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for any M > 0 and wherefor any a and b for which the linear filter is asymptoti-(31) cally stable. Realizing this fact, the designer using the rms bound can be certain that the maximum value of the limit cycle d l not exceed this bound by a factor of more than two.4b for either b 5 0 or b > 0 and I a 1 2 -One should make the important observation that (32) is References identical to (27) over the region where a 2 -4 b 2 0 . Therefore, the rms bound (32) is also an absolute bound over the region a2-4b>0. Comparison of (29) and (33) over the region 6 > 0 and I a1 <4b/(l +b) shows that they differ by a factor of (1 +&. Note that (29) cannot exceed (33) by more than a factor of two over the specified region and for small values of b they are approximately the same. The following important result can be shown by careful comparison of the bounds:( 3 4 ) [l] I.

We have developed a precise airborne temperature-sensing technology to detect buried objects for use by law enforcement. Demonstrations have imaged the sites of buried foundations, walls and trenches; mapped underground waterways and aquifers; and been used to locate underground military objects. Our patented methodology is incorporated in a commercially available, high signal-to-noise, dual-band infrared scanner with real-time, 12-bit digital image processing software and display. Our method creates color-coded images based on surface temperature variations of 0.2°C. Unlike other less-sensitive methods, it maps true (corrected) temperatures by removing the (decoupled) surface emissivity mask equivalent to 1°C or 2°C; this mask hinders interpretation of apparent (blackbody) temperatures. Once removed, we are able to identify surface temperature patterns from small diffusivity changes at buried object sites which heat and cool differently from their surroundings. Objects made of different materials and buried at different depths are identified by their unique spectral, spatial, thermal, temporal, emissivity and diffusivity signatures. We have successfully located the sites of buried (inert) simulated land mines 0.1 to 0.2 m deep; sodcovered rock pathways alongside dry ditches, deeper than 0.2 m; pavement covered burial trenches and cemetery structures as deep as 0.8 m; and aquifers more than 6 m and less than 60 m deep. Our technology could be athpted for drug interdiction and pollution control. For the former, we would locate buried tunnels, underground structures built beneath typical surface structures, mof-tops disguised by jungle canopies and covered containers used for contraband. For the latter, we would depict buried waste containers, sludge migration pathways from faulty containers and the juxtaposition of groundwater channels, if present, nearby. Our precise airborne temperature-sensing technology has a promising potential to detect underground epicenters of smuggling and pollution. BACKGROUND AND TECHNICAL APPROACHThe first successful demonstration of the precise temperature survey technology recently adapted for buried land mine detection was for geothermal resource investigations in 1977. Predawn surface temperature patterns were spatially correlated with sub-surface heat flow anomalies, soil moisture differences and variations in solar heat retained by near-surface rock tcroJ2 A 2 square km aquifer was depicted by the corrected thermal imagery. The aquifer was covered by more than 6 m of soil and was less than 60 m deep. See Figure 1 and Figure 2.Other applications of the technology were conducted at the Mercury Nevada Test Site for treaty verification. We investigated the surface temperature signatures near "ground zero" before and after an underground nuclear explosion. Soil "fluffmg" changed the soil thermal diffusivity and consequently the heating and cooling properties of near-surface materials near "ground zero." All else being equal, the ground nearby had a different temperature variation with...

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