<\/p>\n
From the Abstract:-<\/p>\n
\u00e2\u20ac\u0153<\/i>Here we propose that a reformulation of the global climate feedback in terms of its\u00c2\u00a0<\/i>contributions from regional climate feedbacks provides a clear physical insight into this behaviour\u00c2\u00a0 [the time-variation of global feedback]. \u00c2\u00a0<\/i>Using (i) a state-of-the-art global climate model and (ii) a low-order energy balance model, we\u00c2\u00a0<\/i>show that the global climate feedback is fundamentally linked to the geographic pattern of regional\u00c2\u00a0<\/i>climate feedbacks and the geographic pattern of surface warming at any given time. Time-variation\u00c2\u00a0<\/i>of the global climate feedback arises naturally when the pattern of surface warming evolves,\u00c2\u00a0<\/i>actuating regional feedbacks of different strengths. This result has substantial implications for our\u00c2\u00a0<\/i>ability to constrain future climate changes from observations of past and present climate states. \u00c2\u00a0<\/i>The regional climate feedbacks formulation reveals fundamental biases in a widely-used method for\u00c2\u00a0<\/i>diagnosing climate feedbacks and radiative forcing, the regression of the global top-of-atmosphere\u00c2\u00a0<\/i>radiation flux on global surface temperature.\u00e2\u20ac\u009d<\/i><\/p>\n
<\/p>\n
The implications of this paper are important and wide-ranging.\u00c2\u00a0\u00c2\u00a0 It sends a number of sacred cows to the abattoir without being too concerned about the religion of the owners.\u00c2\u00a0 In a certain sense it offers a unifying theory which should allow extremists on both sides of the climate sensitivity debate to moderate their views, and bring some calm reflection to the question of observation vs model results.<\/p>\n
I would emphasize that the fact of there being a good explanation for the curvilinear relationship in outgoing flux exhibited by the GCMs does not per se<\/i> mean that such a relationship must hold in the real world.\u00c2\u00a0 \u00c2\u00a0And if the relationship in the real world is curvilinear, there is no reason to believe at present that any model has the correct degree of curvature.<\/p>\n
However, there are a number of reasons to believe that some<\/span> curvature is likely for simple physical reasons, and Armour\u00e2\u20ac\u2122s elegant explanation takes us one step closer to being able to test for whether the \u00e2\u20ac\u0153degree of curvature\u00e2\u20ac\u009d exhibited by the models is real or \u00c2\u00a0artifactual.<\/p>\n If one accepts that the curvilinear response is<\/span><\/b> a real world phenomenon and that it is sufficient to bring into question the common assumption of constant linear feedback, \u00c2\u00a0one can reasonably conclude\u00c2\u00a0 that a zero-dimensional linear feedback model should never<\/b> be used by either skeptics or mainstream scientists – other than for local feedbacks or short-term feedbacks \u00e2\u20ac\u201c and yet this is a common\u00c2\u00a0 assumption that has been broadly applied to global response in hundreds of climate science papers.\u00c2\u00a0 \u00c2\u00a0\u00c2\u00a0Here are just a few of the possible inferences to be drawn from Armour 2012:-<\/p>\n <\/p>\n Armour 2012 developed what they called a \u00e2\u20ac\u0153parsimonious model\u00e2\u20ac\u009d to explain the curvilinear flux behaviour in very simple terms.\u00c2\u00a0 Here I want to examine the mathematical characteristics of (a slightly expanded version of) this simple model \u00e2\u20ac\u201c one which subdivides the world into N latitude zones, instead of the three heating elements used in Armour 2012.\u00c2\u00a0\u00c2\u00a0 Please note that my sole objective here is to add insight into how differences in regional temperature gain can combine to define a curvilinear flux response.\u00c2\u00a0 I am not trying to estimate any \u00e2\u20ac\u0153true\u00e2\u20ac\u009d climate parameters.<\/span><\/p>\n In my simple model, each of the latitude zones has an area, Ai<\/sub>, and is assigned a value of feedback, \u00c2\u00a0\u00ce\u00bbi<\/sub> in Watts.m-2<\/sup>.o<\/sup>C-1<\/sup>, and an \u00e2\u20ac\u0153e-folding time\u00e2\u20ac\u009d, \u00cf\u201ei<\/sub>, here with units of years;\u00c2\u00a0 the first is the control knob for the final equilibrium temperature in each latitude zone while the latter controls how long it takes to get there.<\/p>\n The flux balance (in Watts) for each latitude zone, is given by<\/p>\n Ai<\/sub> \u00ce\u00bbi<\/sub> \u00cf\u201ei<\/sub> dTi<\/sub>\/dt\u00c2\u00a0 (+\u00c2\u00a0 \u00ce\u201dQi<\/sub> -\u00c2\u00a0 \u00ce\u201dQi+1<\/sub> \u00c2\u00a0) =\u00c2\u00a0 Ai<\/sub> F\u00c2\u00a0 – Ai<\/sub> \u00ce\u00bbi<\/sub> Ti<\/sub>\u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0Eq\u00c2\u00a0 (1)<\/p>\n Where:-<\/p>\n Ti<\/sub> = temperature change in the ith latitude zone for i = 1 to N<\/p>\n \u00ce\u201dQi<\/sub>\u00c2\u00a0\u00c2\u00a0 = the change<\/span> in meridonial heat flux (ocean plus atmosphere) expressed in Watts flowing from the ith zone to the (i-1)th zone. \u00c2\u00a0This has zero value when i = 0 or i = N+1 .<\/p>\n F = the (cumulative) global forcing which varies as a function of time (Watts.m-2<\/sup>)<\/p>\n For simplicity here, although not necessary, \u00c2\u00a0we will make all of the Ai<\/sub> values equal.\u00c2\u00a0 If we sum up Eq (1) for all zones and divide throughout by the total global area, \u00c2\u00a0noting that the sensible heat flux terms must sum to zero, we obtain:-<\/p>\n (1\/N) \u00ce\u00a3 \u00ce\u00bbi<\/sub> \u00cf\u201ei<\/sub> dTi<\/sub>\/dt\u00c2\u00a0 =\u00c2\u00a0\u00c2\u00a0 F \u00c2\u00a0-\u00c2\u00a0 \u00c2\u00a0(1\/N) \u00ce\u00a3 \u00ce\u00bbi<\/sub> Ti<\/sub>\u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 Eq (2)<\/p>\n The aggregate net outgoing radiative flux\u00c2\u00a0 = (1\/N) \u00ce\u00a3 \u00ce\u00bbi<\/sub> Ti<\/sub>\u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0Eq (3)<\/p>\n The average surface temperature \u00c2\u00a0=\u00c2\u00a0 (1\/N) \u00ce\u00a3 Ti<\/sub>\u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 Eq (4)<\/p>\n We can now use Eq (1) to solve for temperature in each latitude zone and we can plot the net outgoing flux from Eq (3) against the average surface temperature from Eq (4).\u00c2\u00a0\u00c2\u00a0 In practice, the values of\u00c2\u00a0\u00c2\u00a0 \u00ce\u201dQi<\/sub> in Eq (1)\u00c2\u00a0 are non-negligible for the local temperature solutions, even though we are interested only in the change<\/span> in horizontal flux relative to initial steady-state conditions.\u00c2\u00a0 Like Armour 2012, I am going to ignore the horizontal flux terms here, since it complicates the solution routine and the picture, \u00c2\u00a0and I can match the effect on zonal temperatures at equilibrium by changing the local values of the feedback, \u00ce\u00bbi<\/sub> .\u00c2\u00a0 This simplification misses the richness of induced changes in the functional form of the local temperature solutions with time, but not the endpoint temperature values.\u00c2\u00a0 However, since the objective here is to gain insight rather than to achieve an accurate match to actual data, it is not a big problem here.<\/p>\n With this simplification, the solution to Eq 1 for each latitude zone is given by:-<\/p>\n Ti<\/sub> = F*(1 \u00e2\u20ac\u201c exp(-t\/ \u00cf\u201ei<\/sub>))\/ \u00ce\u00bbi<\/sub>\u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 Eq (5)<\/p>\n And at equilibrium (or rather steady-state) as t gets very large, we see that Ti<\/sub> ->\u00c2\u00a0\u00c2\u00a0 F\/ \u00ce\u00bbi<\/sub> for the zone.<\/p>\n We are now ready to test our new model.<\/p>\n I am going to use three forcing datasets to test the model.\u00c2\u00a0 These are:-<\/p>\n The F3 dataset comes from the inversion of the Hadcrut3 temperature series into the flux domain using a global linear feedback model.\u00c2\u00a0\u00c2\u00a0 In the context of this article, it can be thought of as just an arbitrary forcing dataset with high frequency content at about the right magnitude\u00c2\u00a0 to reflect 20th<\/sup> century variation.\u00c2\u00a0 The three datasets are shown below.<\/p>\n <\/p>\n This first test is the simplest of all.\u00c2\u00a0 We are going to make the values of lambda all equal to each other, and the values of tau all equal to each other.<\/p>\n <\/p>\n <\/p>\n These results are not surprising in any way.\u00c2\u00a0 The model is behaving as a global linear feedback model.\u00c2\u00a0 The values of lambda (2.95 Watts.m-2<\/sup>.o<\/sup>C-1<\/sup>) and of tau (3.05 years) have been chosen here so as to exactly reproduce the Hadcrut3 temperature data from which the forcing dataset was first derived.\u00c2\u00a0\u00c2\u00a0 The important thing to note is that the outgoing flux relationship with temperature is strictly linear, as it will always be if the latitude zones all have the same values of lambda and tau.<\/p>\n <\/p>\n <\/p>\n Now let\u00e2\u20ac\u2122s retain fixed values of tau \u00e2\u20ac\u201c equal response times in each latitude zone \u00e2\u20ac\u201c but assign differing \u00c2\u00a0lambda values and hence varying equilibrium temperatures to the latitude zones.<\/p>\n <\/p>\n Again, we will use the F3 forcing dataset.\u00c2\u00a0 The results are as follows.<\/p>\n <\/p>\n <\/p>\n <\/p>\n You will note that the results are identical to\u00c2\u00a0 Test #1.\u00c2\u00a0 This is because, although the lambda values are very different for each zone in this test, they were scaled here to yield the same total feedback as in Test #1 (i.e. so that the harmonic mean is equal to 2.95 Watts.m-2<\/sup>.o<\/sup>C-1<\/sup>).\u00c2\u00a0 The important thing to note is that the outgoing flux response is again strictly linear with temperature.\u00c2\u00a0 It is easily shown analytically that this is always true i.e if the response times are the same in each latitude zone then for any and every set of feedback values, the outgoing flux response will always be strictly linear with temperature. <\/span><\/b><\/p>\n This is an important result in its own right.\u00c2\u00a0 It means inter alia<\/i> that polar amplification alone cannot explain the curvature in outgoing flux seen in the GCMs.<\/span><\/p>\n <\/p>\n Now let\u00e2\u20ac\u2122s set the lambda values in each latitude zone equal to each other, but allow tau to take up differing values.<\/p>\n <\/p>\n Once again we see that the relationship between outgoing flux and temperature is strictly linear, and once again, we can show analytically that this is always true i.e. if the latitude zones all have the same value of lambda, then for any and every set of tau values the relationship between outgoing flux and temperature will be strictly linear<\/b>.\u00c2\u00a0\u00c2\u00a0 We may also note in passing that it is no longer possible to tune the result to match perfectly the Hadcrut3 temperature data with the F3 forcing dataset when we have a randomly assigned set of tau values.<\/p>\n <\/p>\n Well so far, we have not been able to generate any curvature in the flux response.\u00c2\u00a0 In order to do so we need a specific combination of circumstances; specifically, we need some zones which have a high relative temperature at equilibrium (low value of lambda) and a very slow relative response time (high value of tau).\u00c2\u00a0\u00c2\u00a0 The high temperature response is expected at high latitudes (\u00e2\u20ac\u0153polar amplification\u00e2\u20ac\u009d);\u00c2\u00a0 we need to postulate that the response times for the high latitudes are much higher than for the tropics and subtropical regions.\u00c2\u00a0 So here are my assigned values of lambda and tau, together with the resulting equilibrium temperatures by latitude zone.<\/p>\n The ECS for this system is 3.2 deg C.\u00c2\u00a0\u00c2\u00a0 The parameter values have been chosen so that they still give a fair match to historical temperatures under the F3 forcing dataset, and so that they also yield the approximate regional temperature contrasts observed in the GCMs (polar amplification).<\/p>\n The large contrasts in temperature and in response times between the latitude zones are shown below for forcing dataset F2 (1% doubling).\u00c2\u00a0\u00c2\u00a0 Tropical and subtropical responses are small and fast, polar responses are large\u00c2\u00a0and slow.<\/p>\n A comparison with the temperature vs latitude plots shown in Armour2012 reveals that the above results are not wildly different in form from the results abstracted from the CCSM4 model examined in the paper.<\/p>\n With these parameter values the outgoing flux response now looks as follows:-<\/p>\n So there it is.\u00c2\u00a0 We see that with the strong relative contrasts in both temperature and response time, we have now introduced a significant curvature into the outgoing flux response, in fact one that is slightly exaggerated relative to the CCSM4 model.<\/p>\n The behaviour of the system under the historic forcing dataset F3 still reveals little about the curvature.\u00c2\u00a0 A regression on the data from the F3 run would suggest an ECS value of 1.96 deg C, under the assumption of linear behaviour, whereas the actual ECS of this system is 3.2 deg C.\u00c2\u00a0 This highlights the danger of using short-term data to estimate ECS if the system really does have a nonlinear flux response.<\/p>\n We additionally note that we have now introduced a historic dependence into the relationship between the outgoing flux response and temperature;\u00c2\u00a0 such dependence (on forcing history) cannot exist\u00c2\u00a0 if the flux response is linear.\u00c2\u00a0 This has important implications for selection of the appropriate\u00c2\u00a0 methodology to analyze observational data or GCM results.\u00c2\u00a0\u00c2\u00a0 In particular it raises an important question about the bias introduced by the application of simple regression methods or estimates of climate feedbacks from secant gradients. \u00c2\u00a0Quite simply, the outgoing flux response for this system \u00c2\u00a0is multivalued<\/b> against average surface temperature, and we will see this more clearly in Test #5.<\/p>\n <\/p>\n A simple answer is \u00e2\u20ac\u0153yes\u00e2\u20ac\u009d.\u00c2\u00a0\u00c2\u00a0 Using the same model parameters as in Test #4, I have run out a series of step-forcing tests for forcing values of 1, 2 \u00c2\u00a0and 4 Watts\/m2.<\/p>\n The results are shown below.<\/p>\n <\/p>\n We see that this system maintains a strictly linear relationship between the input step-forcings and the final equilibrium temperatures (the red spots), by virtue of a series of self-similar curves in flux-temperature.\u00c2\u00a0 This is compatible with results reported from GCMs.\u00c2\u00a0 In fact the simple system modelled here is a linear system for temperature in the time domain.\u00c2\u00a0\u00c2\u00a0 (Temperature solutions from different forcing scenarios are additive in the time domain.)<\/p>\n We can also see rather more clearly that the flux response in this system is multi-valued with respect to average surface temperature.\u00c2\u00a0 The gradient of any line picked out by regression (or a secant gradient estimated between two temperature points) is dependent on the specific forcing history.\u00c2\u00a0\u00c2\u00a0\u00c2\u00a0 This raises some serious questions about the validity of regression methods of the form (F \u00e2\u20ac\u201c Net Flux) vs Temperature to estimate feedback from models or from long-term history with variable forcing.\u00c2\u00a0\u00c2\u00a0 Equally, it raises questions about the validity of plots of Forcing vs Temperature to estimate Transient Climate Response (TCR) \u00e2\u20ac\u201c the temperature gain at the point of doubling CO2 after a 1% per year growth.<\/p>\n <\/p>\n <\/p>\n Armour 2012 offers a simple, elegant and coherent explanation for the curvature in outgoing flux response with temperature gain observed in the GCMs.\u00c2\u00a0 It proposes a model which is linear in temperature, and so which yields a linear relationship between the surface average temperature at equilibrium and applied (step) forcing.\u00c2\u00a0\u00c2\u00a0 This is consistent with GCM results.\u00c2\u00a0\u00c2\u00a0\u00c2\u00a0 It also very neatly matches\u00c2\u00a0 and explains\u00c2\u00a0 the \u00e2\u20ac\u0153recalcitrant heating\u00e2\u20ac\u009d observed by Held et al 2010 in the CM2.1 model (see here<\/a>), although, for brevity, \u00c2\u00a0I have not included the tests here.<\/p>\n In an ideal world Armour 2012 should cause sceptics to recognise that\u00c2\u00a0 many observation-based estimates of climate sensitivity, founded on a global linear feedback assumption,\u00c2\u00a0 might represent no more than a lower-bound.\u00c2\u00a0 It should also prompt some soul-searching with respect to regression methods for estimating climate sensitivity and related parameters.\u00c2\u00a0 These methods can give misleading results even with small deviation from a linear relationship between outgoing flux and temperature.<\/p>\n The landmark papers on the partitioning of feedbacks in the GCMs produce\u00c2\u00a0 arbitrary and misleading results.\u00c2\u00a0 Note that this is true whether or not the curvature is a realworld phenomenon, since the curvature most definitely does exist in the models.<\/p>\n On the other hand, the question of how much curvature is correct in the real world is still open.\u00c2\u00a0 Just demonstrating that polar amplification is likely on physical grounds is not alone sufficient to justify the curvature shown by many of the models.\u00c2\u00a0 As shown above in Test #2, it is also necessary to prove by reference to real data that the pace of heating is an order of magnitude slower in those regions with expected amplified temperatures than in the tropical to mid-latitudes.\u00c2\u00a0\u00c2\u00a0 This should signal a requirement for\u00c2\u00a0modelers\u00c2\u00a0to shift \u00c2\u00a0focus from the global aggregate response to careful examination of the regional and latitude responses.\u00c2\u00a0 This is an imperative.\u00c2\u00a0 Armour 2012 demonstrates that ECS estimates derived from models are only as good or as bad as the model\u00e2\u20ac\u2122s ability to match latitudinal behaviour with respect to changes in temperature and TOA flux in time. \u00c2\u00a0There is still time for an adult conversation.<\/p>\n Update 17th February – Communication with Kyle Armour<\/strong><\/p>\n Nic Lewis in a couple of comments pushed back against Armour’s geometric explanation of curvilinear flux response on the grounds that the distribution of feedbacks ran in the “wrong” direction in the models.<\/p>\n Nic Lewis comment: “I find the fact that their CCSM4 GCM seems to have exactly the opposite pattern of latitudinal variation of the climate feedback parameter to what I understand to be the correct pattern (lambda higher in the extratropics) very disconcerting. If both their GCM and their 3-zone model results are based on the opposite of the actual latitudinal pattern of lambda, why should one think they are correct?<\/em>”<\/p>\n and later from Nic:<\/p>\n “The models have only small negative, or even positive, feedback in the tropics, as with constant relative humidity the water vapour feedback is extremely strong there. Water vapour feedback decreases much faster than temperature feedback with latitude, and net feedbacks become, on the whole, increasingly negative towards the poles.<\/em>”<\/p>\n I subsequently responded:<\/p>\n ” Polar amplification seems to be a near-universal feature of the GCMs, but I think I am going to have to accept your pushback. If Figure 3 in Zelinka is valid, then I guess we conclude that many models explain polar amplification not with a low relative magnitude feedback, but with a high magnitude feedback and an enhanced meridonial heat flux.<\/em>”<\/p>\n I asked Dr Armour by e-mail if he could explain this apparent discrepancy in feedback distribution with latitude, and received a thoughtful response which addresses the question directly. \u00c2\u00a0I reproduce the main body of his response in full below (with his permission).<\/p>\n …You (and commenters) raise some very good points concerning the distribution of feedbacks across different GCMs. I have a few thoughts on the subject that I hope will clarify things:<\/p>\n 1) There is an important difference in how we calculate local feedbacks compared to previous studies. Specifically, our local feedbacks are a linearization about local surface temperature change (as in W\/m^2 per degree local warming), which allows us to assess how the global-mean feedback varies with evolving patterns of surface temperature. The local feedbacks in Zelinka and Hartmann (2012) are instead normalized with respect to global-mean surface temperature change (as in W\/m^2 per degree global warming, following the methods of Brian Soden and Karen Shell). Unfortunately, this means that you can’t directly compare our feedback patterns with those of Zelinka.<\/p>\n One of the points we make is that with the Zelinka\/Soden\/Shell normalization, the feedback pattern itself will depend on the pattern of surface warming, which is different across models and forcing scenarios. We propose that linearizing about local temperature provides a more steady measure of local feedbacks, and hypothesize that this may help to narrow the large feedback spread across models (e.g., Zelinka Fig. 3). I’m working on such a feedback re-normalization using CMIP5 models, in a followup to this study, but unfortunately don’t have results just yet.<\/p>\n <\/p>\n 2) Keeping the above in mind, it may still be the case that the feedback pattern in CCSM4 is a bit of an outlier. The cloud feedbacks in particular seem to be less positive in the tropics than in most models, leading to a net feedback that is more negative in the tropics than in the model average. This large meridional feedback gradient (increasing toward higher latitudes) implies that effective climate sensitivity should vary more in CCSM4 than in most models, and this seems to be the case (diagnosing Teff from Table 2 of Winton et al 2010). As we note in the paper, those models with a substantially more “flat” meridional feedback structure should show much less time variation of effective climate sensitivity.<\/p>\n Thus, I wouldn’t say that our physical explanation is model-specific, but instead that the degree to which effective climate sensitivity varies is model-specific due their highly-variable feedback patterns. Of critical importance for future climate prediction is of course understanding the pattern of regional feedbacks in nature.<\/p>\n 3) While I like our linear regional feedbacks framework for its simplicity and ability to explain the CCSM4 behavior, there is the possibility that other mechanisms are at work as well. An interesting possibility is that nonlinearities in local cloud feedbacks may contribute to the “curvature” of global TOA flux with global surface temperature. While we found this to be a small effect in CCSM4, I am open to the idea that it is a larger effect in other models. I’m looking forward to repeating this analysis across a range of GCMs to understand this better.<\/p>\n <\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":" As I have noted before, most of the AOGCMs exhibit a curvilinear\u00c2\u00a0 response in outgoing global flux with respect to average temperature change.\u00c2\u00a0 One of the consequences of this is that there is a well-reported apparent increase in the effective climate sensitivity with time and temperature in the models; in particular, the effective climate sensitivity … Continue reading Observation vs Model – Bringing Heavy Armour into the War<\/span> \n
A slightly less parsimonious model<\/h2>\n
Testing the Model<\/h2>\n
\n
![\"forcingdata\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/forcingdata-500x375.jpg\")
<\/a><\/p>\nTest #1 \u00e2\u20ac\u201c Equal values of lambda and equal values of tau<\/h3>\n
![\"test1\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test1-500x375.jpg\")
<\/a><\/p>\nTest #2 Latitude Zones have differing values of Lambda but equal values of Tau<\/h3>\n
![\"test2\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test2-500x375.jpg\")
<\/a><\/p>\nTest #3 Equal values of lambda but differing values of tau<\/h3>\n
![\"test3\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test3-500x375.jpg\")
<\/a><\/p>\nTest #4 Varying values of lambda and of tau<\/h3>\n
![\"Assignedvalues\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/Assignedvalues-500x375.jpg\")
<\/a><\/p>\n![\"test4tempmatch\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test4tempmatch-500x375.jpg\")
<\/a><\/p>\n![\"test4tempbehaviour\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test4tempbehaviour-500x375.jpg\")
<\/a><\/p>\n![\"test4allFdatasets\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test4allFdatasets-500x375.jpg\")
<\/a><\/p>\nTest #5\u00c2\u00a0 Is this System a Linear System?<\/h3>\n
![\"test5diffstepF\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/test5diffstepF-500x375.jpg\")
<\/a><\/p>\nMain Conclusions<\/h2>\n
![\"Zelinkafeedbacks\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2013\/02\/Zelinkafeedbacks-500x375.jpg\")
<\/a><\/p>\n