A Diurnal Animation of Thermal Images froma Day–Night Pair

Ken Watson*

Interpretation of thermal images is often complicated which can provide clues to structural controls and mate-because the physical property information is contained in rial property differences. This ‘visual change’ approachboth the spatial and temporal variations of the data and could significantly increase the use of thermal data forthermal models are necessary to extract and display this environmental, hazard, and resource studies. Publishedinformation. A linearized radiative transfer solution to by Elsevier Science Inc., 2000the surface flux has been used to derive a function thatis invariant with respect to thermal inertia. This relation-ship makes it possible to predict the temperature variation INTRODUCTIONat any time in the diurnal cycle using only two distinct

Repetitive, global remote-sensing data are used to studymeasurements (e.g., noon and midnight). An animationimportant problems related to sustainable developmentcan then be constructed from a pair of day–night imagesand environmental monitoring, assessment, and man-to view both the spatial and temporal temperatureagement.changes throughout the diurnal cycle. A more complete

As higher spatial resolution aircraft and satellite datasolution for the invariant function, using the method ofbecome available to measure diurnal surface temperatureLaplace transforms and based on the linearized solution,variations, effective ways to analyze these data are re-was introduced. These results indicate that the linearquired. Thermal modeling can be used to explore rela-model does not provide a sufficiently accurate estimate.tionships between surface temperature and surface heatUsing standard conditions (latitude 308, solar declinationfluxes to extract thermal property information about the08, acquisition times at noon and midnight), this new re-ground surface and to determine the presence of heatlationship was used to predict temperature throughoutsources. For the purposes of this study, areas were ex-the diurnal cycle to an rms error of 0.28C, which is closecluded where significant vegetation cover precluded ob-to the system noise of most thermal scanners. The methodtaining information about the soil. (A comprehensivewas further extended to include the primary effects ofthermal model has been developed for forest and grass-topographic slope with similar accuracy. The tempera-land vegetation canopies: Kimes et al., 1981; Smith et al.,ture was computed at 48 equally spaced times in the di-1981.) The model in this paper builds on previous stud-urnal cycle with this algorithm using a co-registered dayies (e.g., Watson 1971, 1975, 1982b; Kahle, 1977; Crack-and night TIMS (Thermal Infrared Multispectral Scan-nell and Xue, 1996) and gathers together concepts thatner) data pair (330 pixels, 450 lines) acquired of the Car-have been scattered over a number of papers.lin, Nevada, area and a co-registered DEM (Digital Ele-

We begin by considering a bare surface devoid ofvation Model). (Any reader can view the results byvegetation. The surface flux includes the absorbed solardownloading the animation file from an identified ftpradiation, the net long-wavelength radiation between thesite.) The results illustrate the power of animation to dis-ground and the sky, sensible heat transfer, latent heating,play subtle temporal and spatial temperature changes,and geothermal heating (including the effects of endo-and exothermic reactions). Because the solar flux is the

* U.S. Geological Survey, P.O. Box 25046, MS 964, Denver main driving force for the diurnal surface temperatureAddress correspondence to K. Watson, U.S. Geological Survey, change, we shall assume that the heating flux is in the

P.O. Box 25046, MS 964, Denver, CO 80225. E-mail: kwatson@corona. form of a periodic half wave, linearly proportional to thecr.usgs.govReceived 9 June 1999; revised 19 October 1999. cosine of the zenith angle of the sun, with arbitrary coef-

REMOTE SENS. ENVIRON. 72:237–243 (2000)Published by Elsevier Science Inc., 2000 0034-4257/00/$–see front matter655 Avenue of the Americas, New York, NY 10010 PII S0034-4257(99)00106-6

238 Watson

shape of the solar flux under clear sky conditions and isexpressed as f(t)5max{cos(z),0}, where cos(z)5cos(k)cos(d)cos(xt)1sin(k)sin(d) and k is the site latitude, d isthe solar declination, x is the diurnal angular frequency,and t is the local solar time. Because of its periodicityand symmetry, f(t) can also be written (see Appendix Afor An coefficients) as

on50

An cos (nxt)

An exact discrete solution can be obtained using themethod of Laplace transforms. The basic equation is

Figure 1. Half-wave heating flux with linear coef-ficients a and b. Fi5

P

√psom

s51vsui2s11 (i51, 2, . . . , m) (2)

where Fi is the flux into the ground at the ith interval,ficients. Internal heating sources will be assumed to beP is the thermal inertia of the ground, P5K/√j, s is thetime invariant. Conditions that differ from these can bediurnal period, vs is the surface temperature at the sthtreated as transients superimposed on the periodic so-interval, and ui are a set of coefficients determined solelylutions.by the number of intervals, m. The iterative solution re-Because the diurnal heating wave (primarily due toquires an initial guess to vs. The linearized solution thatsolar heating) is reduced by a factor of 2 by the time itis introduced in the next section can be used to providehas penetrated one-tenth of a wavelength into thethis initial guess. Generally the Laplace transform solu-ground (about 5 cm in dry sandy soil, 11 cm in rock),tion is useful for testing algorithms and developing em-the ground will be treated as a homogeneous half space.pirical relationships between variables. It is too time con-The effects of surface coatings or thin dust layers can besuming, however, to apply to each pixel in a large image.examined using a ’layer over half-space’ model (Watson,

1973). However, the effects of limited vegetation coverSolution to Linearized Boundary Conditiongenerally cannot be handled in such a simplified manner.

The heterogeneity introduced by temperature differ- The quartic radiative term in the boundary condition,ences between soil and vegetation and by the effects of Eq. (2), can be linearized by introducing an arbitraryradiation between shaded and sunlit soil and vegetation temperature vo and rewritten ascomponents requires a more complex treatment involv-

2K]v/]x5af(t)1b2er(4v.v3o23v4

o) at x5o (3)ing knowledge of the vegetation canopy distributionThe solution to Eq. (1), subject to the revised boundary(density, structure) and the ground and vegetation emis-

sivities. There is an extensive literature examining several condition, Eq. (3), is (Watson, 1982a)aspects of this problem (Caselles et al., 1992; Kimes et

v(0,t)5Bo1os51

Bsh cos (xst2ds)/√(h1ws)21w2s (4)al., 1981). An energy budget model of a simple vegeta-

tion layer has been developed (Balick et al., 1981) thatwhereprovides useful guidance on the multiplicity of parame-

ters that must to be specified and of the greater impor- ws5√xs/2jtance of foliage cover on daytime temperatures. h54erv3

o /K

ds5tan21 {ws /(h1ws)}Initial Problem: Periodic HeatingConsider the periodic heating of a homogeneous half Bo5(4erv3

o)21{aAo1b}1.75vospace by a half-wave flux (Fig. 1) with arbitrary linear

Bs5(4erv3o)21{aAs}coefficients.

Now, since ws /h.1, thenk] 2v/]x25]v/]th/[(h1ws)21w2

s]1/2<4erv30 /P√xswhere

and ds<p/4, where P5K/√j is the thermal inertia of the2K]v/]x5af(t)1b2erv4 @ x50 (1)ground. Therefore

and v(x,t) is the temperature in the half space (ground)v(o,t)<Bo1(a/P√x)o

s51As cos (xst2p/4)/√s (5)at depth x and time t; j is the thermal diffusivity; K is

the thermal conductivity; e is the emissivity of theWe shall call this solution the linearized heating by a pe-ground; r is the Stefan–Boltzmann constant; and a andriodic half wave. As mentioned previously, it can be usedb are arbitrary coefficients of the half-wave flux, f(t).

The half-wave heating flux closely approximates the to generate an initial guess for the Laplace transform

Diurnal Animation of Thermal Images 239

Figure 2. The function h(t) using linearmodel.

method. More significantly it can be used to derive a maximize the temperature difference), we can introducevMEAN5(vD1vN)/2 and vDIF5vD2vN. Then an expressionnew function that is invariant with respect to thermal in-

ertia and thus can be used to estimate continuous diurnal can be constructed to compute the continuous diurnaltemperature changes from thermal observations at only variation:two different times (e.g., noon and midnight).

v(t)2vMEAN5ha /P√xj h f(t)2fMEANj (6a)

Estimating continuous diurnal temperature vDIF5ha /P√xj fDIF (6b)variation from two discrete measurements

whereAssume the previous solution for the linearized heatingby a periodic half wave. If we take measurements at two f(t)5o

s51hAs /√sj cos (xst2p/4)

different times in the diurnal cycle (generally a daytimetemperature, vD, and a nighttime temperature, vN, to fMEAN5(f(tD)1f(tN))3.5

Figure 3. The function h(t) from Laplacemethod (points) compared with linearmodel (curve).

240 Watson

Figure 4. Least squares coefficients (ai) for Laplace solution with topography.

Testing h(t) function using LaplacefDIF5f(tD)2f(tN)transform solution

Therefore The Laplace transform method was then used to com-pute the h function for a range of thermal inertias (1000,{v(o,t)2vMEAN}/vDIF5h(t) (7)1250, . . . , 3000). The results for the nominal case arewhereshown in Figure 3. Although the shape of the function

h(t)5(f(t)2fMEAN)/fDIF is similar to the linear model, the differences are signifi-cant. More importantly the results show that the functionis invariant with respect to thermal inertia. We can re-is not invariant with respect to thermal inertia.write Eq. (7) as

A revised form was then computed based on Eq. (8):v(o,t)5vMEAN1h(t)vDIF (8)

v(t)5c1(t)1c2(t)vMEAN1c3(t)vDIF (9)The term h(t) is directly computed using tD, tN, and the

where the cs are computed by least-squares fitting atFourier components (As) of the half-wave flux. Enougheach time. This fitted form introduces an rms error interms must be used in forming the sums to ensure thattemperature of 0.28C for standard conditions (latitudethe residuals will be less than some arbitrary value, such308, solar declination 08, noon and midnight), which isas the system temperature noise. The times tD and tNonly slightly greater than the NEDT (noise equivalentneed to be sufficiently distinct so that fDIF.0. For a largetemperature difference) of current thermal scanners.geographic region, As will also vary with latitude. An in-

terpolation table can be generated for h(t,latj) at discreteExtending solution to include effects of slopevalues and linearly interpolated for intermediate lat-The previous solution assumes that the surface is flat.itudes.Extension to inclined surfaces is based on an update ofThe function h(t), developed using the linear model,previous work (Watson, 1982b) and includes a more ac-is plotted in Figure 2. Two cases are shown: nominalcurate evaluation of the effect of slope.conditions with k5308, d508, tD50, tN512 and condi-

The heating flux for a surface with slope d, azimuthtions for the Carlin data used in the animation withk5418, d519.868, tD521.34, tN516. u is assumed to be of the form

Diurnal Animation of Thermal Images 241

Figure 5. Day and night thermal image pair together with selected images from the animation. The times (t)are hours in the diurnal cycle starting from local solar noon. The structural feature discussed in the textoccurs about 40% of the image height down from the top and extends across half the image width. It ismost evident on the three middle images. The changes associated with the unconsolidated materials can beseen by examining the changes in the upper left-hand octant of the images. Note the dark (cool) patternthat disappears during the daytime.

242 Watson

{a2QDIR(12A)} cos z1QDIR(12A) cos zd1b in the animation than is evident when examining just aday and a night image. Partly this occurs because the im-whereages are superimposed during the animation (see carlin.

cos zd5A9 cos xt1B9 sin xt1C91cos z animation.gif), but mainly it is due to the eye’s superiorskill at detecting complex spatial and temporal patterns.A95(cos d21) cos k cos d1sin d cos d cos u sin kThe animation is particularly useful for showing the sub-

B952sin d sin u cos d tle thermal expression of structures. An example is aneast–west structure near the center of the image. In ad-C95(cos d21) sin k sin d2sin d cos usin d cos kdition, the patterns of temperature changes in the un-We computed vDIF and vMEAN using the Laplace transform consolidated materials (for an example, see the northeastmethod for a slope of 58 and a range of azimuths (u50, corner of the image) are also more evident in the anima-15, . . . , 345). We then least-squares fitted v(t) to the tion than the single images. Additionally the data areform useful for studying problems dealing with weathering, in-

v(t,d,u)5vMEAN{a1d cos u1a2d sin u1a3} filtration, and mass wasting.For optimum analysis, the author found it instructive1vDIF{a4d cos u1a5d sin u1a6}1a7 (10)

to focus on a single area of the image throughout thewhere ai5ai(t). The coefficients are plotted in Figure 4 animation rather than the entire image. One additionaland are used in conjunction with the day and night im- but important feature of the animation is that it providesage pair and the DEM image to generate the animation. an instructive illustration of the form that the diurnalThe rms error in the inferred temperature computed us- variation takes and why a single thermal image or a pairing standard conditions was 0.28C. of images are not as likely to provide all the critical infor-

mation needed to study a site. For example, it is oftenerroneously stated that the effects of solar heating are re-EXAMPLEmoved by examining night images. The animation clearly

This algorithm was used to generate an animated diurnal shows topographic related variations during the nighttime,thermal image of the Carlin, NV, area. First a day and although the greatest change occurs at sunrise.a night thermal image were selected using TIMS band3 of data acquired in 1988 (Watson et al., 1990). These

I would like to express my appreciation to S. Miller and E. Livodata were co-registered to the 7.5 min DEM using a of the U.S. Geological Survey and to two journal reviewers forthin-plate spline algorithm (Hutchinson, 1993). The ani- improving this paper.mation was generated by forming 48 images that arespaced equally in time throughout the diurnal cycle. A

APPENDIX A1% dynamic range stretch was applied to each image(frame), to enhance spatial details (A histogram is used Fourier Components of the Half Waveto identify the 1%, mean, and 99% values, and a linear (Source: Appendix A, Watson, 1982a)stretch is used to map these values to 0, 127, and 255)

Ao5(cos(k) cos(d) sin(xt1)1sin(k) sin(d)xt1)/pThis was necessary because the range of temperaturesin the warmest image did not overlap with the range of A15[cos(k) cos(d) {xt11sin(2xt1)/2}temperatures in the coldest image. Much of the spatial

12sin(k) sin(d) sin(xt1)]/p for m.1detail would be lost if the same stretch were applied toall the images. The animation (carlin.animation.gif) and Am5[cos(k) cos(d)E(mxt1)the day–night image pair (daynight.tif) can be down-

12 sin(k) sin(d) sin(mxt1)/m]p (A-1)loaded from the USGS anonymous ftp site ftp://musette.

wherecr.usgs.gov/tmp/kwatson (see Appendix B for viewingguidance). A selection of the animation images is shown E(mxt1)5sin((m21)xt1)/(m21)1sin((m11)xt1)/(m11)in Figure 5, together with the original day and night im-

xt15p/21sin21(tan k tan d).age pair.

APPENDIX BCONCLUSIONS

Animation ViewingThe animation illustrates several important points. Theoriginal day and night images (see daynight.tif) do not The files (carlin.animation.gif and daynight.tif) can be

viewed on PCs using commercial programs such as Paintreveal all the information that is contained in the diurnalvariation. Areas where temperature changes are associ- Shop Pro. An evaluation copy can be downloaded from

their web site at http://www.paintshoppro.com. To runated with structural controls and with changes in uncon-solidated materials appear to be more easily recognized the animation, go to ‘Animation Shop’, select ‘file’ and in

Diurnal Animation of Thermal Images 243

Hutchinson, M. F. (1993), On thin plate splines and kriging.‘View’ select ‘animation’. An alternative method of view-In Computing and Science in Statistics 25 (M. E. Tarter anding is possible using the commercial program ENVI. TheM. D. Lock, Eds.), Interface Foundation of North America,file diurnal.pix can be downloaded from the same ftp siteUniv. of Calif., Berkeley.and imported into ENVI using ‘File/Open External File/

Kahle, A. B. (1977), A simple thermal model of the earth’s sur-Image Processing Formats/PCI’. Enter the appropriateface for geologic mapping by remote sensing. J. Geophys.path to the diurnal.pix file and load the first band. Inside Res. 82:1673–1680.

the full resolution window select ‘Functions/Interactive Kimes, D. S., Smith, J. A., and Link, L. E. (1981), A thermalAnalysis/Animation’. A larger animation view than the IR emittance model of a plant canopy. Appl. Optics 20(4):default of 220 by 300 (say, 440 by 600) can be used if 623–632.sufficient memory is available. The animation speed can Smith, J. A., Ranson, K. J., Nguyen, D., Balick, L. K., Link,

L. E., Fritschen, L. J., and Hutchison, B. A. (1981), Ther-also be adjusted.mal vegetation canopy model studies. Remote Sens. Envi-ron. 11:311–326.

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Watson, K. (1973), Periodic heating of a layer over a semi-infi-of a simple layer of vegetation in terrain temperature mod-nite solid. J. Geophys. Res. 78(26):5904–5910.els for thermal infrared IR signature prediction. IEEE

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Watson, K. (1982b), Topographic slope correction for analysisSens. Environ. 39:203–211.of thermal images. NTIS Report E82-10214. pp. 1–13.Cracknell, A. P., and Xue, Y. (1996), Dynamic aspects study of

Watson, K., Kruse, F., and Hummer-Miller, S. (1990), Thermalsurface temperature from remotely-sensed data using ad-infrared exploration in the Carlin trend. Geophysics 55(1):vanced thermal inertia model. Int. J. Remote Sens. 17(13):

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