain reflectometry (TDR). We set the measurement uncertainty to a standard deviation of 0.007 (e.g., Jaumann and Roth2017).

As material we choose sandy loam from : θs=0.41, θr=0.065, $\mathit{\alpha }=-\mathrm{7.5}$ m−1, n=1.89, ${K}_{\mathrm{0}}=\mathrm{1.23}×{\mathrm{10}}^{-\mathrm{5}}$ m s−1, and τ=0.5. For the Miller scaling we choose ξ1=0.32 at the upper measurement position and ξ2=3.2 at the lower measurement position. We reduce the description of the heterogeneity to these two parameters. The full function of the scaling factors is calculated by linearly interpolating between the measurement positions and constantly extrapolating to the boundaries.

The forward simulations are performed using MuPhi with a spatial resolution of 1 cm. This corresponds to a state with 50 dimensions.

To test the inflation method, we perform a perfect model experiment. With the EnKF we estimate the water content state and four parameters (ξ1, ξ2, K0, and τ) through the augmented state u=[θ, log10(ξ1), log10(ξ2), log10(K0),τ]. We choose to include the logarithm of ξ1, ξ2, and K0, because we expect a more linear relation to the water content state than for the actual parameters. For the water content state, we use the correct initial condition as the mean with an uncertainty of 0.005. The uncertainty is spatially correlated using the fifth-order piecewise rational function by with the length scale c=5 cm. As an initial guess for the parameters, we start with unknown heterogeneity ${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{1}}\right)={\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{2}}\right)=\mathrm{0.0}±\mathrm{0.25}$, corresponding to 2 SD (standard deviations) away from the true values of ${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{1}}\right)=-\mathrm{0.5}$ and log10(ξ2)=0.5. For the saturated hydraulic conductivity, we choose a too small value of ${\mathrm{log}}_{\mathrm{10}}\left({K}_{\mathrm{0}}\right)=-\mathrm{5.5}±\mathrm{0.5}$, K0 in (m s−1), which is about 1 SD away from the true value of ${\mathrm{log}}_{\mathrm{10}}\left({K}_{\mathrm{0}}\right)=-\mathrm{4.9}$. For the tortuosity $\mathit{\tau }=\mathrm{0.5}±\mathrm{0.5}$ we start from the true value.

Through the unrepresented heterogeneity, we can mimic a model error, leading to a bias towards smaller values for the estimation of K0 during times without dynamics, which may necessitate inflation. The parameter τ is expected to have a smaller influence, since the uncertainty is chosen small and it is already at the true value. This way it can act as an indicator parameter for the inflation as it does not require inflation.

The EnKF is set up with a total of 25 ensemble members and a damping vector of γ=[1.0, 0.3, 0.3, 0.3, 0.3], which we also apply to the inflation. The damping factor of 0.3 is applied to the parameters to alleviate issues of nonlinear relations between observations and parameters. For the uncertainty of the inflation factors we choose σλ=1.0.

3 Results

We estimate the water content state together with the four parameters ξ1, ξ2, K0 and τ with the EnKF as described in Sect. 2.3. The development of the water content at the two measurement locations at a depth of 9.5 and 19.5 cm is shown together with the inflation factor at these locations in Fig. 1. The inflation factor is applied to the forecast ensemble before the analysis. The standard deviation of the inflated ensemble should describe the distance of the estimated mean to the synthetic truth. Note, that the inflation factor is not based on this distance and relies on the noisy measurements. Therefore, it is only an indicator.

Figure 1Water content estimation at the two measurement locations. The standard deviation of the inflated ensemble should be able to explain the differences between the inflated mean and the synthetic truth. The inflation factor is increased when the ensemble uncertainty is too small.

During the first 3 days without any dynamics, the uncertainty for the upper measurement is slightly underestimated, while the uncertainty in the lower measurement is slightly overestimated. This leads to an inflation factor of basically 1 for the lower measurement (factors smaller than 1 are not allowed), while the inflation factor for the upper measurement is larger. However, due to correlations between the measurement locations a stronger inflation to fully explain the difference to the truth is prevented.

The deviation from the synthetic truth is induced through the initial guess of no heterogeneity and can also be seen in the systematic deviation of the inflated mean (which is equal to the forecast mean) from the analysis mean. When the infiltration front reaches the measurements, the deviations from the truth, underestimation of the uncertainty, and inflation factors increase rapidly. All of them are more pronounced for the upper measurement location. After the main peak, the differences and also the inflation factors decrease rapidly again.

The inflation factor for the state is shown in Fig. 2. It shows the strong increase in the inflation factor during the infiltration and its fast decrease afterwards. The inflation is strongest at the measurement location at a depth of 9.5 cm. The inflation factor is transferred to the other state locations through the correlations, which decrease with distance. Directly below the measurement locations the inflation factors are increased less than above. This is due to the chosen interpolation of the Miller scaling factors. Through the interpolation between the measurement locations and extrapolation to the boundaries, the dynamics changes at the measurement locations. During the infiltration the dynamics is mainly influenced by the water content above and the correlations with these locations are stronger.

The development of the Miller scaling factors ξ1 and ξ2 at the two measurement positions (9.5 and 19.5 cm depth) is shown in Fig. 3a and b together with the estimated inflation factor for these parameters. Both initial conditions assume no heterogeneity and start at ${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{1}}\right)={\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{2}}\right)=\mathrm{0.0}±\mathrm{0.25}$, corresponding to 2 SD away from the true value. At the upper location the true value of ${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\xi }}_{\mathrm{1}}\right)=-\mathrm{0.5}$ corresponds to a finer material. Consequently, the water content drops, as seen in Fig. 1, leading to a strong correlation with the scaling factor, and log10(ξ1) is adjusted rapidly to lower values. Accordingly, the inflation factor is increased quickly in the beginning and then reduced back to 1 when the estimation of log10(ξ1) reaches and eventually underestimates the true value. The underestimation of the scaling factor corresponds to a too fine material, which leads to slower changes in the water content state and therefore smaller correlations. The scaling factor is corrected during the rain event on the fourth day, which also leads to an inflation.

Figure 2Inflation factor for the water content state. The inflation is strongest at the upper measurement location during the infiltration, when the uncertainty is underestimated the most. The inflation factor is transferred to the other measurement locations through the correlations in the Kalman gain. The used interpolation of Miller scaling factors impacts these correlations and leads to the smaller inflation directly below the measurement locations.

The initial guess for the scaling factor for the depth of 19.5 cm underestimates the scaling factor, which corresponds to a too fine material. Again, the correlations are small. The value increases slowly during the dry period in the beginning, but is inflated and adjusted strongly during the rain event.

The saturated hydraulic conductivity K0 (Fig. 3c) was chosen to start a little more than 1 SD below the true value. Due to the unrepresented heterogeneity in the beginning, the value decreases even further. The inflation remains very small due to correlations with both measurement locations. However, as soon as the infiltration event reaches the first measurement location, the value is corrected towards the true value. At the same time, the inflation factor is increased due to the too small uncertainty. After the rain event the inflation factor drops rapidly back to 1. The hydraulic conductivity remains below the true value. Another rain event would be required to improve the estimation further.

The tortuosity τ (Fig. 3d) also influences the hydraulic conductivity function, but has in this case a much smaller impact and consequently smaller correlations with the measurements than K0. We use it as an indicator parameter and start at the true value. During the infiltration event the value is changed due to its correlation. The corresponding inflation factor is increased as well, but remains small enough and drops back to 1 quickly enough to not cause any over-inflation of the parameter.

Figure 3Development of Miller scaling factors ξ1 and ξ2, saturated hydraulic conductivity K0, and tortuosity τ together with their corresponding inflation factors during estimation with the EnKF.

To emphasize the need for a fast-adapting inflation factor, we reduce the uncertainty of the inflation factors to σλ=0.5 to slow down their adjustment. The results are summarized in Fig. 4. The inflation of the water content state (Fig. 4a) shows that the inflation factor does not reach as high values as before (see Fig. 2). To compensate for this, the inflation acts over a longer period of time. The same effect is also observed in the inflation of the parameters (Fig. 4b and c). This leads to a smaller inflation during the rain event and consequently a too small uncertainty. At later times, when the cause of the error is not active any more, the correlations with measurement locations are reduced, leading to a slower reduction of the inflation in the parameters. In the indicator parameter τ the beginning of an over-inflation can be seen towards later times. This necessitates a more rapid inflation when correlations are used to update inflation information.

The results for the parameters K0 and τ of a run without inflation (and only damping) are shown in Fig. 5. Again, K0 moves further away from the true value due to the unrepresented heterogeneity and comes closer to the true value during the infiltration event. However, since the Miller scaling factor is not inflated in the beginning, it is adjusted slower. Consequently, the K0 is corrected longer in the wrong direction. The uncertainty eventually becomes too small and in the end the mean is more than 5 SD away from the true value, since the uncertainty cannot be increased any more.

Figure 4Development of the inflation factor for the water content state and saturated hydraulic conductivity K0 and tortuosity τ together with their corresponding inflation factors for an estimation with a reduced inflation factor uncertainty of σλ=0.5.

Figure 5Development of saturated hydraulic conductivity K0 and tortuosity τ for an estimation without inflation.

4 Discussion

The proposed inflation method uses a Kalman filter to estimate inflation factors within the EnKF. It is based on the difference between measurements and mean forecast state. It transfers correlations from the forecast of the augmented state to the inflation. Consequently, the performance will be limited if model errors are structurally not represented in the forecast error covariance matrix. The estimation of the inflation factors with a Kalman filter is, like the EnKF itself, based on a linearized analysis. The use of a damping factor can alleviate issues with estimating nonlinear dependent parameters. To keep the inflation consistent with the analysis in the EnKF, we apply the same damping factor for both.

We designed a small synthetic hydrologic test case for the inflation. This test case mimics a model error through initially unrepresented heterogeneity. We designed the test case so that a strong temporally varying inflation is necessary, as it can occur with real data. We choose a short time so that the details of the behavior of the method can be explored. The method showed that it is capable of inflating states and parameters. The inflation is adjusted quickly and differentiates between parameters with strong and not so strong correlations. No over-inflation of weakly correlated parameters occurred. In this specific test case the estimation with inflation is far superior to an estimation without inflation.

Fast-dropping correlations between measurements and parameters are a limit for the method. An example could be a parameter only acting on an infiltration boundary condition. After the infiltration is over, correlations with this parameter would drop to zero and the inflation factor for this parameter will not be changed any more. If the inflation factor is not equal to 1 at this time, the parameter spread will keep increasing. In such a case, when there is no correlation, the parameter should be excluded from the estimation and consequently also from the inflation.

The method is in principle capable of compensating unrepresented model errors. However, it relies on correlations calculated from the forecast ensemble of the augmented state. If parameters have correlations with measurement locations with underestimated forecast uncertainties, the inflation will keep increasing the parameter spread until the forecast uncertainties are increased sufficiently. Therefore the correlations have to contain useful information. This means that inflating the parameters based on their correlations with measurement locations has to increase the forecast spread at these measurement locations. If the parameters have an insufficient influence on the state uncertainty, an over-inflation of the parameters can occur. An example are measurements with underestimated measurement uncertainties and short times between measurements compared to the timescale of the dynamics. Then the parameters are not able to increase the state uncertainty in the short forecast time between measurements and the forecast dynamics is not able represent the measurement noise. If such errors occur intermittently, e.g., the closed-eye period as proposed by could be used to bridge these times. A rather heuristic solution could be a decay of the inflation factor towards values of 1, as already proposed by .

5 Conclusions

In this work we propose a novel spatiotemporally adaptive inflation method, specifically designed for soil hydrology, which nevertheless is expected to work in similar systems as well. The inflation method is based on a Kalman filter acting within the EnKF. The method is capable of rapid adjustments of inflation factors, treating each augmented state dimension individually. This rapid adjustment is required due to temporally varying model errors, as they can appear through violation of the local equilibrium assumption of the Richards equation, hysteresis, or unrepresented heterogeneity.

We demonstrate the use of our inflation method in combination with a damping factor on a small hydrologic example. We choose heterogeneity as a possible model error, but allow the heterogeneity to be estimated along with the soil hydrologic parameters K0 and τ of the Mualem–van Genuchten parameterization. Our proposed inflation method proved to be stable in combination with parameter estimation. The performance of the estimation improved and parameter uncertainty remained consistent. The method requires that the correlations from in the forecast ensemble contain useful information for the inflation. However, we demonstrate that it even works for only weakly correlated parameters. We expect the inflation method to generally improve data assimilation with the EnKF and to thus lead to better state and parameter estimations in soil hydrology.

Data availability
Data availability.

The synthetic data are available upon request from the corresponding author.

Appendix A: Jacobian in the inflation method

We briefly show the derivation of the Jacobian matrix Hλ for the inflation (Eq. 15). Again, the entrywise product is denoted by  and the entrywise square root of λ by $\sqrt{\mathbit{\lambda }}$:

$\begin{array}{ll}{\left({\mathbf{H}}_{\mathit{\lambda }}\right)}_{ij}& =\frac{\partial }{\partial {\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}\\ & =\frac{\partial }{\partial {\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}}{\left[{\left(\mathbf{R}\right)}_{ii}+{\left(\mathbf{H}\left[{\mathbf{P}}^{\mathrm{f}}\cdot \left[\sqrt{{\mathbit{\lambda }}^{\mathrm{f}}}{\sqrt{{\mathbit{\lambda }}^{\mathrm{f}}}}^{T}\right]\right]{\mathbf{H}}^{T}\right)}_{ii}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}},\\ & \mathrm{with}\phantom{\rule{0.25em}{0ex}}{\mathbf{P}}^{\mathrm{f}}\phantom{\rule{0.25em}{0ex}}\mathrm{symmetric}=\frac{\partial }{\partial {\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}}\left[{\left(\mathbf{R}\right)}_{ii}+\sum _{m}\sum _{k}\left(\mathbf{H}{\right)}_{im}\right\\ & {\left(\mathbf{H}{\right)}_{ik}{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{km}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{k}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}]}^{\frac{\mathrm{1}}{\mathrm{2}}}\\ & ={\left[\mathrm{2}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}\right]}^{-\mathrm{1}}\sum _{m}\sum _{k}\left(\mathbf{H}{\right)}_{im}\left(\mathbf{H}{\right)}_{ik}{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{km}\\ & \frac{\partial }{\partial {\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}}\left[{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{k}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}\right]\\ & ={\left[\mathrm{2}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}\right]}^{-\mathrm{1}}\sum _{m}\sum _{k}\left(\mathbf{H}{\right)}_{im}\left(\mathbf{H}{\right)}_{ik}{\left({\mathbf{P}}^{\mathrm{f}}\right)}_{km}\\ & \frac{\mathrm{1}}{\mathrm{2}}\left[{\mathit{\delta }}_{mj}\frac{{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{k}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}}{{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}}+{\mathit{\delta }}_{kj}\frac{{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}}{{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{k}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}}\right]\\ & ={\left[\mathrm{2}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{j}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}{\left({\mathbit{h}}_{\mathit{\lambda }}\left({\mathbit{\lambda }}^{\mathrm{f}}\right)\right)}_{i}\right]}^{-\mathrm{1}}\sum _{m}\left(\mathbf{H}{\right)}_{ij}\left(\mathbf{H}{\right)}_{im}\\ \text{(A1)}& & {\left({\mathbf{P}}^{\mathrm{f}}\right)}_{jm}{\left[{\left({\mathbit{\lambda }}^{\mathrm{f}}\right)}_{m}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}.\end{array}$
Appendix B: Real-world application

We also applied the inflation method to reanalyze the case presented earlier by , where measurements from 11 TDR probes were assimilated with an EnKF. There, the inflation method confirmed the behavior observed in the small synthetic case presented in this paper. For the details of the real-world case as well as the concept of the closed-eye period please refer to or Bauser (2018, chap. 5), the latter of which also includes the reanalysis of the case.

In this paper, we only show the inflation related to the closed-eye period (Fig. B1), which presents the major challenge to the inflation in that particular application. During this time, preferential flow occurs and the underlying local equilibrium assumption of the Richards equation is violated. With a standard approach, parameters become biased to compensate these errors. To avoid this, introduced the closed-eye period, which pauses the parameter estimation and only guides the water content states through times, when assumptions are violated. Compared to the standard approach, this leads to a reduced bias in the parameters, but effectively increases the model errors during the closed-eye period. A strong inflation is required to compensate this error. The inflation method used in was just able to accomplish this and the authors were concerned that a too slow adjustment speed of the inflation limits the applicability of the closed-eye period for cases with larger model errors.

Figure B1 confirms the fast adjustment speed of the new inflation method proposed in this paper for the real-world application. The strong required inflation stays well within the closed-eye period. This enables the EnKF to pick up the parameter estimation after the period from a water content state consistent with the TDR measurements and facilitates the use of the closed-eye period.

Figure B1Inflation factor for the water content state in a real-world application for a standard and closed-eye EnKF together with atmospheric forcing and the corresponding response of measured water contents. The closed-eye period starts when the infiltration front reaches the topmost TDR. During this time the local equilibrium assumption of the Richards equation is violated and a strong inflation is required. The new inflation method allows a fast adjustment of the inflation factors, which enables the EnKF to effectively guide the water content states with the TDR measurements through the closed-eye period. Modified from .

Author contributions
Author contributions.

HHB designed, implemented, performed and analyzed the presented study. DB provided computational software. All authors participated in continuous discussions. HHB prepared the manuscript with contributions from all authors.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We thank editor Insa Neuweiler and two anonymous reviewers for their comments, which helped to improve this paper.

This research is funded by the Deutsche Forschungsgemeinschaft (DFG) through project RO 1080/12-1 and the Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg (Az 33-7533.-30-20/6/2). HGS MathComp provided travel expenses for Hannes H. Bauser and Daniel Berg.

Edited by: Insa Neuweiler
Reviewed by: two anonymous referees

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text:
# Introducing a product measure from a sequence of transition kernels

Given measurable spaces $$(\Omega_k,\mathcal {F_k })_{k=1 } ^n$$, $$P_1$$ a probability measure on $$\Omega_1$$, $$P_2$$ a transition kernel from $$\mathcal {F_1}$$ to $$\mathcal {F_2 }$$,$$P_3$$ a transition kernel from $$\mathcal {F_1}\otimes \mathcal {F_2}$$ to $$\mathcal {F_3}$$ ...$$P_n$$ a transition kernel from $$\mathcal {F_1}\otimes \ldots\otimes \mathcal {F_{n-1 }}$$ to $$\mathcal {F_n } \$$. Here $$\ P_m$$ is a transition kernel from $$\mathcal {F_1}\otimes \ldots \otimes \mathcal {F_{m-1 }}$$ to $$\mathcal F_m$$ means that for every $$B \in \mathcal {F_m } , \ P_m( \ , B)$$ is a measurable function from $$\Omega_1 \times \ldots \times \Omega_{m-1 }$$ to $$[0,1]$$ and for every $$(\omega_1,\ldots,\omega_{m-1 } ) \in \Omega_1 \times \ldots \times \Omega_{m-1 }$$ a probability measure on $$\mathcal {F_m}$$.

I would like to show that $$P(B)=\int P_1(d\omega_1) \int P_2(\omega_1,d \omega_2) \ldots \int 1_B(\omega_1,\ldots,\omega_n)P_n(\omega_1,\ldots,\omega_{n-1 } , d \omega_n)$$ defines a measure on $$\Omega_1 \times \ldots\times \Omega_n$$

This should be done by an argument using induction. That is one shows that $$P$$ is a countably additive set function on the set of measurable rectangles $$B_0\times \ldots\times B_n$$ (and then extend this set function by the usual arguments). To show countable additivity of $$P$$ on measurable rectangles we derive this for $$n=1$$ and then extended the result to an arbitrary number $$n$$ with an argument using induction. And this inductive step is what I need help with.

For the base case $$n=1$$ one do as follows:

First one shows that $$\omega_1 \mapsto \int 1_B(\omega_1,\omega_2)P_2(\omega_1,d \omega _2)$$ is $$\mathcal F_1$$ measurable and this is done exactly as usually done in a first step in Fubini's theorem. Then noting that $$B \mapsto \int P_1(d \omega_1)\int 1_B(\omega_1,\omega_2)P_2(\omega_1,d \omega _2)$$ is nonnegative and monotone all we need to verify is the $$\sigma$$-additivity. This follows from the monotone convergence theorem:

\begin{align*}\int P_1(d \omega_1)\int 1_{\cup_{k=1}^{\infty }F_k} (\omega_1,\omega_2)P_2(\omega_1,d \omega _2) &=\int P_1(d \omega_1)\sum_{k=1 }^ {\infty } \int 1_{F_k} (\omega_1,\omega_2)P_2(\omega_1,d \omega _2) \\ &=\sum_{k=1 }^ {\infty }\int P_1(d \omega_1) \int 1_{F_k} (\omega_1,\omega_2)P_2(\omega_1,d \omega _2). \end{align*}

And the inductive step should be something like: assume that $$Q(B)=\int P_1(d\omega_1) \int P_2(\omega_1,d \omega_2)\ldots\int 1_B(\omega_1,\ldots,\omega_{n-1 } )P_{n-2 } (\omega_1,\ldots,\omega_{n-2 } , d \omega_{n-1 } )$$ defines a measure on $$\Omega_1 \times \ldots\times \Omega_{n-1 }$$ then

\begin{align*} & \int P_1(d\omega_1) \int P_2(\omega_1,d \omega_2)\ldots\int 1_{B_0 \times \ldots \times B_n}(\omega_1,\ldots,\omega_n)P_n(\omega_1,\ldots,\omega_{n-1 } , d \omega_n) \\ &=\int 1_{B_0 \times \ldots \times B_{n-1}} \left(\int 1_{B_n } P_n(\omega_1,\ldots,\omega_{n-1 } , d \omega_n) \right) Q(d \omega_1,\ldots,d \omega_{n-1 } ) \end{align*} And we would then be in a similar situation as in the base case.

My question is: If this is correct, how is the equality motivated? Or if not, how should the inductive step be?

Let's start with a technical lemma which we will use several times:

Lemma. Let $$(\Omega_i,\mathcal{F}_i)$$, $$i=1,2$$,be measurable spaces, and let $$P$$ a transition kernel from $$\mathcal{F}_1$$ to $$\mathcal{F}_2$$. If $$X:(\Omega_1 \times \Omega_2, \mathcal{F}_1 \otimes \mathcal{F}_2) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$$ is a bounded measurable mapping then $$(\Omega_1,\mathcal{F}_1) \ni \omega_1 \mapsto \int_{\Omega_2} X(\omega_1,\omega_2) \, P(\omega_1,d\omega_2) \in (\mathbb{R},\mathcal{B}(\mathbb{R}))$$ is measurable.

Idea of proof: For $$X=1_A 1_B$$ with $$A \in \mathcal{F}_1$$ and $$B \in \mathcal{F}_2$$ the assertion is immediate and then we can extend it using a classical monotone class argument.

For measurable spaces $$(\Omega_i,\mathcal{F}_i)$$, $$i \in \{0,\ldots,n\}$$, and transition kernels $$P_i$$ from $$\mathcal{F}_0 \otimes \ldots \otimes \mathcal{F}_{i-1}$$ to $$\mathcal{F}_i$$ set

$$Q_n(\omega_0,B) := \int P_1(\omega_0,d\omega_1) \int P_2(\omega_0,\omega_1,d\omega_2) \ldots \int 1_B(\omega_1,\ldots,\omega_n) \, P_n(\omega_0,\ldots,\omega_{n-1},d\omega_n)$$

Claim: For each $$n \in \mathbb{N}$$ it holds for any choice of measurable spaces and transition kernels that $$Q_n$$ is a transition kernel from $$\mathcal{F}_0$$ to $$\mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_n$$.

Base: Since $$Q_1(\omega_0,B) = P_1(\omega_0,B)$$, the assertion is obvious for $$n=1$$.

Inductive step: Assume that the assertion holds for some $$n \in \mathbb{N}$$. Let $$(\Omega_k,\mathcal{F}_k)$$, $$k \leq n+1$$, be measurable spaces, and $$P_i$$ transition kernels from $$\mathcal{F}_0 \otimes \ldots \otimes \mathcal{F}_{i-1}$$ to $$\mathcal{F}_i$$. For fixed $$(\omega_0,\omega_1) \in \Omega_0 \times \Omega_1$$ consider

\begin{align*} &U((\omega_0,\omega_1),A) \\ &:= \int P_2(\omega_0,\omega_1,d\omega_2) \int P_3(\omega_0,\omega_1,\omega_2,d\omega_3) \ldots \int 1_{A}(\omega_2,\ldots,\omega_{n+1}) \, P_{n+1}(\omega_0,\ldots,\omega_n,d\omega_{n+1}) \end{align*}

for $$A \in \mathcal{F}_2 \otimes \ldots \otimes \mathcal{F}_{n+1}$$. Since $$P_i(\omega_0,\omega_1,\cdot)$$, $$i \in \{2,\ldots,n+1\}$$ are transition kernels from $$\mathcal{F}_2 \otimes \ldots \otimes \mathcal{F}_{i-1}$$ to $$\mathcal{F}_i$$, it follows from our induction hypothesis that $$U$$ is a transition kernel from $$\mathcal{F}_0 \otimes \mathcal{F}_1$$ to $$\mathcal{F}_2 \otimes \ldots \times \mathcal{F}_{n+1}$$. By the above lemma, this implies that $$(\Omega_0 \times \Omega_1, \mathcal{F}_0 \otimes \mathcal{F}_1) \ni (\omega_0,\omega_1) \mapsto \int_{\Omega_2 \times \ldots \times \Omega_{n+1}} 1_A(\omega_0,\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega}) \tag{1}$$ is measurable for any $$A \in \mathcal{F}_0 \otimes \ldots \otimes \mathcal{F}_{n+1}$$. In particular, $$Q_{n+1}(\omega_0,B)$$ is well defined and

\begin{align*} Q_{n+1}(\omega_0,B) &= \int_{\Omega_1} \int_{\Omega_2 \times \ldots \times \Omega_{n+1}} 1_B(\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega}) \, P_1(\omega_0,d\omega_1) \tag{2} \end{align*}

for any $$B \in \mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_{n+1}$$ and $$\omega_0 \in \Omega_0$$.

Now let's show that $$Q_{n+1}$$ has all the desired properties. Let $$(B_j)_{j \in \mathbb{N}} \subseteq \mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_{n+1}$$ be pairwise disjoint sets and set $$B:= \bigcup_{j \geq 1} B_j$$. Since $$1_B = \sum_{j \geq 1} 1_{B_j}$$ it follows from the monotone convergence theorem that

$$\int_{\Omega_2 \times \ldots \times \Omega_n} 1_B(\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega}) = \sum_{j \geq 1} \int_{\Omega_2 \times \ldots \times \Omega_{n+1}} 1_{B_j}(\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega})$$

for any $$(\omega_0,\omega_1) \in \Omega_0 \times \Omega_1$$; note that we can apply the monotone convergence theorem because we have already shown that $$U((\omega_0,\omega_1),\cdot)$$ is a measure. Integrating both sides with respect to $$P_1$$ we find from $$(2)$$ that

$$Q_{n+1}(\omega_0,B) = \int_{\Omega_1} \sum_{j \geq 1} \left( \int_{\Omega_2 \times \ldots \times \Omega_{n+1}} 1_{B_j}(\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega}) \right) P(\omega_0,d\omega_1).$$

Applying once more the monotone convergence theorem we get

\begin{align*} Q_{n+1}(\omega_0,B) & = \sum_{j \geq 1} \int_{\Omega_1}\left( \int_{\Omega_2 \times \ldots \times \Omega_{n+1}} 1_{B_j}(\omega_1,\tilde{\omega}) \, U((\omega_0,\omega_1),d\tilde{\omega}) \right) P(\omega_0,d\omega_1) \\ &= \sum_{j \geq 1} Q_{n+1}(\omega_0,B_j). \end{align*}

This proves the $$\sigma$$-addivity of $$Q_{n+1}(\omega_0,\cdot)$$. Finally, we note that it follows from $$(1)$$, $$(2)$$ and the above lemma that $$(\Omega_0,\mathcal{F}_0) \ni \omega \mapsto Q_{n+1}(\omega_0,B)$$ is measurable for any $$B \in \mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_{n+1}$$. Consequently, $$Q_{n+1}$$ is a transition kernel from $$\mathcal{F}_0$$ to $$\mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_{n+1}$$.

Choosing $$\Omega_0$$ as a trivial space (e.g. consisting of a single element), it follows immediately that

$$P_n(B) = \int P_1(d\omega_1) \int P_2(\omega_1,d\omega_2) \ldots \int 1_B(\omega_1,\ldots,\omega_n) \, P_n(\omega_1,\ldots,\omega_{n-1},d\omega_n)$$

is a measure on $$(\Omega_1 \times \ldots \times \Omega_n,\mathcal{F}_1 \otimes \ldots \otimes \mathcal{F}_n)$$ for any probability measure $$P_1$$ on $$(\Omega_1,\mathcal{F}_1)$$ and any transition kernels $$P_i$$.

• Thank you! Clear and detailed as ever! I have a small question for the countable additivity, how do you get $Q_{n+1}(\omega_0,\cup _{k=1 } ^{\infty} B^{(k) } ) = \int \sum _{k=1 } ^{\infty } 1_{B^{(k)} _1}(\omega_1) U((\omega_0,\omega_1),B_2 ^{(k) } \times \ldots \times B_{n+1}^{(k)} ) \, P_1(\omega_0,d\omega_1).$? On which we then could apply the monotone convergence theorem. – MrFranzén May 17 '19 at 13:07
• On the other hand when I think about it, could we instead apply the monotone convergence theorem $n+1$ times to "pull" the limit out from the $n+1$ integrals. Here using that for any $k$ and for $(\omega_1,...,\omega_k)$ the map $(\omega_{k+1 } ,...,\omega_{n+1 } )\mapsto \int P_{k+1 } ...\int \sum_{k=1 } ^m 1_{B_1^{(k)} \times ... \times B_{n+1 }^{(k)} } P_{n+1 } (\omega_0,...\omega_n, d \omega_{n+1 } )$ is increasing with $m$. (Or maybe this last claim would need some motivation??) – MrFranzén May 17 '19 at 13:08
• @MrFranzén See my edited answer, I hope it's fine now. Let me know if something seems out of place (... there are still some typos, I guess... sorry already in advance) – saz May 17 '19 at 15:35
• Excellent! There is one minor thing. I believe that we get $(2)$ first for sets of the form $B_1\times C$ for sets $B_1 \in \mathcal {F_1 }$ and $C \in \mathcal {F_2} \times ... \times \mathcal {F_{n+1 } }$, since we have to use $1_{B_1 \times C } = 1_{B_1 } 1 _C$ and then we extend the class of sets for which the equation holds using a standard $\pi - \lambda$ argument. – MrFranzén May 18 '19 at 6:35
• And lots of thanks again! – MrFranzén May 18 '19 at 6:37