text: (cid:90) (cid:88) ∞ ∞ − 1 (cid:16)− e(cid:17) iαξ (cid:112) K(α, β) = ( iβ)n dξ (27) n! (cid:88)−∞ (ξ2 + 1)1+n 2 n=0 √ ∞ − n iβ α/2 = 2πα K (α) , (28) (n+1)/2 n!Γ(1 + n/2) n=0 (cid:112) where K (x) is the modiﬁed Bessel function of the second kind of order n. The zeroth n − order result (26) is recovered by observing that K (x) = π e x. For small β the series 1/2 2x converges rapidly, and one can even improve the agreement with the numerically calculated kernel by re-exponentiating the ﬁrst few terms. For example, up to second order we have a polynomial p + p β + p β2 which we wish to write as p exp(a β + a β2). Expanding the 0 1 2 0 1 2 exponential in powers of β and comparing powers up to order β2, one ﬁnds a = p /p and 1 1 0 − a = p /p (p /p )2/2. When plotted together with the exact result, the thus obtained 2 2 0 1 0 (cid:39) approximation agrees with K(α, β) for α = 2 visibly well up to β 4, i.e. well beyond the (cid:28) regime β 1. For the fourth order re-exponentiated form the agreement extends up to (cid:39) about β 5. However, the exponential decay (26) already of the zeroth order term with α (cid:39) indicates that for the values of α 102 to 103 the contribution to the η integral for values such that β is of order of or smaller than one, can be entirely neglected. In the opposite regime of large β, an approximation based on a stationary phase ap- (cid:29) proximation can be found. More precisely, one needs β α. In that case on√e can treat − − e iαξ/(ξ2 + 1) as a slowly varying factor compared to the rapidly oscillating e iβ/ ξ2+1. The point of stationary phase of the latter term is found at ξ = 0 (where the function has a (cid:114) maximum). The second derivative of the phase at ξ = 0 equals 1. With this we get (cid:39) 2π − K(α, β) eiπ/4e iβ , (29) β (cid:29) valid for β/α 1. Interestingly, the integral kernel becomes independent of α in this regime, which is of course a consequence of the fact that the stationary phase point is at ξ = 0, thus eliminating the factor α in the phase of the prefactor. We furthermore see that in this regime there is no exponential suppression of the kernel. For α = 0, the kernel can be evaluated exactly, 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: iαξ

text: K(α,

text: β)

text: iβ)n

text: dξ

text: n!

text: n

text: iβ

text: 2πα

text: K

text: (α)

text: where

text: K

text: (x)

text: is

text: the

text: modiﬁed

text: Bessel

text: function

text: of

text: the

text: second

text: kind

text: of

text: order

text: n.

text: The

text: zeroth

text: n

text: order

text: result

text: is

text: recovered

text: by

text: observing

text: that

text: K

text: (x)

text: π

text: e

text: x.

text: For

text: small

text: β

text: the

text: series

text: converges

text: rapidly,

text: and

text: one

text: can

text: even

text: improve

text: the

text: agreement

text: with

text: the

text: numerically

text: calculated

text: kernel

text: by

text: re-exponentiating

text: the

text: ﬁrst

text: few

text: terms.

text: For

text: example,

text: up

text: to

text: second

text: order

text: we

text: have

text: a

text: polynomial

text: p

text: p

text: β

text: p

text: β2

text: which

text: we

text: wish

text: to

text: write

text: as

text: p

text: exp(a

text: β

text: a

text: Expanding

text: the

text: exponential

text: in

text: powers

text: of

text: β

text: and

text: comparing

text: powers

text: up

text: to

text: order

text: one

text: ﬁnds

text: a

text: p

text: /p

text: and

text: a

text: p

text: /p

text: (p

text: /p

text: When

text: plotted

text: together

text: with

text: the

text: exact

text: result,

text: the

text: thus

text: obtained

text: approximation

text: agrees

text: with

text: K(α,

text: β)

text: for

text: α

text: visibly

text: well

text: up

text: to

text: β

text: i.e.

text: well

text: beyond

text: the

text: regime

text: β

text: For

text: the

text: fourth

text: order

text: re-exponentiated

text: form

text: the

text: agreement

text: extends

text: up

text: to

text: about

text: β

text: However,

text: the

text: exponential

text: decay

text: already

text: of

text: the

text: zeroth

text: order

text: term

text: with

text: α

text: indicates

text: that

text: for

text: the

text: values

text: of

text: α

text: to

text: the

text: contribution

text: to

text: the

text: η

text: integral

text: for

text: values

text: such

text: that

text: β

text: is

text: of

text: order

text: of

text: or

text: smaller

text: than

text: one,

text: can

text: be

text: entirely

text: neglected.

text: In

text: the

text: opposite

text: regime

text: of

text: large

text: β,

text: an

text: approximation

text: based

text: on

text: a

text: stationary

text: phase

text: ap-

text: proximation

text: can

text: be

text: found.

text: More

text: precisely,

text: one

text: needs

text: β

text: α.

text: In

text: that

text: case

text: on√e

text: can

text: treat

text: e

text: as

text: a

text: slowly

text: varying

text: factor

text: compared

text: to

text: the

text: rapidly

text: oscillating

text: e

text: iβ/

text: The

text: point

text: of

text: stationary

text: phase

text: of

text: the

text: latter

text: term

text: is

text: found

text: at

text: ξ

text: (where

text: the

text: function

text: has

text: a

text: maximum).

text: The

text: second

text: derivative

text: of

text: the

text: phase

text: at

text: ξ

text: equals

text: With

text: this

text: we

text: get

text: 2π

text: K(α,

text: β)

text: iβ

text: β

text: valid

text: for

text: β/α

text: Interestingly,

text: the

text: integral

text: kernel

text: becomes

text: independent

text: of

text: α

text: in

text: this

text: regime,

text: which

text: is

text: of

text: course

text: a

text: consequence

text: of

text: the

text: fact

text: that

text: the

text: stationary

text: phase

text: point

text: is

text: at

text: ξ

text: thus

text: eliminating

text: the

text: factor

text: α

text: in

text: the

text: phase

text: of

text: the

text: prefactor.

text: We

text: furthermore

text: see

text: that

text: in

text: this

text: regime

text: there

text: is

text: no

text: exponential

text: suppression

text: of

text: the

text: kernel.

text: For

text: α

text: the

text: kernel

text: can

text: be

text: evaluated

text: exactly,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: resulting resolution, as we will show now. V. ESTIMATION OF GEOMETRICAL RESOLUTION A. Approximate analytical inversion of the integral kernel (cid:29) At ﬁrst sight, the requirement β α appears unnatural given that α can already of the order 102 to 103. And indeed, this leads to a ﬁrst rather stringent condition which must be met in order for the correlation function C to be non-zero. In terms of the original parameters, β/α = ω η/(cκ (η2 + h˜2)). For this to be much larger than one, one needs c x η (cid:29) cκ c ∆ω x = , (31) η2 + h˜2 ω v ω c s c (cid:28) ˜ ˜ or ∆ω/ω (v /(2ch)), where we have used already the maximum value 1/(2h) of the c s function of η on the left hand side (lhs) in (31). For the standard parameters, we ﬁnd (cid:28) · − ∆ω/ω 1.66 10 9. When operating at ω in the GHz regime, this means that the c c correlation function essentially vanishes for ∆ω larger than a few Hertz and thus bears no more information for the measurement of the position dependent brightness temperature. Another way of seeing this is to observe that (31) limits the integration range for η: The lhs of (31) is a function that starts of at 0 for η = 0, increases linearly, reaches a maximum ˜ ˜ of 1/2h at η = h, and decays as 1/η for large η. Condition (31) then limits the integration ≤ ≤ (cid:112) range of η to an interval η η η with 1 2 ± − 1 1 4δ2h˜2 ≡ η = η (κ), (32) 1,2 1,2 2δ ≡ ≡ (cid:28) where δ c∆ω/(v ω ) = cκ /ω = cκ/(ω ∆r) = χκ, κ κ ∆r, and χ = c/(ω ∆r) 1 s c x c c x c ˜ (see introduction and Fig.2). A ﬁnite real integration range exists only for δ < 1/(2h), ≡ ≤ ≡ ≤ equivalent to κ < ωc∆r κ . For given η, we have κ κ (η) ωc∆r η κ . 2ch˜ Max max c η2+h˜2 Max A ﬁnite minimal value of κ can be deduced from a maximum desired snapshot size in − x direction. Also the requirement τ < h/v may bound the relevant values of κ from s below, as it leads to a smallest resolvable frequency, and thus also smallest resolvable ∆ω: ⇒ ≡ ∆ω > 2π/τ κ = ∆r∆ω/v > ∆r 2π κ . s τvs min 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: resulting

label: paragraph

text: resolution,

text: as

text: we

text: will

text: show

text: now.

text: V.

label: section

text: ESTIMATION

text: OF

text: GEOMETRICAL

text: RESOLUTION

text: A.

text: Approximate

text: analytical

text: inversion

text: of

text: the

text: integral

text: kernel

text: At

text: ﬁrst

text: sight,

text: the

text: requirement

text: β

text: α

text: appears

text: unnatural

text: given

text: that

text: α

text: can

text: already

text: of

text: the

text: order

text: to

text: And

text: indeed,

text: this

text: leads

text: to

text: a

text: ﬁrst

text: rather

text: stringent

text: condition

text: which

text: must

text: be

text: met

text: in

text: order

text: for

text: the

text: correlation

text: function

text: C

text: to

text: be

text: non-zero.

text: In

text: terms

text: of

text: the

text: original

text: parameters,

text: β/α

text: ω

text: η/(cκ

text: For

text: this

text: to

text: be

text: much

text: larger

text: than

text: one,

text: one

text: needs

text: c

text: x

text: η

text: cκ

text: c

text: ∆ω

text: x

text: η2

text: ω

text: v

text: ω

text: c

text: s

text: c

text: or

text: ∆ω/ω

text: (v

text: where

text: we

text: have

text: used

text: already

text: the

text: maximum

text: value

text: of

text: the

text: c

text: s

text: function

text: of

text: η

text: on

text: the

text: left

text: hand

text: side

text: (lhs)

text: in

text: For

text: the

text: standard

text: parameters,

text: we

text: ﬁnd

text: ∆ω/ω

text: When

text: operating

text: at

text: ω

text: in

text: the

text: GHz

text: regime,

text: this

text: means

text: that

text: the

text: c

text: c

text: correlation

text: function

text: essentially

text: vanishes

text: for

text: ∆ω

text: larger

text: than

text: a

text: few

text: Hertz

text: and

text: thus

text: bears

text: no

text: more

text: information

text: for

text: the

text: measurement

text: of

text: the

text: position

text: dependent

text: brightness

text: temperature.

text: Another

text: way

text: of

text: seeing

text: this

text: is

text: to

text: observe

text: that

text: limits

text: the

text: integration

text: range

text: for

text: η:

text: The

text: lhs

text: of

text: is

text: a

text: function

text: that

text: starts

text: of

text: at

text: for

text: η

text: increases

text: linearly,

text: reaches

text: a

text: maximum

text: of

text: at

text: η

text: h,

text: and

text: decays

text: as

text: for

text: large

text: η.

text: Condition

text: then

text: limits

text: the

text: integration

text: range

text: of

text: η

text: to

text: an

text: interval

text: η

text: η

text: η

text: with

text: η

text: η

text: (κ),

text: 2δ

text: where

text: δ

text: c∆ω/(v

text: ω

text: cκ

text: /ω

text: cκ/(ω

text: ∆r)

text: χκ,

text: κ

text: κ

text: ∆r,

text: and

text: χ

text: c/(ω

text: ∆r)

text: s

text: c

text: x

text: c

text: c

text: x

text: c

text: (see

text: introduction

text: and

text: A

text: ﬁnite

text: real

text: integration

text: range

text: exists

text: only

text: for

text: δ

text: equivalent

text: to

text: κ

text: ωc∆r

text: κ

text: For

text: given

text: η,

text: we

text: have

text: κ

text: κ

text: (η)

text: ωc∆r

text: η

text: κ

text: Max

text: max

text: c

text: Max

text: A

text: ﬁnite

text: minimal

text: value

text: of

text: κ

text: can

text: be

text: deduced

text: from

text: a

text: maximum

text: desired

text: snapshot

text: size

text: in

text: x

text: direction.

text: Also

text: the

text: requirement

text: τ

text: h/v

text: may

text: bound

text: the

text: relevant

text: values

text: of

text: κ

text: from

text: s

text: below,

text: as

text: it

text: leads

text: to

text: a

text: smallest

text: resolvable

text: frequency,

text: and

text: thus

text: also

text: smallest

text: resolvable

text: ∆ω:

text: ∆ω

text: 2π/τ

text: κ

text: ∆r∆ω/v

text: ∆r

text: 2π

text: κ

text: s

text: τvs

text: min

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ (cid:39) hence ∆y = ∆rπ(η2+h˜2)3/2/(k˜ h˜2). For y h, this is of the order 2 2πhc/(∆rω ) = χh. s c0 c (cid:39) The numerical value for the standard parameters gives ∆y 2.1 km, i.e. a resolution of the order of a kilo-meter. However, for actually achieving this resolution for an extended source, one has to face two issues: i.) The reconstructed point-source image should be brought as close as possible to a single narrow peak; and ii.) one has to deal with the diﬀerent phases from sources at positive or negative η. The ﬁrst issue can be addressed by superposing correlation functions from pairs of antennas at diﬀerent separation, and/or changing the considered central frequency. This shifts the pattern of peaks due to the cos-function, and one can engineer a rather narrow central peak (see [14] for details). The second issue should be absent in a nu√merically exact inversion of the integral kernel. The Gaussian envelope has a width hc/( 2πB∆r) given by the inverse bandwidth, which is much larger than the (cid:39) width of a single peak, namely by a factor ω /(4πB) 35 for the standard parameters. c The resolution in x-direction follows from the eﬀective wave vector κ in the sinc max function. It depends on the position of the source and reaches its maximum possible value ˜ κ for η = h (i.e. y = h). The inverse of κ thus gives the best possible resolution in Max s s Max x-direction: ≥ ∆r 2hc ∆x = . (48) κ ω ∆r Max c We conclude that both in x- and y-direction one can expect a geometric resolution of the order hχ = c/(∆rω ) for sources close to y = h. For sources close to y = 0, κ(η ) c s ∝ ∝ goes to zero η , whereas for larger y the decay of κ(η ) is 1/η . The geometric s s s s resolution in x-direction deteriorates correspondingly. The resolution in y-direction, on the other hand, depends only weakly o√n the source-position, as (ηs2 + h˜2)3/2/(k˜ch˜2) increases ˜ monotonically from h at y = 0 to 2 2 at y = h, and keeps growing slowly beyond y = h. s s s It is remarkable that correlating electric ﬁelds at two diﬀerent frequencies can lead to a resolution that is given by the central frequency. The deﬁnition of κ is based on the request that the stationary phase approximation max (cid:29) (SPA) holds in the regime β α. In practice, the SPA is almost always better than 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: hence

text: ∆y

text: For

text: y

text: h,

text: this

text: is

text: of

text: the

text: order

text: χh.

text: s

text: c

text: The

text: numerical

text: value

text: for

text: the

text: standard

text: parameters

text: gives

text: ∆y

text: km,

text: i.e.

text: a

text: resolution

text: of

text: the

text: order

text: of

text: a

text: kilo-meter.

text: However,

text: for

text: actually

text: achieving

text: this

text: resolution

text: for

text: an

text: extended

text: source,

text: one

text: has

text: to

text: face

text: two

text: issues:

text: i.)

text: The

text: reconstructed

text: point-source

text: image

text: should

text: be

text: brought

text: as

text: close

text: as

text: possible

text: to

text: a

text: single

text: narrow

text: peak;

text: and

text: ii.)

text: one

text: has

text: to

text: deal

text: with

text: the

text: diﬀerent

text: phases

text: from

text: sources

text: at

text: positive

text: or

text: negative

text: η.

text: The

text: ﬁrst

text: issue

text: can

text: be

text: addressed

text: by

text: superposing

text: correlation

text: functions

text: from

text: pairs

text: of

text: antennas

text: at

text: diﬀerent

text: separation,

text: and/or

text: changing

text: the

text: considered

text: central

text: frequency.

text: This

text: shifts

text: the

text: pattern

text: of

text: peaks

text: due

text: to

text: the

text: cos-function,

text: and

text: one

text: can

text: engineer

text: a

text: rather

text: narrow

text: central

text: peak

text: (see

text: for

text: details).

text: The

text: second

text: issue

text: should

text: be

text: absent

text: in

text: a

text: nu√merically

text: exact

text: inversion

text: of

text: the

text: integral

text: kernel.

text: The

text: Gaussian

text: envelope

text: has

text: a

text: width

text: hc/(

text: given

text: by

text: the

text: inverse

text: bandwidth,

text: which

text: is

text: much

text: larger

text: than

text: the

text: width

text: of

text: a

text: single

text: peak,

text: namely

text: by

text: a

text: factor

text: ω

text: for

text: the

text: standard

text: parameters.

text: c

text: The

text: resolution

text: in

text: x-direction

text: follows

text: from

text: the

text: eﬀective

text: wave

text: vector

text: κ

text: in

text: the

text: sinc

text: max

text: function.

text: It

text: depends

text: on

text: the

text: position

text: of

text: the

text: source

text: and

text: reaches

text: its

text: maximum

text: possible

text: value

text: κ

text: for

text: η

text: h

text: (i.e.

text: y

text: h).

text: The

text: inverse

text: of

text: κ

text: thus

text: gives

text: the

text: best

text: possible

text: resolution

text: in

text: Max

text: s

text: s

text: Max

text: x-direction:

text: ∆r

text: ∆x

text: κ

text: ω

text: ∆r

text: Max

text: c

text: We

text: conclude

text: that

text: both

text: in

text: x-

text: and

text: y-direction

text: one

text: can

text: expect

text: a

text: geometric

text: resolution

text: of

text: the

text: order

text: hχ

text: c/(∆rω

text: for

text: sources

text: close

text: to

text: y

text: h.

text: For

text: sources

text: close

text: to

text: y

text: κ(η

text: c

text: s

text: goes

text: to

text: zero

text: η

text: whereas

text: for

text: larger

text: y

text: the

text: decay

text: of

text: κ(η

text: is

text: The

text: geometric

text: s

text: s

text: s

text: s

text: resolution

text: in

text: x-direction

text: deteriorates

text: correspondingly.

text: The

text: resolution

text: in

text: y-direction,

text: on

text: the

text: other

text: hand,

text: depends

text: only

text: weakly

text: o√n

text: the

text: source-position,

text: as

text: increases

text: monotonically

text: from

text: h

text: at

text: y

text: to

text: at

text: y

text: h,

text: and

text: keeps

text: growing

text: slowly

text: beyond

text: y

text: h.

text: s

text: s

text: s

text: It

text: is

text: remarkable

text: that

text: correlating

text: electric

text: ﬁelds

text: at

text: two

text: diﬀerent

text: frequencies

text: can

text: lead

text: to

text: a

text: resolution

text: that

text: is

text: given

text: by

text: the

text: central

text: frequency.

text: The

text: deﬁnition

text: of

text: κ

text: is

text: based

text: on

text: the

text: request

text: that

text: the

text: stationary

text: phase

text: approximation

text: max

text: (SPA)

text: holds

text: in

text: the

text: regime

text: β

text: α.

text: In

text: practice,

text: the

text: SPA

text: is

text: almost

text: always

text: better

text: than

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ˜ ˜ the reconstructed temperature depends on the precise value of a as well as the ratio k /b. c0 Outside the mentioned interval, T (x, y) oscillates again as function of ζ, which can be rec understood from the fact that the box is cut-oﬀ when ζ gets within a distance aσ/2 of 0 or ˆ ζ. The sought-for order of magnitude can be estimated from the maximum value of (70) as (cid:39) function of a. As for standard parameters k˜ /˜b 70, w(cid:113)e can bound the sin-function by c0 ∼ (cid:39) ˜ one (while still having a 1), in which case we obtain T (x, y) T /k = T χ in the rec 0 c0 0 (cid:39) mentioned ζ-interval. With all this, and approximating h˜2 + ηˆ2 ηˆ in the logarithm in ∼ ∼ (67), we ﬁnd the order of magnitude σ(T (x, y))/T (x, y) ln(2ηˆ/h˜)/χ3/2 For ζ 1, this rec (cid:112)rec ∼ is of order 105 for standard parameters, i.e. a catastrophically large uncertainty. A small | | − value of σ(T (x, y))/T (x, y) is possible only if ζ (1 ζ2) is very small, but apart from rec rec the fact that one should not rely on this image-distorting factor, it could only be suﬃciently small for y unrealistically close to the nadir. If one traces back the diﬀerence to the single-point source, one realizes that while σ(T ) √ rec scales in both cases as 1/ χ, the diﬀerence comes from T itself: for the single point rec source it is of order 1/χ, but for uniform T in the upper half plane of order χ, which explains a factor 1/χ2 worse relative RR for the latter compared to the former. The factor 1/χ in the single-point T arises from the cut-oﬀ of the κ integral: κ scales as 1/χ, and rec Max ∼ for x = 0 the κ-integral in (44) just gives a factor 2κ 1/χ, as the correlation-function Max is independent of κ in this case. On the other hand, for constant temperature in the upper half-plane, the cut-oﬀs κ do not play a role, as the δ(κ)-function picks up only κ = 0. Max This leads to the loss of one factor 1/χ in T . The second one comes from the integration rec (cid:48) ˜ over ζ in (68): the rapidly oscillating cos-function leads to a factor 1/k = χ, whereas c0 for the point-source only a single point ζ = ζ contributes, such that the cosine is of order one. s √ ∝ In the light of the RR of standard radiometers that typically scales as σ(T ) 1/ Bt , int where t is the integration time, the fact that σ(T ) in eqs.(60,67) is independent of the int rec ˜ bandwidth is rather surprising. Formally, the disappearing of b can be traced back to using the lowest order in the Laplace approximation of (58). The next order corrections are of order → O ˜ ˜ b, such that σ(T ) σ(T )(1+ (b)). One expects the sign of the correction to be positive, rec rec as the integrand is positive everywhere, and the lowest order approximations amounts to replacing the Gaussians by normalized Dirac-delta functions. Hence, for small but ﬁnite 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: reconstructed

text: temperature

text: depends

text: on

text: the

text: precise

text: value

text: of

text: a

text: as

text: well

text: as

text: the

text: ratio

text: k

text: /b.

text: Outside

text: the

text: mentioned

text: interval,

text: T

text: (x,

text: y)

text: oscillates

text: again

text: as

text: function

text: of

text: ζ,

text: which

text: can

text: be

text: rec

text: understood

text: from

text: the

text: fact

text: that

text: the

text: box

text: is

text: cut-oﬀ

text: when

text: ζ

text: gets

text: within

text: a

text: distance

text: of

text: or

text: ˆ

text: ζ.

text: The

text: sought-for

text: order

text: of

text: magnitude

text: can

text: be

text: estimated

text: from

text: the

text: maximum

text: value

text: of

text: as

text: function

text: of

text: a.

text: As

text: for

text: standard

text: parameters

text: k˜

text: /˜b

text: can

text: bound

text: the

text: sin-function

text: by

text: one

text: (while

text: still

text: having

text: a

text: in

text: which

text: case

text: we

text: obtain

text: T

text: (x,

text: y)

text: T

text: /k

text: T

text: χ

text: in

text: the

text: rec

text: mentioned

text: ζ-interval.

text: With

text: all

text: this,

text: and

text: approximating

text: ηˆ2

text: ηˆ

text: in

text: the

text: logarithm

text: in

text: we

text: ﬁnd

text: the

text: order

text: of

text: magnitude

text: σ(T

text: (x,

text: y))/T

text: (x,

text: y)

text: For

text: ζ

text: this

text: rec

text: is

text: of

text: order

text: for

text: standard

text: parameters,

text: i.e.

text: a

text: catastrophically

text: large

text: uncertainty.

text: A

text: small

text: value

text: of

text: σ(T

text: (x,

text: y))/T

text: (x,

text: y)

text: is

text: possible

text: only

text: if

text: ζ

text: is

text: very

text: small,

text: but

text: apart

text: from

text: rec

text: rec

text: the

text: fact

text: that

text: one

text: should

text: not

text: rely

text: on

text: this

text: image-distorting

text: factor,

text: it

text: could

text: only

text: be

text: suﬃciently

text: small

text: for

text: y

text: unrealistically

text: close

text: to

text: the

text: nadir.

text: If

text: one

text: traces

text: back

text: the

text: diﬀerence

text: to

text: the

text: single-point

text: source,

text: one

text: realizes

text: that

text: while

text: σ(T

text: rec

text: scales

text: in

text: both

text: cases

text: as

text: χ,

text: the

text: diﬀerence

text: comes

text: from

text: T

text: itself:

text: for

text: the

text: single

text: point

text: rec

text: source

text: it

text: is

text: of

text: order

text: but

text: for

text: uniform

text: T

text: in

text: the

text: upper

text: half

text: plane

text: of

text: order

text: χ,

text: which

text: explains

text: a

text: factor

text: worse

text: relative

text: RR

text: for

text: the

text: latter

text: compared

text: to

text: the

text: former.

text: The

text: factor

text: in

text: the

text: single-point

text: T

text: arises

text: from

text: the

text: cut-oﬀ

text: of

text: the

text: κ

text: integral:

text: κ

text: scales

text: as

text: and

text: rec

text: Max

text: for

text: x

text: the

text: κ-integral

text: in

text: just

text: gives

text: a

text: factor

text: 2κ

text: as

text: the

text: correlation-function

text: Max

text: is

text: independent

text: of

text: κ

text: in

text: this

text: case.

text: On

text: the

text: other

text: hand,

text: for

text: constant

text: temperature

text: in

text: the

text: upper

text: half-plane,

text: the

text: cut-oﬀs

text: κ

text: do

text: not

text: play

text: a

text: role,

text: as

text: the

text: δ(κ)-function

text: picks

text: up

text: only

text: κ

text: Max

text: This

text: leads

text: to

text: the

text: loss

text: of

text: one

text: factor

text: in

text: T

text: The

text: second

text: one

text: comes

text: from

text: the

text: integration

text: rec

text: over

text: ζ

text: in

text: the

text: rapidly

text: oscillating

text: cos-function

text: leads

text: to

text: a

text: factor

text: χ,

text: whereas

text: for

text: the

text: point-source

text: only

text: a

text: single

text: point

text: ζ

text: ζ

text: contributes,

text: such

text: that

text: the

text: cosine

text: is

text: of

text: order

text: one.

text: s

text: In

text: the

text: light

text: of

text: the

text: RR

text: of

text: standard

text: radiometers

text: that

text: typically

text: scales

text: as

text: σ(T

text: Bt

text: int

text: where

text: t

text: is

text: the

text: integration

text: time,

text: the

text: fact

text: that

text: σ(T

text: in

text: is

text: independent

text: of

text: the

text: int

text: rec

text: bandwidth

text: is

text: rather

text: surprising.

text: Formally,

text: the

text: disappearing

text: of

text: b

text: can

text: be

text: traced

text: back

text: to

text: using

text: the

text: lowest

text: order

text: in

text: the

text: Laplace

text: approximation

text: of

text: The

text: next

text: order

text: corrections

text: are

text: of

text: order

text: O

text: b,

text: such

text: that

text: σ(T

text: σ(T

text: (b)).

text: One

text: expects

text: the

text: sign

text: of

text: the

text: correction

text: to

text: be

text: positive,

text: rec

text: rec

text: as

text: the

text: integrand

text: is

text: positive

text: everywhere,

text: and

text: the

text: lowest

text: order

text: approximations

text: amounts

text: to

text: replacing

text: the

text: Gaussians

text: by

text: normalized

text: Dirac-delta

text: functions.

text: Hence,

text: for

text: small

text: but

text: ﬁnite

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: To see this, one ﬁrst shows with the help of (52) and in a few lines of calculation that (cid:48) ∗ (cid:48) V = C (r , r , ω , ω )C (r , r , ω , ω ) . (75) zz 1 1 1 1 zz 2 2 2 2 (cid:48) (cid:48) M = C (r , r , ω , ω )C (r , r , ω , ω ) . (76) zz 1 2 1 2 zz 1 2 1 2 (cid:48) (cid:48) − We have C (r , r , ω , ω ) from (56), where now κ = (ω ω )∆r/v , and correspondingly zz 1 1 1 1 1 1 s (cid:48) for C (r , r , ω , ω ). Whether V, M are large or small can be judged by comparing it to the zz 2 2 2 2 product of the stand(cid:113)ard deviations of each factor. This corresponds to calculating Pearson’s ≡ ≡ product-moment coeﬃcients [16] V res V and Mres M where we deﬁne, for σ(Cˆ)σ(Cˆ(cid:48)) σ(Cˆ)σ(Cˆ(cid:48)) ≡ (cid:60) (cid:61) ≡ (cid:48) ≡ (cid:48) (cid:48) complex Cˆ, σ(Cˆ) σ2( Cˆ) + σ2( Cˆ), and Cˆ Cˆ (r , r , ω , ω ), Cˆ Cˆ (r , r , ω , ω ) zz 1 2 1 2 zz 1 2 1 2 for short. Going through the same calculation as for V , we ﬁnd after some algebra σ2(Cˆ) = C (r , r , ω , ω )C (r , r , ω , ω ) (77) zz 1 1 1 1 zz 2 2 2 2 = (πK )2I2(0) , (78) 6 where (cid:90) (cid:113) √ ∞ −| | ≡ T˜ (κ)e κ η2+h˜2 0,η I(κ) dη , (79) (cid:82) −∞ η2 + h˜2 ∞ √ and hence I(0) = −∞ T˜0,η(0) dη. This implies η2(cid:12)+h˜2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:48) − (cid:48) − | | I((ω ω )∆r/v )I((ω ω )∆r/v ) V res = 1 1 s 2 2 s (80) I2(0) For M we have (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:48) (cid:12) (cid:48) |Mres| = (cid:12)(cid:12)Czz(r1, r2, ω1, ω2)Czz(r1, r(cid:12)(cid:12)2, ω1, ω2) (81) (cid:12)C (r , r , ω , ω )C (r , r(cid:12) , ω , ω ) zz 1 1 1 1 zz 2 2 2 2 J(κ (cid:48), ω (cid:48))J(κ (cid:48) , ω (cid:48) ) 12 c12 1 2 c1 2 = (82) I2(0) 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: To

text: see

text: this,

text: one

text: ﬁrst

text: shows

text: with

text: the

text: help

text: of

text: and

text: in

text: a

text: few

text: lines

text: of

text: calculation

text: that

text: V

text: C

text: (r

text: r

text: ω

text: ω

text: )C

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: M

text: C

text: (r

text: r

text: ω

text: ω

text: )C

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: We

text: have

text: C

text: (r

text: r

text: ω

text: ω

text: from

text: where

text: now

text: κ

text: (ω

text: ω

text: )∆r/v

text: and

text: correspondingly

text: zz

text: s

text: for

text: C

text: (r

text: r

text: ω

text: ω

text: Whether

text: V,

text: M

text: are

text: large

text: or

text: small

text: can

text: be

text: judged

text: by

text: comparing

text: it

text: to

text: the

text: zz

text: product

text: of

text: the

text: deviations

text: of

text: each

text: factor.

text: This

text: corresponds

text: to

text: calculating

text: Pearson’s

text: product-moment

text: coeﬃcients

text: V

text: res

text: V

text: and

text: Mres

text: M

text: where

text: we

text: deﬁne,

text: for

text: complex

text: Cˆ,

text: σ(Cˆ)

text: Cˆ)

text: Cˆ),

text: and

text: Cˆ

text: Cˆ

text: (r

text: r

text: ω

text: ω

text: Cˆ

text: Cˆ

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: for

text: short.

text: Going

text: through

text: the

text: same

text: calculation

text: as

text: for

text: V

text: we

text: ﬁnd

text: after

text: some

text: algebra

text: C

text: (r

text: r

text: ω

text: ω

text: )C

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: (πK

text: where

text: T˜

text: (κ)e

text: κ

text: I(κ)

text: dη

text: η2

text: and

text: hence

text: dη.

text: This

text: implies

text: I((ω

text: ω

text: )∆r/v

text: )I((ω

text: ω

text: )∆r/v

text: V

text: res

text: s

text: s

text: For

text: M

text: we

text: have

text: |Mres|

text: (r

text: r

text: ω

text: ω

text: )C

text: (r

text: ω

text: ω

text: zz

text: zz

text: J(κ

text: ω

text: ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where (cid:90) (cid:113) (cid:113) (cid:113) ≡ dη ∆rω (cid:48) η J(κ (cid:48), ω (cid:48)) K(κ (cid:48) η2 + h˜2, c12 )T˜ (κ (cid:48)) (83) 12 c12 12 0,η 12 c η2 + h˜2 η2 + h˜2 ≡ (cid:48) − ≡ − (cid:48) ≡ (cid:48) ≡ (cid:48) with κ (cid:48) (ω ω )/v , κ (cid:48) (ω ω )/v , ω (cid:48) (ω +ω )/2, and ω (cid:48) (ω +ω )/2. 12 2 1 s 1 2 2 1 2 c12 1 2 c1 2 1 2 (cid:48) From the properties of the integration kernel K we know that C (r , r , ω , ω ) vanishes iﬀ zz 1 2 1 2 | − (cid:48) | (cid:38) (cid:48) (cid:48) | | (cid:28) ˜ ω ω (ω +ω )v /(2ch), and correspondingly for C (r , r , ω , ω ). Hence, for M 1 1 2 1 2 s zz 1 2 1 2 (cid:48) | − (cid:48) | (cid:38) | (cid:48) − | (cid:38) ˜ ˜ and ω and ω all of order ω , we need ω ω ω v /(ch) or ω ω ω v /(ch). 1,2 1,2 0 1 2 0 s 1 2 0 s (cid:29) ˜ Note that ω v /(ch) = (1/χ)v /h v /h. 0 s s s For determining the properties of V , we consider our two previous cases of sources. ˜ Case 1: Single point source. Here we have T independent of κ, see eq.(41), which inserted into (79) yields (cid:113) √ T ∆r −| | I(κ) = √ 0 e κ ηs2+h˜2 (84) 2π η2 + h˜2 s and hence √ | | − | (cid:48) − | | (cid:48) − | V res = e ∆vsr ηs2+h˜2( ω1 ω1 + ω2 ω2 ) . (85) (cid:113) ∼ | | (cid:28) | (cid:48) − | | (cid:48) − | For sources at η h˜, we have therefore V res 1 iﬀ ω ω > δω or ω ω > δω s 1 1 2 2 ≡ ∼ ∼ − where δω v /(∆r η2 + h˜2) v /(∆rh˜) = v /h 10 2 Hz. s s s s ˜ Case 2: Constant temperature ﬁeld in the positive upper half plane. Here, T (κ , η) is given x (cid:48) (cid:54) (cid:48) (cid:54) by eq.(64). Hence, V res = 0 as soon as ω = ω or ω = ω . Of course, the δ(κ ) function 1 1 2 2 x in (64) arises from the complete lack of structure of the temperature proﬁle in x-direction. (cid:0) (cid:1) More realistic is at least a cut-oﬀ at the size of Earth, which we take as the same as in y direction. In that case one ﬁnds T˜ (cid:48)(κ (cid:18)) = (yˆ(cid:19)T(cid:90)/π)sinc κxyˆ and hence r x 0 π (cid:113) √ κ yˆ ηˆ dη −| | I(κ) = (yˆT /π)sinc x e κ η2+h˜2 . (86) 0 π 0 η2 + h˜2 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: dη

text: ∆rω

text: η

text: J(κ

text: ω

text: K(κ

text: η2

text: )T˜

text: (κ

text: c

text: η2

text: η2

text: with

text: κ

text: (ω

text: ω

text: )/v

text: κ

text: (ω

text: ω

text: )/v

text: ω

text: (ω

text: +ω

text: and

text: ω

text: (ω

text: +ω

text: s

text: From

text: the

text: properties

text: of

text: the

text: integration

text: kernel

text: K

text: we

text: know

text: that

text: C

text: (r

text: r

text: ω

text: ω

text: vanishes

text: iﬀ

text: zz

text: ω

text: ω

text: (ω

text: +ω

text: )v

text: and

text: correspondingly

text: for

text: C

text: (r

text: r

text: ω

text: ω

text: Hence,

text: for

text: M

text: s

text: zz

text: and

text: ω

text: and

text: ω

text: all

text: of

text: order

text: ω

text: we

text: need

text: ω

text: ω

text: ω

text: v

text: /(ch)

text: or

text: ω

text: ω

text: ω

text: v

text: /(ch).

text: s

text: s

text: Note

text: that

text: ω

text: v

text: /(ch)

text: /h

text: v

text: /h.

text: s

text: s

text: s

text: For

text: determining

text: the

text: properties

text: of

text: V

text: we

text: consider

text: our

text: two

text: previous

text: cases

text: of

text: sources.

text: Case

text: Single

text: point

text: source.

text: Here

text: we

text: have

text: T

text: independent

text: of

text: κ,

text: see

text: which

text: inserted

text: into

text: yields

text: T

text: ∆r

text: I(κ)

text: e

text: κ

text: 2π

text: η2

text: s

text: and

text: hence

text: V

text: res

text: e

text: ∆vsr

text: ω1

text: ω1

text: ω2

text: ω2

text: For

text: sources

text: at

text: η

text: h˜,

text: we

text: have

text: therefore

text: V

text: res

text: iﬀ

text: ω

text: ω

text: δω

text: or

text: ω

text: ω

text: δω

text: s

text: where

text: δω

text: v

text: /(∆r

text: η2

text: v

text: /(∆rh˜)

text: v

text: /h

text: Hz.

text: s

text: s

text: s

text: s

text: Case

text: Constant

text: temperature

text: ﬁeld

text: in

text: the

text: positive

text: upper

text: half

text: plane.

text: Here,

text: T

text: (κ

text: η)

text: is

text: given

text: x

text: by

text: Hence,

text: V

text: res

text: as

text: soon

text: as

text: ω

text: ω

text: or

text: ω

text: ω

text: Of

text: course,

text: the

text: δ(κ

text: function

text: x

text: in

text: arises

text: from

text: the

text: complete

text: lack

text: of

text: structure

text: of

text: the

text: temperature

text: proﬁle

text: in

text: x-direction.

text: More

text: realistic

text: is

text: at

text: least

text: a

text: cut-oﬀ

text: at

text: the

text: size

text: of

text: Earth,

text: which

text: we

text: take

text: as

text: the

text: same

text: as

text: in

text: y

text: direction.

text: In

text: that

text: case

text: one

text: ﬁnds

text: T˜

text: κxyˆ

text: and

text: hence

text: r

text: x

text: π

text: κ

text: yˆ

text: ηˆ

text: dη

text: I(κ)

text: (yˆT

text: /π)sinc

text: x

text: e

text: κ

text: π

text: η2

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text: ≡ K (T (ω ), T (ω )) K(T (ω ), T (ω ))/K(T (ω ), T (ω )) (88) rel rec 01 rec 02 rec 01 rec 02 rec 01 rec 01 that obviously satisﬁes K (T (ω ), T (ω )) = 1. We have rel rec 01 rec 01 (cid:113) (cid:18) (cid:19) (cid:90) | | ˜ ˜ 1 2 ˜ ˜ kc1kc2 K(T (ω ), T (ω )) = dκ dκ dk dk rec 01 rec 02 1 2 c1 c2 2πK ∆r 2π 6 × (cid:0)− − − − √˜b √˜b e i(κ1 κ2)(x˜ x˜0)ei(k˜c1 k˜c2)ζF (k˜ , k˜(1), )F (k˜ , k˜(2), ) c1 c0 c1 c0 2 2 (cid:1) × (cid:104) ∗ (cid:48) (cid:48) (cid:105) ˆ ˆ C (r , r , ω , ω )C (r , r , ω , ω ) zz 1 2 1 2 zz 1 2 1 2 − (cid:104) ∗ (cid:105)(cid:104) (cid:48) (cid:48) (cid:105) ˆ ˆ C (r , r , ω , ω ) C (r , r , ω , ω ) , (89) zz 1 2 1 2 zz 1 2 1 2 − (cid:48) − (cid:48) (cid:48) ˜ ˜ where κ = (ω ω )∆r/v , κ = (ω ω )∆r/v , k = (ω + ω )∆r/(2c), k = (ω + 1 2 1 s 2 2 1 s c1 1 2 c2 1 ω(cid:48) )∆r/(2c), k˜(i) = ω(i) (i = 1, 2), F (k˜ , k˜ , √˜b ) = A(ω , ω )A∗(ω , ω ) with k˜ = ∆rω /c 2 c0 0 c c0 2 1 0 2 0 c0 0 ∈ R (see eq.(36)), and we have used that T . We evaluate K(T (ω ), T (ω )) for rec rec 01 rec 02 the case of constant temperature in the upper half plane. Using (73), (65), and switching (cid:48) (cid:48) ˜ momentarily to integration variables ω , ω , ω , ω , and then back to κ k , we are led to (cid:90) 1 2 1 2 1 c1 T 2v | | √˜b √˜b K(T (ω ), T (ω )) = 0 s dκ dk˜ k˜ e2iζk˜c1F (k˜ , k˜(1), )F (k˜ , k˜(2), ) .(90) rec 01 rec 02 2πc 1 c1 c1 c1 c0 2 c1 c0 2 The integral is clearly real, as it should. The integral over κ leads, when integrated 1 −∞ ∞ from to to a divergent factor, but that factor cancels (together with the remaining prefactor T 2v /(2πc)) when we consider the re-scaled version of the correlation function 0 s Krel(Trec(ω01√), Trec(ω02)). If we set k˜c(02) = k˜c(01) + δk˜c0, it is clear that the only remaining scale ˜ ˜ ˜ for δk is b/ 2. The remaining integral over k in (90) can in fact be evaluated analytically. c0 c1 ˜ The result is too cumbersome to be reported here, but plotting it as function of δk shows c0 ˜ that indeed the correlations decay only on a scale of order b. This proves that by shifting the center frequency within the available bandwidth one cannot gain independent estimates of T that would allow one to improve substantially the signal-to-noise ratio. rec C. Additional antennas 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K

text: (T

text: (ω

text: T

text: (ω

text: K(T

text: (ω

text: T

text: (ω

text: ))/K(T

text: (ω

text: T

text: (ω

text: rel

text: rec

text: rec

text: rec

text: rec

text: rec

text: rec

text: that

text: obviously

text: satisﬁes

text: K

text: (T

text: (ω

text: T

text: (ω

text: We

text: have

text: rel

text: rec

text: rec

text: K(T

text: (ω

text: T

text: (ω

text: dκ

text: dκ

text: dk

text: dk

text: rec

text: rec

text: ∆r

text: 2π

text: √˜b

text: √˜b

text: e

text: (k˜

text: )F

text: (k˜

text: ˆ

text: ˆ

text: C

text: (r

text: r

text: ω

text: ω

text: )C

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: ˆ

text: ˆ

text: C

text: (r

text: r

text: ω

text: ω

text: C

text: (r

text: r

text: ω

text: ω

text: zz

text: zz

text: where

text: κ

text: (ω

text: ω

text: )∆r/v

text: κ

text: (ω

text: ω

text: )∆r/v

text: k

text: (ω

text: ω

text: k

text: (ω

text: s

text: s

text: k˜(i)

text: ω(i)

text: (i

text: F

text: (k˜

text: k˜

text: √˜b

text: A(ω

text: ω

text: )A∗(ω

text: ω

text: with

text: k˜

text: ∆rω

text: /c

text: c

text: R

text: (see

text: and

text: we

text: have

text: used

text: that

text: T

text: We

text: evaluate

text: K(T

text: (ω

text: T

text: (ω

text: for

text: rec

text: rec

text: rec

text: the

text: case

text: of

text: constant

text: temperature

text: in

text: the

text: upper

text: half

text: plane.

text: Using

text: and

text: switching

text: momentarily

text: to

text: integration

text: variables

text: ω

text: ω

text: ω

text: ω

text: and

text: then

text: back

text: to

text: κ

text: k

text: we

text: are

text: led

text: to

text: T

text: √˜b

text: √˜b

text: K(T

text: (ω

text: T

text: (ω

text: s

text: dκ

text: dk˜

text: k˜

text: (k˜

text: )F

text: (k˜

text: rec

text: rec

text: The

text: integral

text: is

text: clearly

text: real,

text: as

text: it

text: should.

text: The

text: integral

text: over

text: κ

text: leads,

text: when

text: integrated

text: from

text: to

text: to

text: a

text: divergent

text: factor,

text: but

text: that

text: factor

text: cancels

text: (together

text: with

text: the

text: remaining

text: prefactor

text: T

text: when

text: we

text: consider

text: the

text: re-scaled

text: version

text: of

text: the

text: correlation

text: function

text: s

text: If

text: we

text: set

text: it

text: is

text: clear

text: that

text: the

text: only

text: remaining

text: scale

text: for

text: δk

text: is

text: b/

text: The

text: remaining

text: integral

text: over

text: k

text: in

text: can

text: in

text: fact

text: be

text: evaluated

text: analytically.

text: The

text: result

text: is

text: too

text: cumbersome

text: to

text: be

text: reported

text: here,

text: but

text: plotting

text: it

text: as

text: function

text: of

text: δk

text: shows

text: that

text: indeed

text: the

text: correlations

text: decay

text: only

text: on

text: a

text: scale

text: of

text: order

text: b.

text: This

text: proves

text: that

text: by

text: shifting

text: the

text: center

text: frequency

text: within

text: the

text: available

text: bandwidth

text: one

text: cannot

text: gain

text: independent

text: estimates

text: of

text: T

text: that

text: would

text: allow

text: one

text: to

text: improve

text: substantially

text: the

text: signal-to-noise

text: ratio.

text: rec

text: C.

text: Additional

text: antennas

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text: The thermal ﬂuctuations of the electric ﬁeld are isotropic in their intensity, such that (cid:104)| | (cid:105) (cid:104)| | (cid:105) one third of the energy is in a given polarization direction i, i.e. E˜ (ω) 2 = 1 E˜ (ω) 2 . i,r1 3 r1 Inserting eq.(100) for the latter quantity,(cid:90)we are led to (cid:104)| | (cid:105) τ k ω2T (x, y) cos θ(x, y, h) E˜ (ω) 2 = a B dx dy B . (104) i,r1 3π3(cid:15) c3 h2 + x2 + y2 0 Comparison with eq.(103) allows one to identify (cid:104)| | (cid:105) j˜(x, y, ω) 2 = K T (x, y), (105) i 2 eﬀ ≡ with K = 32τ τ k /(3τ l3dµ c) and T (x, y) T (x, y) cos θ(x, y, h). Thus, the 2 a c B F c 0 eﬀ B current ﬂuctuations are given directly by the brightness temperature (rescaled by the directional-cos θ(x, y, h)), up to a constant prefactor. As mentioned in the Introduction, we write T for short for T in the rest of the article. The constant prefactor depends on eﬀ the time intervals for averaging and the Fourier transforms, but in the end we will always be interested in relative radiometric resolution, i.e. σ(T (x, y))/T (x, y), where σ(T (x, y)) denotes the standard deviation of the reconstructed temperatures over the thermal ensemble of the radiation ﬁeld, such that the constant prefactor cancels out. 2. Circular symmetry A Gaussian distribution of a complex jointly-Gaussian random vector z = ∈ C ∀ (z , z , . . . , z ) n is fully characterized by the expectation values, E(z ) i, the covariance 1 2 n i † matrix K = E[z z ], and the pseudo-covariance matrix M = E[z zt]. Both matrices together specify the correlations between the four diﬀerent combinations of real and imaginary parts of the z . The Gaussian distribution is called circularly symmetric, if P (z) is invariant under i (cid:55)→ the transformation z zeiφ with an arbitrary real phase φ. One shows that a distribution ∀ is Gaussian symmetric if and only if M = 0. This implies immediately that E[z ] = 0 i [17]. i The corresponding deﬁnitions and statements for complex Gaussian processes are easily obtained by replacing the discrete index i in z by a continuous one, e.g. a time argument, i 40 ##LTLine## ##LTLine## ##LTLine##

text: The

text: thermal

text: ﬂuctuations

text: of

text: the

text: electric

text: ﬁeld

text: are

text: isotropic

text: in

text: their

text: intensity,

text: such

text: that

text: one

text: third

text: of

text: the

text: energy

text: is

text: in

text: a

text: given

text: polarization

text: direction

text: i,

text: i.e.

text: E˜

text: (ω)

text: E˜

text: (ω)

text: Inserting

text: for

text: the

text: latter

text: are

text: led

text: to

text: τ

text: k

text: (x,

text: y)

text: cos

text: θ(x,

text: y,

text: h)

text: E˜

text: (ω)

text: a

text: B

text: dx

text: dy

text: B

text: Comparison

text: with

text: allows

text: one

text: to

text: identify

text: j˜(x,

text: y,

text: ω)

text: K

text: T

text: (x,

text: y),

text: i

text: eﬀ

text: with

text: K

text: τ

text: k

text: c)

text: and

text: T

text: (x,

text: y)

text: T

text: (x,

text: y)

text: cos

text: θ(x,

text: y,

text: h).

text: Thus,

text: the

text: a

text: c

text: B

text: F

text: c

text: eﬀ

text: B

text: current

text: ﬂuctuations

text: are

text: given

text: directly

text: by

text: the

text: brightness

text: temperature

text: (rescaled

text: by

text: the

text: directional-cos

text: θ(x,

text: y,

text: h)),

text: up

text: to

text: a

text: constant

text: prefactor.

text: As

text: mentioned

text: in

text: the

text: Introduction,

text: we

text: write

text: T

text: for

text: short

text: for

text: T

text: in

text: the

text: rest

text: of

text: the

text: article.

text: The

text: constant

text: prefactor

text: depends

text: on

text: eﬀ

text: the

text: time

text: intervals

text: for

text: averaging

text: and

text: the

text: Fourier

text: transforms,

text: but

text: in

text: the

text: end

text: we

text: will

text: always

text: be

text: interested

text: in

text: relative

text: radiometric

text: resolution,

text: i.e.

text: σ(T

text: (x,

text: y))/T

text: (x,

text: y),

text: where

text: σ(T

text: (x,

text: y))

text: denotes

text: the

text: standard

text: deviation

text: of

text: the

text: reconstructed

text: temperatures

text: over

text: the

text: thermal

text: ensemble

text: of

text: the

text: radiation

text: ﬁeld,

text: such

text: that

text: the

text: constant

text: prefactor

text: cancels

text: out.

text: Circular

text: symmetry

text: A

text: Gaussian

text: distribution

text: of

text: a

text: complex

text: jointly-Gaussian

text: random

text: vector

text: z

text: C

text: (z

text: z

text: z

text: n

text: is

text: fully

text: characterized

text: by

text: the

text: expectation

text: values,

text: E(z

text: i,

text: the

text: covariance

text: n

text: i

text: matrix

text: K

text: E[z

text: z

text: and

text: the

text: pseudo-covariance

text: matrix

text: M

text: E[z

text: zt].

text: Both

text: matrices

text: together

text: specify

text: the

text: correlations

text: between

text: the

text: four

text: diﬀerent

text: combinations

text: of

text: real

text: and

text: imaginary

text: parts

text: of

text: the

text: z

text: The

text: Gaussian

text: distribution

text: is

text: called

text: circularly

text: symmetric,

text: if

text: P

text: (z)

text: is

text: invariant

text: under

text: i

text: the

text: transformation

text: z

text: zeiφ

text: with

text: an

text: arbitrary

text: real

text: phase

text: φ.

text: One

text: shows

text: that

text: a

text: distribution

text: is

text: Gaussian

text: symmetric

text: if

text: and

text: only

text: if

text: M

text: This

text: implies

text: immediately

text: that

text: E[z

text: i

text: i

text: The

text: corresponding

text: deﬁnitions

text: and

text: statements

text: for

text: complex

text: Gaussian

text: processes

text: are

text: easily

text: obtained

text: by

text: replacing

text: the

text: discrete

text: index

text: i

text: in

text: z

text: by

text: a

text: continuous

text: one,

text: e.g.

text: a

text: time

text: argument,

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ˆ or in our case of E (ω), a 4-component real vector with a “continuous index” ω, r. In z,r ˆ order to show that E (ω) is a circularly symmetric complex Gaussian process over ω, r, z,r ≡ (cid:104) (cid:105) ˆ ˆ we need to prove that 0 = M(r , r , ω , ω ) E (ω )E (ω ) , at least in the narrow 1 2 1 2 z,r1 1 z,r2 2 frequency band that we are interested in. In view of eq.(5), for this it is enough to show ≡ (cid:104) (cid:105) that M ˜j (ω )˜j (ω ) = 0. Expressed as Fourier transforms of the time-dependent J z,r1 1 z,r2 2 current-densities, this correlator equa(cid:90)ls ∞ (cid:104) (cid:105) 1 − (cid:104) (cid:105) ˜j (ω )˜j (ω ) = dt dt e i(ω1t1+ω2t2) j (t )j (t ) . (106) z,r1 1 z,r2 2 2π −∞ 1 2 z,r1 1 z,r2 2 The physical origin of the current ﬂuctuations are thermal ﬂuctuations, and the condition of thermal equilibrium implies that the current correlator is invariant under global time- translation (i.e. a shift of the origin of the time axis of t and t by the same amount) and 1 2 − (cid:104) (cid:105) − hence depends only on t t , j (t )j (t ) = f(r , r , τ), where τ = t t , and we 2 1 z,r1 1 z,r2 2 1 2 2 1 will also use t = (t + t )/2. With this we get 2 1 (cid:90) (cid:90) ∞ ∞ 1 − − − M = dt e i(ω1+ω2)t dτ e i(ω2 ω1)τ/2f(r , r , τ) J √2π −∞ −∞ 1 2 − ˜ = 2πδ(ω + ω )f(r , r , (ω ω )/2) , (107) 1 2 1 2 2 1 ˜ where f(r , r , ω) is the Fourier transform of f(r , r , t) with respect to time. The δ- 1 2 1 2 − function implies that M vanishes unless ω = ω . But we are interested only in frequencies J 1 2 ∈ − in the small interval ω [ω ∆ω, ω + ∆ω], centered close to ω of the order of 1.4GHz. 2 1 1 0 Hence, in this frequency interval we have indeed M = 0, and the complex Gaussian process J ˆ E (ω) over ω, r can be considered as circularly symmetric and eq.(53) valid. In particular, z,r ∗ it does not contain any correlator of the type E E, but only of the type E E . [1] Thompson, A. R., Moran, J. M., & George W. Swenson, J., Interferometry and Synthesis in Radio Astronomy, 2nd Edition (WILEY-VCH, 2001). [2] Kerr, Y. H. et al., Soil Moisture Retrieval from Space: The Soil Moisture and Ocean Salinity (SMOS) Mission, IEEE Transactions on geoscience and remote sensing 39 (8) (2001). 41 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ˆ

text: or

text: in

text: our

text: case

text: of

text: E

text: (ω),

text: a

text: real

text: vector

text: with

text: a

text: “continuous

text: index”

text: ω,

text: r.

text: In

text: z,r

text: ˆ

text: order

text: to

text: show

text: that

text: E

text: (ω)

text: is

text: a

text: circularly

text: symmetric

text: complex

text: Gaussian

text: process

text: over

text: ω,

text: r,

text: z,r

text: ˆ

text: ˆ

text: we

text: need

text: to

text: prove

text: that

text: M(r

text: r

text: ω

text: ω

text: E

text: (ω

text: )E

text: (ω

text: at

text: least

text: in

text: the

text: narrow

text: frequency

text: band

text: that

text: we

text: are

text: interested

text: in.

text: In

text: view

text: of

text: for

text: this

text: it

text: is

text: enough

text: to

text: show

text: that

text: M

text: ˜j

text: (ω

text: )˜j

text: (ω

text: Expressed

text: as

text: Fourier

text: transforms

text: of

text: the

text: time-dependent

text: J

text: current-densities,

text: this

text: correlator

text: ˜j

text: (ω

text: )˜j

text: (ω

text: dt

text: dt

text: e

text: j

text: (t

text: )j

text: (t

text: 2π

text: The

text: physical

text: origin

text: of

text: the

text: current

text: ﬂuctuations

text: are

text: thermal

text: ﬂuctuations,

text: and

text: the

text: condition

text: of

text: thermal

text: equilibrium

text: implies

text: that

text: the

text: current

text: correlator

text: is

text: invariant

text: under

text: global

text: time-

text: translation

text: (i.e.

text: a

text: shift

text: of

text: the

text: origin

text: of

text: the

text: time

text: axis

text: of

text: t

text: and

text: t

text: by

text: the

text: same

text: amount)

text: and

text: hence

text: depends

text: only

text: on

text: t

text: t

text: j

text: (t

text: )j

text: (t

text: f(r

text: r

text: τ),

text: where

text: τ

text: t

text: t

text: and

text: we

text: will

text: also

text: use

text: t

text: (t

text: t

text: With

text: this

text: we

text: get

text: M

text: dt

text: e

text: dτ

text: e

text: r

text: τ)

text: J

text: √2π

text: 2πδ(ω

text: ω

text: )f(r

text: r

text: (ω

text: ω

text: where

text: f(r

text: r

text: ω)

text: is

text: the

text: Fourier

text: transform

text: of

text: f(r

text: r

text: t)

text: with

text: respect

text: to

text: time.

text: The

text: δ-

text: function

text: implies

text: that

text: M

text: vanishes

text: unless

text: ω

text: ω

text: But

text: we

text: are

text: interested

text: only

text: in

text: frequencies

text: J

text: in

text: the

text: small

text: interval

text: ω

text: [ω

text: ∆ω,

text: ω

text: ∆ω],

text: centered

text: close

text: to

text: ω

text: of

text: the

text: order

text: of

text: Hence,

text: in

text: this

text: frequency

text: interval

text: we

text: have

text: indeed

text: M

text: and

text: the

text: complex

text: Gaussian

text: process

text: J

text: ˆ

text: E

text: (ω)

text: over

text: ω,

text: r

text: can

text: be

text: considered

text: as

text: circularly

text: symmetric

text: and

text: valid.

text: In

text: particular,

text: z,r

text: it

text: does

text: not

text: contain

text: any

text: correlator

text: of

text: the

text: type

text: E

text: E,

text: but

text: only

text: of

text: the

text: type

text: E

text: E

text: Thompson,

text: A.

text: R.,

text: Moran,

text: J.

text: M.,

text: George

text: W.

text: Swenson,

text: J.,

text: Interferometry

text: and

text: Synthesis

text: in

text: Radio

text: Astronomy,

text: Edition

text: (WILEY-VCH,

text: Kerr,

text: Y.

text: H.

text: et

text: al.,

text: Soil

text: Moisture

text: Retrieval

text: from

text: Space:

text: The

text: Soil

text: Moisture

text: and

text: Ocean

text: Salinity

text: (SMOS)

text: Mission,

text: IEEE

text: Transactions

text: on

text: geoscience

text: and

text: remote

text: sensing

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) ∞ (cid:48) 1 | − − (cid:48)| − | − (cid:48)| exp [. . .] dω = 2πδ(τ + ( r v τ/2 r r + v τ/2 r )) eiτ ω1+ω2 . (18) 1 s 2 s 2 −∞ c We introduce center-of-mass and relative coordinates for r and r , R = (r + r )/2 and 1 2 1 2 − | − (cid:48) ± | ∆r = r r . We further restrict v τ to values much smaller than R r ∆r . This implies 2 1 s a limitation of the integration range for τ when calculating the Fourier components, but it | − (cid:48) ± | ≥ ≤ is a mild one. Since R r ∆r h, it is enough to have τ h/v , which is typically of s order 100s, and therefore gives time to resolve Fourier components down to a hundredth of a Hertz. We can then approximate to ﬁrst order in v , s | − − (cid:48)| − | − (cid:48)| (cid:39) − · r v τ/2 r r + v τ/2 r eˆ − (cid:48) (∆r + v τ) . (19) 1 s 2 s R r s | | | − (cid:48)| Neglecting terms of order ∆r + v τ / R r and of order β = v /c in the prefactor of s s the exponential, as well as a second order term of order βω ∆r/c in the phase, the integral c over the Dirac δ-function gives (cid:90) (cid:90) ∞ ∞ − (cid:48) (cid:20) (cid:48) (cid:21)(cid:48) T˜ (cid:48)(κ )e iκxx Cij(r1, r2, ω1, ω2) = K6δij −∞ dx −∞ dy 0,y|R −x r(cid:48)|2 · × ∆r eˆ − (cid:48) R r exp i ω , (20) c c and K = (2π)3/2ω2K /v . The unit vector eˆ − (cid:48) is obtained by taking the original 6 0 5 s R r (cid:48) (cid:48) (cid:48) center of mass position of the antennas at R = (x , 0, h), and r = (x , y , 0). Eq.(20) is one 0 of the central results of this paper. It shows that the two-frequency correlation function of the ﬁelds at diﬀerent antenna positions is related linearly via a 2D integral-transformation to the brightness temperature ﬁeld in the source plane, or more precisely to the Fourier transform of that ﬁeld in x-direction. With T (x, y) deﬁned on a 2D grid, the reconstruction of T (x, y) from the measured correlation function thus becomes a matrix inversion problem that has to be performed numerically in general. A crucial question is the conditioning of the inversion problem. It will be studied in more detail in a subsequent paper dedicated to a numerical approach [14]. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: exp

text: dω

text: 2πδ(τ

text: r

text: v

text: r

text: r

text: v

text: r

text: eiτ

text: s

text: s

text: c

text: We

text: introduce

text: center-of-mass

text: and

text: relative

text: coordinates

text: for

text: r

text: and

text: r

text: R

text: (r

text: r

text: and

text: ∆r

text: r

text: r

text: We

text: further

text: restrict

text: v

text: τ

text: to

text: values

text: much

text: smaller

text: than

text: R

text: r

text: ∆r

text: This

text: implies

text: s

text: a

text: limitation

text: of

text: the

text: integration

text: range

text: for

text: τ

text: when

text: calculating

text: the

text: Fourier

text: components,

text: but

text: it

text: is

text: a

text: mild

text: one.

text: Since

text: R

text: r

text: ∆r

text: h,

text: it

text: is

text: enough

text: to

text: have

text: τ

text: h/v

text: which

text: is

text: typically

text: of

text: s

text: order

text: and

text: therefore

text: gives

text: time

text: to

text: resolve

text: Fourier

text: components

text: down

text: to

text: a

text: hundredth

text: of

text: a

text: Hertz.

text: We

text: can

text: then

text: approximate

text: to

text: ﬁrst

text: order

text: in

text: v

text: s

text: r

text: v

text: r

text: r

text: v

text: r

text: eˆ

text: (∆r

text: v

text: τ)

text: s

text: s

text: R

text: r

text: s

text: Neglecting

text: terms

text: of

text: order

text: ∆r

text: v

text: τ

text: R

text: r

text: and

text: of

text: order

text: β

text: v

text: /c

text: in

text: the

text: prefactor

text: of

text: s

text: s

text: the

text: exponential,

text: as

text: well

text: as

text: a

text: second

text: order

text: term

text: of

text: order

text: βω

text: ∆r/c

text: in

text: the

text: phase,

text: the

text: integral

text: c

text: over

text: the

text: Dirac

text: δ-function

text: gives

text: T˜

text: )e

text: iκxx

text: dx

text: dy

text: −x

text: ∆r

text: eˆ

text: R

text: r

text: exp

text: i

text: ω

text: c

text: c

text: and

text: K

text: /v

text: The

text: unit

text: vector

text: eˆ

text: is

text: obtained

text: by

text: taking

text: the

text: original

text: s

text: R

text: r

text: center

text: of

text: mass

text: position

text: of

text: the

text: antennas

text: at

text: R

text: (x

text: h),

text: and

text: r

text: (x

text: y

text: is

text: one

text: of

text: the

text: central

text: results

text: of

text: this

text: paper.

text: It

text: shows

text: that

text: the

text: two-frequency

text: correlation

text: function

text: of

text: the

text: ﬁelds

text: at

text: diﬀerent

text: antenna

text: positions

text: is

text: related

text: linearly

text: via

text: a

text: integral-transformation

text: to

text: the

text: brightness

text: temperature

text: ﬁeld

text: in

text: the

text: source

text: plane,

text: or

text: more

text: precisely

text: to

text: the

text: Fourier

text: transform

text: of

text: that

text: ﬁeld

text: in

text: x-direction.

text: With

text: T

text: (x,

text: y)

text: deﬁned

text: on

text: a

text: grid,

text: the

text: reconstruction

text: of

text: T

text: (x,

text: y)

text: from

text: the

text: measured

text: correlation

text: function

text: thus

text: becomes

text: a

text: matrix

text: inversion

text: problem

text: that

text: has

text: to

text: be

text: performed

text: numerically

text: in

text: general.

text: A

text: crucial

text: question

text: is

text: the

text: conditioning

text: of

text: the

text: inversion

text: problem.

text: It

text: will

text: be

text: studied

text: in

text: more

text: detail

text: in

text: a

text: subsequent

text: paper

text: dedicated

text: to

text: a

text: numerical

text: approach

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: theoretical results. Indeed, there is a big disconnect between theory and practice [2]. One could interpret Babai’s recent breakthrough, the quasipolynomial time algorithm [1], as a ﬁrst step of convergence. The result implies that if graph isomorphism is NP-complete then all problems in NP have quasi-polynomial time algorithms, which may lead one to also theoretically believe that graph isomorphism is not NP-complete. In this paper we are interested in creating diﬃcult benchmark instances for the graph isomor- phism problem. It would be a misconception to think that for unstructured randomly generated instances graph isomorphism should be hard in practice. Quite the opposite is true. Instances that encompass suﬃcient randomness usually turn out to be among the easiest instances (see [15] for extensive tests suites on various kinds of graphs and [3, 11] for theoretical arguments). In fact, it is non-trivial to create challenging instances for eﬃcient isomorphism solvers and so far the number of diﬃcult benchmark graphs available is quite limited. The prime source for diﬃcult instances are graphs arising from combinatorial structures. In 1978, Mathon [14] provided a set of benchmark graphs arising from combinatorial objects such as strongly regular graphs, block designs and coherent conﬁgurations. The actual graphs that are provided, having less than 50 vertices, are small by todays standards. Nowadays, larger combinatorial objects yielding diﬃcult graphs are known. The most challenging examples are for instance incidence graphs of projective planes, Hadamard matrices, or Latin squares (see [15]). It is important to understand that not all such incidence structures yield diﬃcult graphs. Indeed, typically there is an algebraic version that can readily be generated. For example to obtain a projective plane it is possible to consider incidences between one- and two-dimensional subspaces of a three-dimensional vector space over a ﬁnite ﬁeld. If however one uses such an algebraic construction then the resulting graph will automatically have a large automorphism group. The practical isomorphism solvers are tuned to ﬁnding such automorphisms and to exploit them for search space contraction. On top of that, there are also theoretical reasons that lead one to believe that graphs with automorphisms make easier examples for the practical tools (see [22, Theorem 9]). The non-algebraic counterparts of combinatorial structures often do not have the shortcoming of having a large automorphism group, but in contrast to the algebraic constructions they are rare and diﬃcult to generate. The second source of diﬃcult instances is based on modiﬁcations of the so called Cai-Fürer- Immerman construction [5]. This construction is connected to the family of Weisfeiler-Leman al- gorithms, which for every integer k contains a k-dimensional version that constitutes a polynomial time algorithm used to distinguish non-isomorphic graphs. The 1-dimensional variant (usually called color reﬁnement) is a subroutine in all competitive practical isomorphism solvers. For larger k, how- ever, the k-dimensional variant becomes impractical due to excessive space consumption. For each k, Cai, Fürer and Immerman construct pairs of non-isomorphic graphs that are not distinguished by the k-dimensional variant of algorithm. In 1996, the construction was adapted by Miyazaki to explic- itly show that the then current version of nauty has exponential running time in the worst case [17]. The Cai-Fürer-Immerman graphs and the Miyazaki graphs constitute families of benchmark graphs that are infeasible for simplistic algorithms for the graph isomorphism problem. Nowadays, however, most state-of-the-art solvers scale reasonably well on these instances (Figures 6.9 and 6.10). Contribution. We describe a construction to eﬃciently generate instances on any desired number of vertices for the graph isomorphism problem that are diﬃcult or even infeasible for all state-of- the-art graph isomorphism solvers. Experiments show the graphs pose by far the most challenging graphs to date. Already for 1500 vertices our new instances are by several orders of magnitude more diﬃcult than all previously 2

text: theoretical

text: results.

text: Indeed,

text: there

text: is

text: a

text: big

text: disconnect

text: between

text: theory

text: and

text: practice

text: One

text: could

text: interpret

text: Babai’s

text: recent

text: breakthrough,

text: the

text: quasipolynomial

text: time

text: algorithm

text: as

text: a

text: ﬁrst

text: step

text: of

text: convergence.

text: The

text: result

text: implies

text: that

text: if

text: graph

text: isomorphism

text: is

text: NP-complete

text: then

text: all

text: problems

text: in

text: NP

text: have

text: quasi-polynomial

text: time

text: algorithms,

text: which

text: may

text: lead

text: one

text: to

text: also

text: theoretically

text: believe

text: that

text: graph

text: isomorphism

text: is

text: not

text: NP-complete.

text: In

text: this

text: paper

text: we

text: are

text: interested

text: in

text: creating

text: diﬃcult

text: benchmark

text: instances

text: for

text: the

text: graph

text: isomor-

text: phism

text: problem.

text: It

text: would

text: be

text: a

text: misconception

text: to

text: think

text: that

text: for

text: unstructured

text: randomly

text: generated

text: instances

text: graph

text: isomorphism

text: should

text: be

text: hard

text: in

text: practice.

text: Quite

text: the

text: opposite

text: is

text: true.

text: Instances

text: that

text: encompass

text: suﬃcient

text: randomness

text: usually

text: turn

text: out

text: to

text: be

text: among

text: the

text: easiest

text: instances

text: (see

text: for

text: extensive

text: tests

text: suites

text: on

text: various

text: kinds

text: of

text: graphs

text: and

text: for

text: theoretical

text: arguments).

text: In

text: fact,

text: it

text: is

text: non-trivial

text: to

text: create

text: challenging

text: instances

text: for

text: eﬃcient

text: isomorphism

text: solvers

text: and

text: so

text: far

text: the

text: number

text: of

text: diﬃcult

text: benchmark

text: graphs

text: available

text: is

text: quite

text: limited.

text: The

text: prime

text: source

text: for

text: diﬃcult

text: instances

text: are

text: graphs

text: arising

text: from

text: combinatorial

text: structures.

text: In

text: Mathon

text: provided

text: a

text: set

text: of

text: benchmark

text: graphs

text: arising

text: from

text: combinatorial

text: objects

text: such

text: as

text: strongly

text: regular

text: graphs,

text: block

text: designs

text: and

text: coherent

text: conﬁgurations.

text: The

text: actual

text: graphs

text: that

text: are

text: provided,

text: having

text: less

text: than

text: vertices,

text: are

text: small

text: by

text: todays

text: standards.

text: Nowadays,

text: larger

text: combinatorial

text: objects

text: yielding

text: diﬃcult

text: graphs

text: are

text: known.

text: The

text: most

text: challenging

text: examples

text: are

text: for

text: instance

text: incidence

text: graphs

text: of

text: projective

text: planes,

text: Hadamard

text: matrices,

text: or

text: Latin

text: squares

text: (see

text: It

text: is

text: important

text: to

text: understand

text: that

text: not

text: all

text: such

text: incidence

text: structures

text: yield

text: diﬃcult

text: graphs.

text: Indeed,

text: typically

text: there

text: is

text: an

text: algebraic

text: version

text: that

text: can

text: readily

text: be

text: generated.

text: For

text: example

text: to

text: obtain

text: a

text: projective

text: plane

text: it

text: is

text: possible

text: to

text: consider

text: incidences

text: between

text: one-

text: and

text: two-dimensional

text: subspaces

text: of

text: a

text: three-dimensional

text: vector

text: space

text: over

text: a

text: ﬁnite

text: ﬁeld.

text: If

text: however

text: one

text: uses

text: such

text: an

text: algebraic

text: construction

text: then

text: the

text: resulting

text: graph

text: will

text: automatically

text: have

text: a

text: large

text: automorphism

text: group.

text: The

text: practical

text: isomorphism

text: solvers

text: are

text: tuned

text: to

text: ﬁnding

text: such

text: automorphisms

text: and

text: to

text: exploit

text: them

text: for

text: search

text: space

text: contraction.

text: On

text: top

text: of

text: that,

text: there

text: are

text: also

text: theoretical

text: reasons

text: that

text: lead

text: one

text: to

text: believe

text: that

text: graphs

text: with

text: automorphisms

text: make

text: easier

text: examples

text: for

text: the

text: practical

text: tools

text: (see

text: Theorem

text: The

text: non-algebraic

text: counterparts

text: of

text: combinatorial

text: structures

text: often

text: do

text: not

text: have

text: the

text: shortcoming

text: of

text: having

text: a

text: large

text: automorphism

text: group,

text: but

text: in

text: contrast

text: to

text: the

text: algebraic

text: constructions

text: they

text: are

text: rare

text: and

text: diﬃcult

text: to

text: generate.

text: The

text: second

text: source

text: of

text: diﬃcult

text: instances

text: is

text: based

text: on

text: modiﬁcations

text: of

text: the

text: so

text: called

text: Cai-Fürer-

text: Immerman

text: construction

text: This

text: construction

text: is

text: connected

text: to

text: the

text: family

text: of

text: Weisfeiler-Leman

text: al-

text: gorithms,

text: which

text: for

text: every

text: integer

text: k

text: contains

text: a

text: k-dimensional

text: version

text: that

text: constitutes

text: a

text: polynomial

text: time

text: algorithm

text: used

text: to

text: distinguish

text: non-isomorphic

text: graphs.

text: The

text: variant

text: (usually

text: called

text: color

text: reﬁnement)

text: is

text: a

text: subroutine

text: in

text: all

text: competitive

text: practical

text: isomorphism

text: solvers.

text: For

text: larger

text: k,

text: how-

text: ever,

text: the

text: k-dimensional

text: variant

text: becomes

text: impractical

text: due

text: to

text: excessive

text: space

text: consumption.

text: For

text: each

text: k,

text: Cai,

text: Fürer

text: and

text: Immerman

text: construct

text: pairs

text: of

text: non-isomorphic

text: graphs

text: that

text: are

text: not

text: distinguished

text: by

text: the

text: k-dimensional

text: variant

text: of

text: algorithm.

text: In

text: the

text: construction

text: was

text: adapted

text: by

text: Miyazaki

text: to

text: explic-

text: itly

text: show

text: that

text: the

text: then

text: current

text: version

text: of

text: nauty

text: has

text: exponential

text: running

text: time

text: in

text: the

text: worst

text: case

text: The

text: Cai-Fürer-Immerman

text: graphs

text: and

text: the

text: Miyazaki

text: graphs

text: constitute

text: families

text: of

text: benchmark

text: graphs

text: that

text: are

text: infeasible

text: for

text: simplistic

text: algorithms

text: for

text: the

text: graph

text: isomorphism

text: problem.

text: Nowadays,

text: however,

text: most

text: state-of-the-art

text: solvers

text: scale

text: reasonably

text: well

text: on

text: these

text: instances

text: (Figures

text: and

text: Contribution.

text: We

text: describe

text: a

text: construction

text: to

text: eﬃciently

text: generate

text: instances

text: on

text: any

text: desired

text: number

text: of

text: vertices

text: for

text: the

text: graph

text: isomorphism

text: problem

text: that

text: are

text: diﬃcult

text: or

text: even

text: infeasible

text: for

text: all

text: state-of-

text: the-art

text: graph

text: isomorphism

text: solvers.

text: Experiments

text: show

text: the

text: graphs

text: pose

text: by

text: far

text: the

text: most

text: challenging

text: graphs

text: to

text: date.

text: Already

text: for

text: vertices

text: our

text: new

text: instances

text: are

text: by

text: several

text: orders

text: of

text: magnitude

text: more

text: diﬃcult

text: than

text: all

text: previously


text: · · 104 104 timeout timeout 1 Bliss 1 Bliss Traces Bliss 0.8 Nauty 0.8 Conauto Saucy Conauto 0.6 Conauto 0.6 0.4 0.4 0.2 0.2 0 0 0 500 1,000 1,500 2,000 0 500 1,000 1,500 2,000 2,500 3,000 number of vertices number of vertices ∗ Figure 6.4: Performance of the various algorithms on the reduced graph R(B (G , σ)) (left) obtained n by the linear algebra reduction (see Subsection 13). Also shown is a comparison for Bliss and ∗ ∗ Conauto on R(B(G , σ)) and R(B (G , σ)) (right, ‘ ’ indicates experiments for reduced graphs). n n · · 104 104 timeout timeout 1 Bliss 1 Bliss Bliss Bliss 0.8 Conauto 0.8 Conauto Conauto Conauto 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 500 1,000 1,500 2,000 0 500 1,000 1,500 2,000 2,500 3,000 number of vertices number of vertices ∗ ∗ ∗ Figure 6.5: Performance comparison for Bliss and Conauto for R(B (G , σ)) and R (B (G , σ)) ∗ ∗ ∗ ∗ n n (left) and for R (B(G , σ)) and R (B (G , σ)) (right) (‘ ’ indicates experiments for reduced n n graphs). 6.2 Other groups In this second part of the section, we give some experiments for the rigid base construction on other groups (Figure 6.6). We show here the performance of Bliss in order to compare the diﬀerent constructions, but we remark that the behavior of the other programs is similar. Overall, we observe in this part that the original construction based on Z yields the most diﬃcult graphs. 2 16 ces 401 ces 401 ces 401 ces 401 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: timeout

text: timeout

text: Bliss

text: Bliss

text: Traces

text: Bliss

text: Nauty

text: Conauto

text: Saucy

text: Conauto

text: Conauto

text: number

text: of

text: vertices

text: number

text: of

text: vertices

label: caption

text: Figure

text: Performance

text: of

text: the

text: various

text: algorithms

text: on

text: the

text: reduced

text: graph

text: R(B

text: (G

text: σ))

text: (left)

text: obtained

text: n

text: by

text: the

text: linear

text: algebra

text: reduction

text: (see

text: Subsection

text: Also

text: shown

text: is

text: a

text: comparison

text: for

text: Bliss

text: and

text: Conauto

text: on

text: R(B(G

text: σ))

text: and

text: R(B

text: (G

text: σ))

text: (right,

text: indicates

text: experiments

text: for

text: reduced

text: graphs).

text: n

text: n

text: timeout

text: timeout

text: Bliss

text: Bliss

text: Bliss

text: Bliss

text: Conauto

text: Conauto

text: Conauto

text: Conauto

text: number

text: of

text: vertices

text: number

text: of

text: vertices

text: Figure

text: Performance

text: comparison

text: for

text: Bliss

text: and

text: Conauto

text: for

text: R(B

text: (G

text: σ))

text: and

text: R

text: (B

text: (G

text: σ))

text: n

text: n

text: (left)

text: and

text: for

text: R

text: (B(G

text: σ))

text: and

text: R

text: (B

text: (G

text: σ))

text: (right)

text: indicates

text: experiments

text: for

text: reduced

text: n

text: n

text: graphs).

text: Other

text: groups

text: In

text: this

text: second

text: part

text: of

text: the

text: section,

text: we

text: give

text: some

text: experiments

text: for

text: the

text: rigid

text: base

text: construction

text: on

text: other

text: groups

text: (Figure

text: We

text: show

text: here

text: the

text: performance

text: of

text: Bliss

text: in

text: order

text: to

text: compare

text: the

text: diﬀerent

text: constructions,

text: but

text: we

text: remark

text: that

text: the

text: behavior

text: of

text: the

text: other

text: programs

text: is

text: similar.

text: Overall,

text: we

text: observe

text: in

text: this

text: part

text: that

text: the

text: original

text: construction

text: based

text: on

text: Z

text: yields

text: the

text: most

text: diﬃcult

text: graphs.

text: ces

text: ces

text: ces

text: ces

text: ##LTLine##

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text: 1 8 2 7 3 6 4 5 Figure 3.2: The graph G 4 Algorithm 1: Rigid Odd Base Graph Construction ∈ N Input : n Output: Graph B = B(G , σ) n × { } ∪ 1 V (B) := V (G ) 0, 1 E(G ) ∅ n n 2 E(B) := ∈ 3 randomly choose permutation σ Sym(E(G )) ∈ × { } n 4 for v V (G ) 0 do n ∪ { | ∈ ∈ } 5 E(B) := E(B) ve e E(G ) with v e n 6 end ∈ × { } 7 for v V (G ) 1 do n ∪ { | ∈ ∈ } 8 E(B) := E(B) ve e E(G ) with v σ(e) n 9 end 10 return B Algorithm 2: The multipede graph construction ∈ Input : Bipartite graph B = (V, W, E) such that every vertex v V has degree 3 Output: Graph G = R(B) { | ∈ ∈ { } } ∪ { | ∈ 1 V (G) := m (v) v V, i 0, 1 3 with number of ones in i is even a(w), b(w) w } i W ∅ 2 E(G) := ∈ 3 for v V do { } 4 w , w , w := N(v) /* fix order on the neighbors of v */ 1 ∈2 { 3 } ∈ { } 5 for j 1, 2, 3 and i 0, 1 3 with number of ones in i is even do 6 if i = 0 then j ∪ {{ }} 7 E(G) := E(G) m (v), a(w ) /* connect m (v) to a(w ) */ i j i j 8 else ∪ {{ }} 9 E(G) := E(G) m (v), b(w ) /* connect m (v) to b(w ) */ i j i j 10 end 11 end 12 end 13 return G 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

text: graph

text: G

text: Algorithm

text: Rigid

text: Odd

text: Base

text: Graph

text: Construction

text: N

text: Input

text: n

text: Output:

text: Graph

text: B

text: B(G

text: σ)

text: n

text: V

text: (B)

text: V

text: (G

text: E(G

text: n

text: n

text: E(B)

text: randomly

text: choose

text: permutation

text: σ

text: Sym(E(G

text: n

text: for

text: v

text: V

text: (G

text: do

text: n

text: E(B)

text: E(B)

text: ve

text: e

text: E(G

text: with

text: v

text: e

text: n

text: end

text: for

text: v

text: V

text: (G

text: do

text: n

text: E(B)

text: E(B)

text: ve

text: e

text: E(G

text: with

text: v

text: σ(e)

text: n

text: end

text: return

text: B

text: Algorithm

text: The

text: multipede

text: graph

text: construction

text: Input

text: Bipartite

text: graph

text: B

text: (V,

text: W,

text: E)

text: such

text: that

text: every

text: vertex

text: v

text: V

text: has

text: degree

text: Output:

text: Graph

text: G

text: R(B)

text: V

text: (G)

text: m

text: (v)

text: v

text: V,

text: i

text: with

text: number

text: of

text: ones

text: in

text: i

text: is

text: even

text: a(w),

text: b(w)

text: w

text: i

text: W

text: E(G)

text: for

text: v

text: V

text: do

text: w

text: w

text: w

text: N(v)

text: fix

text: order

text: on

text: the

text: neighbors

text: of

text: v

text: ∈2

text: for

text: j

text: and

text: i

text: with

text: number

text: of

text: ones

text: in

text: i

text: is

text: even

text: do

text: if

text: i

text: then

text: j

text: E(G)

text: E(G)

text: m

text: (v),

text: a(w

text: connect

text: m

text: (v)

text: to

text: a(w

text: i

text: j

text: i

text: j

text: else

text: E(G)

text: E(G)

text: m

text: (v),

text: b(w

text: connect

text: m

text: (v)

text: to

text: b(w

text: i

text: j

text: i

text: j

text: end

text: end

text: end

text: return

text: G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: b(w, v) and b(v, w) with a(w, v) in case of a twisted connection. This way the number of vertices can be reduced from 10n to 7n. This can further be improved by removing all a and b vertices altogether and connecting inner vertices m (v) to m (w) if both are connected to either a(v, w) or i j b(v, w). In this form the construction only requires 4n vertices. A similar technique can also be applied to the rigid base construction. For this we bypass all vertices of degree 8 by directly joining their neighbors and then removing all such vertices. Given ∗ a bipartite base graph G we denote by R (G) the graph obtained form R(G) by bypassing the outer vertices. When reducing the graphs in that fashion one has to be aware that combinatorial structure of the graph might get lost. In particular it can be the case that vertices of degree 4 (in the base graph) had the same neighborhood before they were removed, leading to inhomogeneity in ∗ the graph R (G). In fact our experiments show that there is wider spread concerning the diﬃculty ∗ of the graphs R (G) than the graphs R(G). In other words, the rigidity and diﬃculty of those graphs depends more heavily on the choice of the matching σ than for the rigid base construction. ∗ However, the graphs R (G) turn out to be more diﬃcult than those obtained from the multipede construction (see Section 13). 4.3 Applying both reductions We can combine the two vertex reduction techniques presented in this section by ﬁrst applying Corollary 4.2 to the base graph and then bypassing the outer vertices. The number of vertices in the combined reduction decreases to 12n. It is not very diﬃcult to see that the graphs have an average degree of at most 24, but the average degree varies among graphs on the same number of vertices since there may be multiple bypass edges that would have the same endpoints. Our experiments show that the combination of the two reductions (Section 13) yields the most ∗ ∗ diﬃcult graphs. We call the resulting graphs R (B (G , σ)) the shrunken multipedes. n 5 Cai-Fürer-Immerman gadgets for other groups The constructions described in the previous sections revolve around the particular CFI-gadget depicted in Figure 2.1. On the outer vertices, the automorphism group of that CFI-gadget induces the set of all permutations that swap an even number of pairs. This corresponds to the subgroup of (Z )3 of elements (g , g , g ) with g g g = 1. A natural idea to push the CFI-construction to its 2 1 2 3 1 2 3 limits is to encode other groups by the use of other gadgets. Indeed, it is not diﬃcult to see that for every permutation group ∆ acting on a set Ω there is∼a graph gadget X with a vertex set containing Ω as Aut(X )-invariant subset such that Aut(X ) = | ∆ ∆ ∆ ∆ and Aut(G) = ∆. (For a description of such a construction see for example [21, Lemma 16].) In Ω other words, every permutation group ∆ can be realized by a graph gadget X . As explained before, ∆ the original CFI-gadget realizes a group that is a subdirect product of Z3, which is in particular an 2 abelian group. In analogy to our previous terminology we call the vertices in Ω the outer vertices \ and the vertices in V (X ) Ω the inner vertices. Note that for our purpose of creating benchmark ∆ graphs, we are interested in keeping the number of inner vertices small. In our constructions, Ω naturally decomposes into sets C , C , C , . . . , C of equal size n as the 1 2 3 t classes of the outer vertices. We think of these sets as colored with diﬀerent colors and as usual we will only consider color preserving isomorphisms, that is permutations that map each vertex to a ∪ ∪ vertex of the same color. If we insert a gadget X with Ω = C C . . . C we obtain a subgroup × × · · · × ∆ 1 2 t of Sym(C ) Sym(C ) Sym(C ), the direct product of symmetric groups. 1 2 t For simplicity, in the rest of this section we focus on the case t = 3. Thus we consider colored ∪ ∪ ⊆ graph gadgets X such that C C C V (X ) and such that the automorphism group Aut(X ) ∆ 1 2 3 ∆ ∆ 10

text: b(w,

text: v)

text: and

text: b(v,

text: w)

text: with

text: a(w,

text: v)

text: in

text: case

text: of

text: a

text: twisted

text: connection.

text: This

text: way

text: the

text: number

text: of

text: vertices

text: can

text: be

text: reduced

text: from

text: to

text: This

text: can

text: further

text: be

text: improved

text: by

text: removing

text: all

text: a

text: and

text: b

text: vertices

text: altogether

text: and

text: connecting

text: inner

text: vertices

text: m

text: (v)

text: to

text: m

text: (w)

text: if

text: both

text: are

text: connected

text: to

text: either

text: a(v,

text: w)

text: or

text: i

text: j

text: b(v,

text: w).

text: In

text: this

text: form

text: the

text: construction

text: only

text: requires

text: vertices.

text: A

text: similar

text: technique

text: can

text: also

text: be

text: applied

text: to

text: the

text: rigid

text: base

text: construction.

text: For

text: this

text: we

text: bypass

text: all

text: vertices

text: of

text: degree

text: by

text: directly

text: joining

text: their

text: neighbors

text: and

text: then

text: removing

text: all

text: such

text: vertices.

text: Given

text: a

text: bipartite

text: base

text: graph

text: G

text: we

text: denote

text: by

text: R

text: (G)

text: the

text: graph

text: obtained

text: form

text: R(G)

text: by

text: bypassing

text: the

text: outer

text: vertices.

text: When

text: reducing

text: the

text: graphs

text: in

text: that

text: fashion

text: one

text: has

text: to

text: be

text: aware

text: that

text: combinatorial

text: structure

text: of

text: the

text: graph

text: might

text: get

text: lost.

text: In

text: particular

text: it

text: can

text: be

text: the

text: case

text: that

text: vertices

text: of

text: degree

text: (in

text: the

text: base

text: graph)

text: had

text: the

text: same

text: neighborhood

text: before

text: they

text: were

text: removed,

text: leading

text: to

text: inhomogeneity

text: in

text: the

text: graph

text: R

text: (G).

text: In

text: fact

text: our

text: experiments

text: show

text: that

text: there

text: is

text: wider

text: spread

text: concerning

text: the

text: diﬃculty

text: of

text: the

text: graphs

text: R

text: (G)

text: than

text: the

text: graphs

text: R(G).

text: In

text: other

text: words,

text: the

text: rigidity

text: and

text: diﬃculty

text: of

text: those

text: graphs

text: depends

text: more

text: heavily

text: on

text: the

text: choice

text: of

text: the

text: matching

text: σ

text: than

text: for

text: the

text: rigid

text: base

text: construction.

text: However,

text: the

text: graphs

text: R

text: (G)

text: turn

text: out

text: to

text: be

text: more

text: diﬃcult

text: than

text: those

text: obtained

text: from

text: the

text: multipede

text: construction

text: (see

text: Section

text: Applying

text: both

text: reductions

text: We

text: can

text: combine

text: the

text: two

text: vertex

text: reduction

text: techniques

text: presented

text: in

text: this

text: section

text: by

text: ﬁrst

text: applying

text: Corollary

text: to

text: the

text: base

text: graph

text: and

text: then

text: bypassing

text: the

text: outer

text: vertices.

text: The

text: number

text: of

text: vertices

text: in

text: the

text: combined

text: reduction

text: decreases

text: to

text: It

text: is

text: not

text: very

text: diﬃcult

text: to

text: see

text: that

text: the

text: graphs

text: have

text: an

text: average

text: degree

text: of

text: at

text: most

text: but

text: the

text: average

text: degree

text: varies

text: among

text: graphs

text: on

text: the

text: same

text: number

text: of

text: vertices

text: since

text: there

text: may

text: be

text: multiple

text: bypass

text: edges

text: that

text: would

text: have

text: the

text: same

text: endpoints.

text: Our

text: experiments

text: show

text: that

text: the

text: combination

text: of

text: the

text: two

text: reductions

text: (Section

text: yields

text: the

text: most

text: diﬃcult

text: graphs.

text: We

text: call

text: the

text: resulting

text: graphs

text: R

text: (B

text: (G

text: σ))

text: the

text: shrunken

text: multipedes.

text: n

text: Cai-Fürer-Immerman

text: gadgets

text: for

text: other

text: groups

text: The

text: constructions

text: described

text: in

text: the

text: previous

text: sections

text: revolve

text: around

text: the

text: particular

text: CFI-gadget

text: depicted

text: in

text: Figure

text: On

text: the

text: outer

text: vertices,

text: the

text: automorphism

text: group

text: of

text: that

text: CFI-gadget

text: induces

text: the

text: set

text: of

text: all

text: permutations

text: that

text: swap

text: an

text: even

text: number

text: of

text: pairs.

text: This

text: corresponds

text: to

text: the

text: subgroup

text: of

text: (Z

text: of

text: elements

text: (g

text: g

text: g

text: with

text: g

text: g

text: g

text: A

text: natural

text: idea

text: to

text: push

text: the

text: CFI-construction

text: to

text: its

text: limits

text: is

text: to

text: encode

text: other

text: groups

text: by

text: the

text: use

text: of

text: other

text: gadgets.

text: Indeed,

text: it

text: is

text: not

text: diﬃcult

text: to

text: see

text: that

text: for

text: every

text: permutation

text: group

text: acting

text: on

text: a

text: set

text: Ω

text: there

text: is∼a

text: graph

text: gadget

text: X

text: with

text: a

text: vertex

text: set

text: containing

text: Ω

text: as

text: Aut(X

text: )-invariant

text: subset

text: such

text: that

text: Aut(X

text: and

text: Aut(G)

text: (For

text: a

text: description

text: of

text: such

text: a

text: construction

text: see

text: for

text: example

text: Lemma

text: In

text: Ω

text: other

text: words,

text: every

text: permutation

text: group

text: can

text: be

text: realized

text: by

text: a

text: graph

text: gadget

text: X

text: As

text: explained

text: before,

text: the

text: original

text: CFI-gadget

text: realizes

text: a

text: group

text: that

text: is

text: a

text: subdirect

text: product

text: of

text: which

text: is

text: in

text: particular

text: an

text: abelian

text: group.

text: In

text: analogy

text: to

text: our

text: previous

text: terminology

text: we

text: call

text: the

text: vertices

text: in

text: Ω

text: the

text: outer

text: vertices

text: and

text: the

text: vertices

text: in

text: V

text: (X

text: Ω

text: the

text: inner

text: vertices.

text: Note

text: that

text: for

text: our

text: purpose

text: of

text: creating

text: benchmark

text: graphs,

text: we

text: are

text: interested

text: in

text: keeping

text: the

text: number

text: of

text: inner

text: vertices

text: small.

text: In

text: our

text: constructions,

text: Ω

text: naturally

text: decomposes

text: into

text: sets

text: C

text: C

text: C

text: C

text: of

text: equal

text: size

text: n

text: as

text: the

text: t

text: classes

text: of

text: the

text: outer

text: vertices.

text: We

text: think

text: of

text: these

text: sets

text: as

text: colored

text: with

text: diﬀerent

text: colors

text: and

text: as

text: usual

text: we

text: will

text: only

text: consider

text: color

text: preserving

text: isomorphisms,

text: that

text: is

text: permutations

text: that

text: map

text: each

text: vertex

text: to

text: a

text: vertex

text: of

text: the

text: same

text: color.

text: If

text: we

text: insert

text: a

text: gadget

text: X

text: with

text: Ω

text: C

text: C

text: C

text: we

text: obtain

text: a

text: subgroup

text: t

text: of

text: Sym(C

text: Sym(C

text: Sym(C

text: the

text: direct

text: product

text: of

text: symmetric

text: groups.

text: t

text: For

text: simplicity,

text: in

text: the

text: rest

text: of

text: this

text: section

text: we

text: focus

text: on

text: the

text: case

text: t

text: Thus

text: we

text: consider

text: colored

text: graph

text: gadgets

text: X

text: such

text: that

text: C

text: C

text: C

text: V

text: (X

text: and

text: such

text: that

text: the

text: automorphism

text: group

text: Aut(X