Keywords: climate; sensitivity; tropical feedback; warming; Pinatubo; volcanism; regression; ERBE; AOD
For the period surrounding the eruption of Mount Pinatubo in the Philippines in June 1991, there are detailed satellite data for both the top-of-atmosphere ( henceforth TOA ) radiation budget and atmospheric optical depth ( henceforth AOD ) that can be used to derive the changes in radiation entering the climate system during that period. Analysis of satellite radiation measurements allows an assessment of the system response to changes in the radiation budget. The Mt Pinatubo eruptions provide a particularly useful natural experiment, since the spread of the effects were centred in the tropics and dispersed fairly evenly between the hemispheres.  fig.6 ( Much detail on the composition and dispersal of the aerosol cloud is given in this meta-study.)
The present study investigates the tropical climate response in the years following the eruption. It is found that a simple relaxation response, driven by volcanic radiative ‘forcing’, provides a close match with the variations in TOA energy budget. The derived scaling factor to convert AOD into radiative flux, supports earlier estimations  based on observations from the 1982 El Chichon eruption. These observationally derived values of the strength of the radiative disturbance caused by major stratospheric eruptions are considerably greater than those currently used as input parameters for general circulation climate models (GCMs).
Two alternative time-constants for the relaxation response are derived, depending on the whether a cyclic, pre-eruption variability is taken into account or not. This is found to have little effect on the derived magnitude of the aerosol forcing.
Changes in net TOA radiative flux, measured by satellite, were compared to volcanic forcing estimated from measurements of atmospheric optical depth. Regional optical depth data, with monthly resolution, are available in latitude bands for four height ranges between 15 and 35 km [DS1] and these values were averaged from 20S to 20N to provide a monthly mean time series for the tropics. Since optical depth is a logarithmic scale, the values for the four height bands were added at each geographic location. Measurements for the period concerned come from SAGE II satellite data. Lacis et al  suggest that aerosol radiative forcing can be approximated by a linear scaling factor of AOD over the range of values concerned. This is the approach usually adopted in IPCC reviewed assessments and is used here. As a result the vertical summations are averaged across the tropical latitude range for comparison with radiation data.
Tropical TOA net radiation flux is provided by Earth Radiation Budget Experiment ( ERBE ) [DS2].
One notable effect of the eruption of Mt Pinatubo on tropical energy balance is a variation in magnitude of the annual cycle, as see by subtracting the pre-eruption, mean annual variation. The sun passes over the tropics twice per year and the mean annual cycle in the tropics shows two peaks: one in March the other in Aug/Sept, with a minimum in June and a lesser dip centred around the new year. The residual annual variability is also basically a six monthly cycle. Following the eruption, the annual variation increased and only returned to similar levels after the 1998 El Nino.
Figure 1 showing the variability of the annual cycle in net TOA radiation flux in tropics. (Click to enlarge)
It follows that using a single period as the basis for the anomaly leaves significant annual residuals. To minimise the residual, three different periods of multiples of 12 months were used to remove the annual variations: pre-1991; 1992-1995; post 1995.
The three resultant anomaly series were combined, ensuring the difference in the means of each period were respected. The mean of the earlier, pre-eruption annual cycles was taken as the zero reference for the whole series. There is a clearly repetitive variation during the pre-eruption period that produces a significant downward trend starting 18 months before the Mt. Pinatubo event. Since it may be important not to confound this with the variation due to the volcanic aerosols, it was characterised by fitting a simple cosine function. Though the degree to which this can be reasonably assumed to continue is speculative, it seems some account needs to be taken of this pre-existing variability. The effect this has on the result of the analysis is assessed.
The break of four months in the ERBE data at the end of 1993 was filled with the anomaly mean for the period to provide a continuous series. This truncates what would have probably been a small peak in the data, marginally lowering the local average, but since this was not the primary period of interest this short defect was considered acceptable.
Figure 2 showing ERBE tropical TOA flux adaptive anomaly. (Click to enlarge)
Figure 2b showing ERBE tropical TOA flux adaptive anomaly with pre-eruption cyclic variability subtracted. (Click to enlarge)
Since the TOA flux represents the net sum of all “forcings” and the climate response, the difference between the volcanic forcing and the anomaly in the energy budget can be interpreted as the climate response to the radiative perturbation caused by the volcanic aerosols. This involves some approximations. Firstly, since the data is restricted to the tropical regions, the vertical energy budget does not fully account for energy entering and leaving the region. There is a persistent flow of energy out of the tropics both via ocean currents and atmospheric circulation. Variations in ocean currents and atmospheric circulation may be part of any climate feedback reaction.
Secondly, taking the difference between TOA flux and the calculated aerosol forcing at the top of the troposphere to represent the top of troposphere energy budget assumes negligible energy is accumulated or lost internally to the upper atmosphere. Although there is noticeable change in stratospheric temperature as a result of the eruption, the heat capacity of the rarefied upper atmosphere means this is negligible in this context.
Figure 3 showing changes in lower stratosphere temperature due to volcanism. (Click to enlarge)
A detailed study on the atmospheric physics and radiative effects of stratospheric aerosols by Lacis, Hansen & Sato  suggested that radiative forcing at the tropopause can be estimated by multiplying optical depth by a factor of 30 W / m2.
This value provides a reasonably close match to the initial change in ERBE TOA flux. However, later studies,  , attempting to reconcile climate model output with the surface temperature record have reduced the presumed magnitude of the effect stratospheric aerosols. With the latter adjustments, the initial effect on net TOA flux is notably greater than the calculated forcing, which is problematic; especially since Lacis et al reported that the initial cooling may be masked by the warming effect of larger particles ( > 1µm ). Indeed, in order for the calculated aerosol forcing to be as large as the initial changes in TOA flux, without invoking negative feedbacks, it is necessary to use a scaling of around 40 W/m2. A comparison of these values is shown in figure 4.
It should be noted that there is a strong commonality of authors in these papers, so rather than being the work of conflicting groups, the more recent weightings reflect the result of a change of approach: from direct physical modelling in 1992 to the later attempts to reconcile general circulation model (GCM) output by altering the inputs.
What is significant is that from just a few months after the eruption, the disturbance in TOA flux is consistently less than the volcanic forcing. This is evidence of a strong negative feedback in the tropical climate system acting to counter the volcanic perturbation. Just over a year after the eruption, it has fully corrected the radiation imbalance despite the disturbance in AOD still being at about 50% of its peak value. The net TOA reaction then remains positive until the “super” El Nino of 1998. This is still the case with reduced forcing values of Hansen et al as can also be seen in figure 4.
Figure 4 comparing volcanic of net TOA flux to various estimations aerosol forcing. ( Click to enlarge )
The fact that the climate is dominated by negative feedbacks is not controversial since this is a pre-requisite for overall system stability. The main stabilising feedback is the Plank response ( about 3.3 W/m2/K at typical ambient temperatures ). Other feedbacks will increase or decrease the net feedback around this base-line value. Where IPCC reports refer to net feedbacks being positive or negative, it is relative to this value. The true net feedback will always be negative.
It is clear that the climate system takes some time to respond to initial atmospheric changes. It has been pointed out that to correctly compare changes in radiative input to surface temperatures some kind of lag-correlation analysis is required : Spencer & Bradwell 2011, Lindzen & Choi 2011. Both show the strongest direct correlation peaks at a lag of about three months. After a few months negative feedbacks begin to have a notable impact and the TOA flux anomaly declines more rapidly than the reduction in AOD.
Figure 5 showing climate feedback response to Mt Pinatubo eruption. Volcanic forcing per Lacis et al. ( Click to enlarge )
It is quickly apparent that a simple, fixed temporal lag is not the most appropriate way to compare the aerosol forcing to its effects on the climate system. The simplest physical response of a system to a disturbance would be a linear relaxation model, or “regression to the equilibrium”, where for a deviation of a variable T from its equilibrium value, there is a restoring action that is proportional to the magnitude of that deviation. The more it is out of equilibrium the quicker its rate of return. This kind of model is common in climatology and is central of the concept of climate sensitivity to changes in various climate “forcings”.
dT/dt= -k*T ; where k is a constant of proportionality.
The solution of this equation for an infinitesimally short impulse disturbance is a decaying exponential. This is called the impulse response of the system. The response to any change in the input can found by convolution with this impulse response. This can be calculated quite simply since it is effectively a weighted running average calculation.
The effect of this kind of system response is a time lag as well as a degree of low-pass filtering which produces a change in the profile of the time series, compared to that of the input forcing.
The speed of the response is characterised by a constant parameter in the exponential function, often referred to as the ‘time-constant’ of the reaction. Once the time-constant parameter has been determined, the time-series of the system response can be calculated from the time-series of the forcing.
The variation of the tropical climate response is compared with a linear relaxation response to the volcanic forcing. The magnitude and time-constant provide two free parameters and are found to provide a good match. This is not surprising since any change in surface temperature will produce a Plank feedback to oppose it. The radiative Plank feedback is the major negative feedback that ensures the general stability of the Earth’s climate. While the Plank feedback is proportional to the fourth power of the absolute temperature, it can be approximated as linear for small changes around typical ambient temperatures of about 300 kelvin.
It is this delayed response curve that needs to be compared to changes in surface temperature to determine the overall climate sensitivity. Regressing the temperature change against the change in radiation is not physically meaningful unless the system can be assumed to equilibrate much faster than the period of the transient being studied, ie on a time scale of a month or less. This is clearly not the case, yet many studies have been published which do precisely this, or worse multivariate regression, which compounds the problem.
The need for this can be seen in figure 6. The in-phase climate feedback response ( shown negated ) that informs about climate sensitivity is confounded with the orthogonal response (the integral of the rate of change of temperature which is what is actually induced by a radiative forcing). Simply shifting the volcanic forcing forward by about a year would line up the “bumps” but not match the profile of the two variables. Therefore simple regression or even a lagged regression would not correctly associate the two variables: the differences in the temporal evolution of the two leads to a lower correlation, a reduced regression result leading to incorrect values of climate sensitivity.
Figure 6 showing tropical feedback as relaxation response to volcanic aerosol forcing ( pre-eruption cycle removed) ( Click to enlarge )
This lagged response to radiative change corresponds to the negative quadrant in Spencer and Braswell’s  figure 3b excerpted below, where temperature lags radiative change. It shows the peak temperature response lagging around 12 months behind the radiative change. The timing of this peak is in close agreement with figure 6 above, despite SB11 being derived from CERES (Clouds and the Earth’s Radiant Energy System) satellite data from the post-2000 period with negligible volcanism.
This emphasises that the value for the correlation in the SB11 graph will be under-estimated, as pointed out by the authors:
Diagnosis of feedback cannot easily be made in such situations, because the radiative forcing decorrelates the co-variations between temperature and radiative flux.
This analysis attempts to address that problem by analysing the fully developed response.
The equation of the relationship of the climate response : ( TOA net flux anomaly – volcanic forcing ) being proportional to an exponentially lagged AOD, is re-arranged to obtain an empirical estimation of the scaling factor by linear regression.
TOA -VF * AOD = -VF * k * exp_AOD eqn. 1
-TOA = VF * ( AOD – k * exp_AOD ) eqn. 2
VF is the scaling factor to convert ( positive ) AOD into a radiation flux anomaly in W/m2. The exp_AOD term is the exponential convolution of the AOD data, a function of the time-constant tau, whose value is also to be estimated from the data. This exp_AOD quantity is multiplied by a constant of proportionality, k. Since TOA net flux is conventionally given as positive downwards, it is negated in equation 2 to give a positive VF comparable to the values given by Lacis, Hansen, etc.
Average pre-eruption TOA flux was taken as the zero for TOA anomaly and, since the pre-eruption AOD was also very small, no constant term was included.
The regression was performed for a range of time-constants from 1 to 24 months. The period for the regression was from just before the eruption, up to 1994.7, when AOD had subsided and the magnitude of the annual cycle was found to change, indicating the end of the initial climatic response ( determined from the adaptive anomaly shown in figure 2 ).
Since the relaxation response effectively filters out ( integrates ) much of high frequency variability giving a less noisy series, this was taken as the independent variable for regression. This choice acts to minimise regression dilution due to the presence of measurement error and non-linear variability in the independent variable. This is a fundamental problem when using regression techniques in this situation. Failure to addressed it leads to spuriously low regression values. For noisey data this can lead to serious errors.
This is an important and pervasive problem that is often overlooked in published work in climatology. Notably in attempts to derive an estimation of climate sensitivity from temperature and radiation measurements and from model output.
Once the scaling factor was obtained by linear regression for each value of tau, the values were checked for regression dilution by examining the correlation of the residual of the fit with the AOD regressor while varying the value of VF in the vicinity of the fitted value. This was found to give a regular curve with a minimum correlation that was very close to the fitted value. It was concluded that the least squares regression results were an accurate estimation of the presumed linear relationship.
The low values of time-constant resulted in very high values of VF ( eg. 154 for tau=1 month ) that were physically unrealistic and way out side the range of credible values. This effectively swamps the TOA anomaly term and is not meaningful.
tau = 6 months gave 54 and this was taken as the lower limit for the range time constant to be considered.
The two free parameters of the regression calculations will ensure an approximate fit of the two curves for each value of the time constant. The latter could then be varied to find the response that best fitted the initial rise after the eruption, the peak and the fall-off during 1993-1995.
This presented a problem since, even with adaptive anomaly approach, there remains a significant, roughly cyclic, six month sub-annual variability that approaches in magnitude the climate response of interest. The relative magnitude of the two also varies depending on the time-constant used, making a simple correlation test unhelpful in determining the best correlation with respect to the inter-annual variability. For this reason it could not be included as a regression variable.
Also the initial perturbation and the response, rise very quickly from zero to maximum in about two months. This means that low-pass filtering to remove the 6 month signal will also attenuate the initial part of the effect under investigation. For this reason a light 3-sigma gaussian filter ( sigma = 2 months ) was used. This has 50% attenuation at 2.3 months. comparable to a 4 month running mean filter ( without the destructive distortions of the latter). This allowed a direct comparison of the rise, duration of the peak and rate of fall-off in the tail that is determined by the value of the time-constant parameter, and thus the value producing the best match to the climate response.
Due to the relatively course granularity of monthly data that limits the choice of the time-constant values, it was possible to determine a single value that produced the best match between the two variables.
The whole process was repeated with and without subtraction of the oscillatory, pre-eruption variability to determine the effects of this adjustment.
Taking the TOA flux, less the volcanic forcing, to represent the climatic reaction to the eruption, gives a response that peaks about twelve months after the eruption, when the stratospheric aerosol load is still at about 50% of its peak value. This implies a strong negative feedback is actively countering the volcanic disturbance.
This delay in the response, due to thermal inertia in the system, also produces an extended period during which the direct volcanic effects are falling and the climate reaction is thus greater than the forcing. This results in a recovery period, during which there is an excess of incoming radiation compared to the pre-eruption period which, to an as yet undetermined degree, recovers the energy deficit accumulated during the first year when the volcanic forcing was stronger than the developing feedback.
Thus if the lagged nature of the climate response is ignored and direct linear regression between climate variables and optical depth are conducted, the later extended period of warming may spuriously attributed to some other factor. This represents a fundamental flaw in multivariate regression studies such as Foster & Rahmstorf 2013  that could lead to seriously erroneous conclusions about the relative contributions of the various regression variables.
For the case where the pre-eruption variation is assumed to continue to underlie the ensuing reaction to the volcanic forcing, the ratio of the relaxation response to the aerosol forcing is found to be 0.86 +/- 0.07%, with a time constant of 8 months. The scaling factor to convert AOD into a flux anomaly was found to be 33 W/m2 +/-11%. With these parameters, the centre line of the remaining 6 month variability ( shown by the gaussian filter ) fits very tightly to the relaxation model.
Figure 6 showing tropical feedback as relaxation response to volcanic aerosol forcing ( pre-eruption cycle removed) ( Click to enlarge )
If the downward trend in the pre-eruption data is ignored (ie its cause is assumed to stop at the instant of the eruption ) the result is very similar ( 0.85 +/-0.09 and 32.4 W/m2 +/- 9% ) but leads to a significantly longer time-constant of 16 months. In this case, the fitted response does not fit nearly as well, as can be seen by comparing figures 6 and 8. The response is over-damped: slow to drop and too slow to rise back up, indicating this time-constant is too long.
Figure 8 showing tropical climate relaxation response to volcanic aerosol forcing, fitted while ignoring pre-eruption variability. ( Click to enlarge )
The analysis with the pre-eruption cycle subtracted provides a generally flat residual ( figure 9 ), showing that it accounts well for the longer term response to the radiative disruption caused by the eruption.
It is also noted that the truncated peak, resulting from substitution of the mean of the annual cycle to fill the break in the ERBE satellite data, lies very close to the zero residual line.
Figure 9 showing the residual of the fitted relaxation response from the satellite derived, top-of-troposphere disturbance. ( Click to enlarge )
Since the magnitude of the pre-eruption variability in TOA flux, while smaller, is of the same order as the volcanic forcing and its period similar to that of the duration of the atmospheric disturbance, the time-constant of the derived response is quite sensitive to whether this cycle is removed or not. However, it does not have much impact on the fitted estimation of the scaling factor ( VF ) required to convert AOD into a flux anomaly or the proportion of the exponentially lagged forcing that matches the TOA flux anomaly.
Assuming that whatever was causing this variability stopped at the moment of the eruption seems unreasonable but whether it was as cyclic as it appears to be, or how long that pattern would continue is speculative. However, approximating it as a simple oscillation seems to be more satisfactory than ignoring it.
In either case, there is a strong support here for values close to the original Lacis et al 1992 calculations of volcanic forcing that were derived from physics-based analysis of observational data, as opposed to later attempts to reconcile the output of general circulation models by arbitrarily adjusting this physical parameter.
Several climate studies ,, have been published which attempt to assess climate sensitivity by lagged linear regression of radiative forcing against temperature, both for observations and for assessing how climate models respond to various forcings. This is not totally satisfactory, since, while recognising the need to account for the delay, it does not account for the fact that the profile of the responsive is a different shape to that of the input. This will reduce the correlation of the two variables and thus increase the error and uncertainty of the regression result and by implication that of the derived climate sensitivity. SB11 makes just this point, showing that observed lag-response is similar to that of a 70/30 mix of radiative and non-radiative interactions and that the peak correlation at 3 months lag will considerably under-estimate the true value.
This analysis models the observed results very closely with a simple relaxation response.
Beyond the initial climate reaction analysed so far, it is noted that the excess incoming flux does not fall to zero. To see this effect more clearly, the deviation of the flux from the chosen pre-eruption reference value is integrated over the full period of the data. The result is shown in figure 10.
Figure 10 showing the cumulative integral of climate response to Mt Pinatubo eruption. ( Click to enlarge )
Pre-eruption variability produces a cumulative sum initially varying about zero. Two months after the eruption, when it is almost exactly zero, there is a sudden change as the climate reacts to the drop in energy entering the troposphere. From this point onwards there is an ever increasing amount of additional energy accumulating in the tropical lower climate system. With the exception of a small drop, apparently in reaction to the 1998 ‘super’ El Nino, this tendency continues to the end of the data.
While the simple relaxation model seems to adequately explain the initial four years following the Mt Pinatubo event, this does not explain it settling to a higher level.
A clue to the this continued excess over the pre-eruption conditions can be found in the temperature of the lower stratosphere, shown in figure 3. Here too, the initial disturbance seems to have stabilised by early 1995 but there is a definitive step change from pre-eruption conditions.
Noting the complementary nature of the effects of impurities in the stratosphere on TLS and the lower climate system, this drop in TLS may be expected to be accompanied by an increase in the amount of incoming radiation penetrating into the troposphere. This is in agreement with the cumulative integral shown in figure 10.
NASA Earth Observatory report  that after Mt Pinatubo, there was a 5% to 8% drop in stratospheric ozone. Presumably a similar removal happened after El Chichon in 1982 which saw an almost identical reduction in TLS.
Whether this is, in fact, the cause or whether other radiation blocking aerosols were flushed out along with the volcanic emissions, the effect seems clear and consistent and quite specifically linked to the eruption event. This is witnessed in both the stratospheric and tropical tropospheric data. Neither effect is attributable to the steadily increasing GHG forcing which did not record a step change in September 1991.
This raises yet another possibility for false attribution in multivariate regression studies and in attempts to arbitrarily manipulate GCM input parameters to engineer a similarity with the recent surface temperature records.
With the fitted scaling factor showing the change in tropical TOA net flux matches 85% of the tropical AOD forcing, the remaining 15% must be dispersed elsewhere within the climate system. That means either storage in deeper waters and/or changes in the horizontal energy budget, ie. interaction with extra-tropical regions.
Since the model fits the data very closely, the residual 15% will have the same time dependency profile as the 85%, so these tropical/ex-tropical variations can also be seen as part of the climate response to the volcanic disturbance. ie. the excess in horizontal net flux, occurring beyond 12 months after the event, is also supporting restoration of the energy deficit in extra-tropical regions by exporting heat energy. Since the major ocean gyres bring cooler temperate waters into the eastern parts of tropics in both hemispheres and export warmer waters at their western extents, this is probably a major vector of this variation in heat transport as are changes in atmospheric processes like Hadley convection.
Extra-tropical regions were previously found to be more sensitive to radiative imbalance that the tropics:
Thus the remaining 15% may simply be the more stable tropical climate acting as a buffer and exerting a thermally stabilising influence on extra-tropical regions.
Both the TLS cooling and the energy budget analysis presented here, show a lasting warming effect on surface temperatures triggered by the Mt Pinatubo event. Unless these effects are recognised, their mechanisms understood and correctly modelled, there is a strong likelihood of this warming being spuriously attributed to some other cause such as AGW.
The values for volcanic aerosol forcing derived here being in agreement with the physics-based assessments of Lacis et al. imply much stronger negative feedbacks must be in operation in the tropics than those resulting from the currently used model “parametrisations” and the much weaker AOD scaling factor.
These two results, taken together indicate that secondary effects of volcanism may have actually contributed to the late 20th century warming, rather than counteracting it. This would go a long way to explaining the discrepancy between climate models and the relative stability of observational temperatures measurements since the turn of the century.
IPCC on Clouds and Aerosols:
IPCC AR5 WG1 Full Report Jan 2014 : Chapter 7 Clouds and Aerosols:
No robust mechanisms contribute negative feedback.
The responses of other cloud types, such as those associated with deep convection, are not well determined.
Satellite remote sensing suggests that aerosol-related invigoration of deep convective clouds may generate more extensive anvils that radiate at cooler temperatures, are optically thinner, and generate a positive contribution to ERFaci (Koren et al., 2010b). The global influence on ERFaci is
WG1 are arguing from a position of self-declared ignorance on this critical aspect of how the climate system reacts to changes in radiative forcing. It is unclear how they can declare confidence levels of 95%, based on such an admittedly poor level of understanding of the key physical processes.
Analysis of satellite radiation measurements allows an assessment of the system response to changes in radiative forcing. This provides an estimation of the aerosol forcing that is in agreement with the range of physics-based calculations presented by Lacis et al in 1992 and is thus brings into question the much lower values currently used in GMC simulations.
Since this scaling factor was lowered specifically to match the output of high sensitivity models to the pre-2000 climate record (Hansen 2002), the higher values of aerosol forcing found here imply the presence of strongly negative feedbacks in tropical climate ( in excess of the base-line Plank feedback ) and hence imply a lower range of sensitivity than those used in the models.
The considerable lag and implied post-eruption recovery period underlines the inadequacy of direct regression and multivariate regression in assessing the importance of various climate ‘forcings’ and their respective climate sensitivities. Use of such methods will suffer from omitted variable bias and can lead to seriously erroneous attributions.
The evidence presented here of a strong, negative feedback in tropical climate countering the radiative disruption caused by major stratospheric volcanic eruptions has profound implications for the impacts of all phenomena that change the earth’s radiation balance, including the effects of anthropogenic greenhouse gases.
 Self et al 1995
“The Atmospheric Impact of the 1991 Mount Pinatubo Eruption”
 Lacis et al 1992 : “Climate Forcing by Stratospheric Aerosols
 Hansen et al 1997 “Forcing and Chaos in interannual to decadal climate change”
 Hansen et al 2002 : Climate forcings in Goddard Institute for Space Studies SI2000 simulations
 Spencer & Braswell 2011: “On the Misdiagnosis of Surface Temperature Feedbacks from Variations in Earth’s Radiant Energy Balance”
 Lindzen & Choi 2011: “On the Observational Determination of Climate Sensitivity and Its Implications”
 Trenberth et al 2010: “Relationships between tropical sea surface temperature and top‐of‐atmosphere radiation”
 Dessler 2011: “Cloud variations and the Earth’s energy budget”
 Foster and Rahmstorf 2011: “Global temperature evolution 1979-2010”
 Mann at al 2014: “On Forced Temperature Changes, Internal Variability and the AMO”
[DS1] AOD data
[DS2] ERBE TOA data
[*] Explanatory notes:
Negative feedback is an engineering term that refers to a reaction that opposes its cause. It is not a value judgement about whether it is good or bad. In fact, negative feedbacks are essential in keeping a system stable. Positive feedbacks lead to instability and are thus generally “bad”.
Convolution is a mathematical process that is used, amongst other things, to implement digital filters. A relaxation response can be regarded as an asymmetric filter with a non-linear phase response. It produces a delay relative to the input and alters it’s shape.
Orthogonal. Variables can be said to be orthogonal in mathematics if they are 90 degrees out of phase with each other, such as is the case for a series and its differential, or more formally, if their inner product is zero.
Regression dilution refers to the reduction in the slope produced by least squares regression when there is significant error or noise in the x-variable. Under negligible x-errors OLS regression can be shown to produce the best unbiased estimation of the slope. However, this essential condition is often over-looked or ignored, leading to erroneously low estimations of the relationship between two quantities.
This is explained in more detail here: