1 Introduction

There are few recent publications about possible UHI influences on global warming. Thus, more up-to-date related studies, including UHI amplification effects that will be discussed in this paper, could offer supporting data for climate change theories and solutions.

One key paper often referred to is by McKitrick and Michael’s [1, 2], who found in 2004 and 2007 using regression trends on socioeconomic, geographical, and temperature indicators that the net warming bias at the global level may explain as much as half the observed land-based warming. Another independent study often quoted by De Laat and Maurellis [3] in 2006 found very similar results. In 2007, IPCC [4] questioned these findings stating the locations of greatest socioeconomic development are also those that have been most warmed by atmospheric circulation changes, which exhibit large-scale coherence.” Therefore, inferring that correlation to warming was not statistically significant but a result of atmospheric oscillations. In 2009, Schmidt [5] agreed and published a paper also suggesting that McKitrick and Michael’s observed correlations were probably spurious. However, in 2010, McKitrick responded with two publications, the first [6] entitled, “Atmospheric Oscillations do not Explain the Temperature-Industrialization Correlation.” The second by McKitrick et al. in 2010 [7] detailed that “evidence for contamination of climatic data is robust across numerous data sets…Consequently, we conclude that important data products used for the analysis of climate change over global land surfaces may be contaminated with socioeconomic patterns related to urbanization and other socioeconomic processes.” In 2013, the IPCC summarized the controversy saying [8], “it is indisputable that UHI and land-use/land-cover are real influences on raw temperature measurements. At question is the extent to which they remain in the global products.” Citations and discussions in the IPCC report suggested the UHI effect would not be more than 10% of observed warming.

However, other authors have also found UHI significance [9,10,11,12,13,14,15,16,17]. For example, Zhao and Huang et al. [14, 16] found that UHIs contributed to warming in China by about 30%. Bian et al. [17] in China at a Shijiazhuang station for periods 1965–2012 found the urban–rural land surface temperatures (LST) trends correlated 100% to urbanization contributions, indicating the yearly increase in annual mean LST at the urban station is entirely caused by urbanization. They concluded the true impact of rising atmospheric CO2 on the global climate may well be vastly overstated. These studies used land-based temperature station data to make assessments. To date, one can conclude that all such studies and findings were not persuasive enough to be influential in the 2015 Paris Climate Accord [18] regarding the need for UHI albedo controls as part of the worldwide effort to mitigate global warming.

This paper provides insight into these controversial findings [1,2,3, 6, 7, 9,10,11,12,13,14,15,16,17] with a WAASU model applied to two time periods, 1950 and 2019. There are currently no papers on the influence of urbanization on climate change using albedo modeling. However, this paper is restricted to UHI and its extent and does not take into account all forms of human land contamination (roads, rural human habitation, deforestation, evapotranspiration loss, anthropogenic heat release, etc.) of which should be roughly correlated in McKitrick and Michael’s socioeconomic - geographical pattern analysis. In this respect, our results are likely conservative.

The WAASU model has advantages as it works from a global view rather than with ground-based studies. There are no concerns about warming oscillations or GHG interference. The model is non-probabilistic and in line with typical energy budgets (IPCC, Hartmann et al. [8]). The model uses only two key parameters: normalized solar effective amplified area and weighted albedo values. Because it is simplistic, it has transparency compared with the complex land-based studies. We also show its utility by extending it to a weighted albedo solar (WAS) model for global warming estimates due to arctic ice melting in Appendix 4.

The contention that UHI effects are primarily of local significance is most likely related to urban area estimates. For example, the IPCC (Satterthwaite et al. [19]) AR5 report references a Schneider et al. [20] study that resulted in urban coverage of 0.148% of the Earth (Table 1). This seemingly small area tends to dismiss the role that the UHI effect can play in large-scale global warming. Furthermore, estimates of how much land has been urbanized vary widely in the literature, in part due to the definition of what is urban and the datasets used. Although such estimates are important for environmental studies, obtaining true estimates for the small urbanized area relative to the total land is very difficult. Compounded by the fact that there is a significant difference in how groups define the term “urban,” Table 1 illustrates several variations from selected papers of interest. Also, global warming UHI amplification effects have not been quantified to a large degree related to area estimates. Urbanized average solar areas remain unknown.

Table 1 Urbanization area extent estimates from various sources

In our study, one key paper listed in Table 1 is that due to Schneider et al. [20] since it is cited by the AR5 2014 IPCC report (Satterthwaite et al. [14]). In Schneider’s paper, the larger area found in the GRUMP [21] study (Table 1) is criticized. Nevertheless, we incorporate the GRUMP area as an upper bound for urbanization. We note that UHI effects have been shown to arise even at very low levels of populations, i.e., towns with fewer than 10,000 people as noted by Karl et al. [25] and Chagnon [26]. The GRUMP study describes datasets with populations greater than 5,000 people while in the Schneider paper population estimates were not included. The GRUMP study combines population statistics and nightlights where the Schneider paper uses a high resolution of illuminated satellite data with decision tree algorithms. The Schneider paper appears to be focused mainly on the “built-up” urbanized area while the GRUMP study “urbanized area” (see the conclusion). Further clarification and guidance is provided in our conclusion where GW estimates are weighted heavily based on Schneider’s value.

Therefore, we use both the Schneider et al. and GRUMP studies for the minimum nominal and maximum worst-case urbanization area estimates, respectively, and provide a weighting method for the final results. Furthermore, these area estimates were done using datasets near the year 2000, a reasonable point in time to extrapolate down to 1950 and up to 2019 (see Sect. 2.5), the two periods of this study.

1.1 UHI amplification effects

Table 2 lists key global warming causes and amplification effects. In general, the complex UHI amplification effects are responsible for the local thermal and related UHI global warming forcing issues. Propagating UHI global warming could further escalate the Earth’s climate feedback response [27,28,29] (see the conclusion). A summary is provided in Appendix 2 of the key UHI effects listed in the table. As well, the conclusions and Appendix 2 includes a discussion on how UHI effects can contribute to climate change issues.

2 Data and methods

The Earth’s solar area has physically grown since 1950 because tall UHI building side area increases. The actual growth in UHI heat intensity though incorporates all solar factors described in Table 2. Besides the tall building solar sides, as shown in the table, many solar effects create a large amplified heat issue. This is a nonlinear problem that could be perhaps impossible to model and is likely best measured instead with what is called the UHI “Footprint” (FP) area, for example. In the discussion below, authors have found that the FP correlates to UHI actual area. Therefore in this section, we expand upon the FP concept. The FP was defined as the continuous extent emanating outward from urban centers to rural areas that have evident UHI effect (i.e., ∆T was statistically larger than zero).

Table 2 Global warming cause and effects

2.1 UHI area amplification effect

We are interested in assessing what we term the UHI complex solar amplification factor. This will only be applied to the UHI component in the WAASU model as an additional weighting factor. To determine this factor, it is logical as we discussed to first look at UHI FP studies as they provide a measure of the UHI amplified heat intensity. Zhang et al. [30] found the ecological FP of the urban land cover extends beyond the perimeter of urban areas, and the FP of urban climates on vegetation phenology they found was 2.4 times the size of the actual urban land cover. In a more recent study by Zhou et al. [31], day-night cycle temperature difference measurements were taken in China. In this study, they found the UHI effect decayed exponentially toward rural areas for the majority of the 32 Chinese cities. Their comprehensive study spanned from 2003 to 2012. They describe China as an ideal area to study since it has experienced the most rapid urbanization in the world in the decade they evaluated. They found that the FP of the UHI effect, including urban areas, was 2.3 and 3.9 times that of urban size for the day and night, respectively. We note that the average day-night amplification footprint coverage factor is 3.1.

To provide some assessment of how the UHI amplification factor scales, we note that Zhou et al. [31] found the FP physical area (km2), correlated tightly and positively with the actual urban area having a correlation coefficient higher than 79% over 32 cities. This correlation suggests that area can be used to provide an initial estimate of this complex amplification factor. Furthermore, the fact that the amplification factor scales with the area are consistent in the calculation of the WAASU model that is weighted by area. This is discussed in Appendix 1 and Sect. 2.5 (Eq. 9).

Therefore, as a model assumption, it is reasonably justified that the amplification factor (AF) should scale with the ratio of areas from 1950 to 2019,

$${\text{AF}}_{{{\text{UHI}}\;{\text{for}}\;2019}} = \frac{{\sum {\left( {{\text{UHI}}\;{\text{Area}}} \right)}_{2019} }}{{\sum {\left( {{\text{UHI}}\;{\text{Area}}} \right)}_{1950} }}$$

Area estimates have been obtained in the next section in Table 3 between 1950 and 2019 time frames, yielding the following results for the Schneider et al. [20] and the GRUMP [21] extrapolated area results:

Table 3 Extrapolated and amplified urbanized coverage estimates
$${\text{AF}}_{{{\text{UHI}}\;{\text{for}}\;2019}} = \frac{{\left( {{\text{Urban}}\,{\text{Size}}} \right)_{2019} }}{{\left( {{\text{Urban}}\,{\text{Size}}} \right)_{1950} }} \approx \left\{ \begin{aligned} \left( {\frac{{\left[ {0.188} \right]_{2019} }}{{\left[ {0.059} \right]_{1950} }}} \right)_{\text{Schneider}} = 3.19 \hfill \\ \left( {\frac{{\left[ {0.952} \right]_{2019} }}{{\left[ {0.316} \right]_{1950} }}} \right)_{\text{GRUMP}} = 3.0 \hfill \\ \end{aligned} \right.$$

From the two studies, area scaling for the UHI solar amplification effect averages 3.1. Coincidently, this factor is the same observed in the Zhou et al. [31] study for the average footprint. This factor may seem high. However, it is likely conservative as other effects would be difficult to assess: increases in global drought due to loss of wetlands, deforestation effects due to urbanization, drought-related fires, and humidity issues. Also difficult to model are factor changes of other impermeable surfaces since 1950, such as city highways, parking lots, event centers, and so forth.

The 3.1 factor is one of the values used to weight the effective UHI area in the WAASU model between 1950 and 2019. It is applied as an UHI effective amplified solar (EAAUHI) area giving more weight to the UHI albedo term. It is initially applied to the UHI area in Table 3 with an example given in Eq. 5. Appendix 1 and Eq. 4 describe the EAAUHI concept.

2.2 Alternate method using the UHI’s dome extent

An alternate approach to check the estimate of Eq. 3 is to look at the UHI’s dome extent. Fan et al. [32] using an energy balance model to obtain the maximum horizontal extent of a UHI heat dome in numerous urban areas found the nighttime extent of 1.5 to 3.5 times the diameter of the city’s urban area (2.5 average) and the daytime value of 2.0 to 3.3 (2.65 average). The horizontal extent of the heat dome is an important parameter for estimating the size of the area it influences and is similar to Zhou et al. [30] footprint.

In the Fan et al. method, the city diameter is multiplied by their derived day (2.65) and night (2.5) factors to obtain the horizontal extent. In our case, we want the diameter change from the area increase in Eq. 2, which is 1.8 \(( = \sqrt {3.1} )\). Therefore, this yields 2.5 × 1.8 = 4.5 higher in the night and 2.65 × 1.8 = 4.8 in the day in 2019 with an average of 4.65. According to Fan et al., this occurs 62.5% of the time. (Their study indicated that transition states are 4 h around sunrise, and about 5 h around sunset, and had less effect, totaling 9 h out of 24.) This yields an effective horizontal extent UHI amplification factor of 2.9. We note this is in good agreement with Zhou et al. footprint and Eq. 2. Fan et al. [32] assessed the heat flux over the urban area extends to its neighboring rural area where the air is transported from the urban heat dome flow. Therefore, the heat dome extends similarly as observed in the footprint studies. If we use the dome concept, we can assume that the actual surface area for the heat flux is increased as the surface area of the dome. This should be considered a measure of the atmospheric UHI vertical and horizontal extents which both are influential in global warming. We do not know the true diameter of the dome, but it is larger than the assessment by Fan et al. Using their dome extend applied to the area diameter D increase from 1950 to 2019, the amplification factor should be correlated to the ratios of the dome spherical surface areas:

$${\text{AF}}_{{{\text{UHI}}\;{\text{for}}\;2019}} = \left( {\frac{{D_{2019} }}{{D_{1950} }}} \right)^{2} = 2.9^{2} = 8.4$$

This value is our second model assumption. Here the ratios of the dome’s surface area are applied as an alternate approach in estimating how the amplification effect scales with UHI growth which provides a measure of vertical and horizontal extent. Therefore, we use both, 3.1 and 8.4, as upper and lower bounds for the solar EAAUHI.

2.3 Applying the amplification factors

In this analysis, 1950 is the reference year. Therefore, it is not subjected to amplification. Only the new UHI solar area is amplified as we are looking at changes since this time frame. The EAAUHI in 2019 (see Sect.2.5) can then be defined as

$${\text{EAA}}_{\text{UHI}} = {\text{AF}}_{\text{UHI}}\times {\text{New}}\;{\text{area}}+{\text{Area}}_{1950} = {\text{AF}}_{\text{UHI}}\times\left( {{\text{Area}}_{2019}-{\text{ Area}}_{1950} } \right)+{\text{Area}}_{1950}$$

Using this, if there were no changes in UHI solar growth, for example, so that the Area2019 = Area1950, the resulting area is just the original Area1950 and if AFUHI = 1, yields the 2019 unamplified area. This result is applied to the new area in Table 3.

2.4 Area extrapolations for 1950 and 2019

To assess the urbanized area, (also used in determining the UHI amplification factor ratios above), we need to project the Schneider [20] and GRUMP [21] area estimates down to 1950 and up to 2019. Both use datasets near 2000, so this is a convenient somewhat middle time frame. Here we decided to use the world population growth rate (World Bank [33]) which varies by year as discussed in Appendix 3 and shown in Fig. 1. We used the average growth rate per ½ decade for iterative projections of about 1.3% (from 2000 to 2019) to 1.8% (from 2000 down to 1955) per year.

To justify this projection, we see that Fig. 2a illustrates that building material aggregates (USGS [34]) as discussed in Appendix 3 used to build cities and roads correlate well to population growth (USGS Population Growth [35]).

It is also interesting to note that building materials for cities and roads also correlate well to global warming trends (NASA [36]) shown in Fig. 2b.

Column 2 in Table 3 shows the projections with the actual year (~ 2000) data point tabulated value also listed in the table (see also Table 1). The UHI area amplification factors (Column 3) are then applied to Schneider [20] and GRUMP [21] studies shown in Column 4 using Eq. 4.

As an example of the EAA calculation in Table 3, using Eq. 4, the 2019 Schneider 3.1 amplification factor is used as follows:

$$\left( {0.188\% - 0.59\% } \right) \times 3.1 + 0.59\% = 0.459\%$$

2.5 Weighted amplification albedo solar urbanization (WAASU) model overview

The WAASU model is very straightforward; the weighted model is rigorously derived in Appendix 1 and is based on a global weighted albedo model. The weighted solar albedo model for 1950 is

$$\alpha_{1950} = \frac{0.33}{{A_{\text{E}} }}\sum {\hat{A}_{i\,} \alpha_{i} } + \frac{0.33}{{A_{\text{E}} }}A_{\text{UHI}} \;\alpha_{\text{UHI}} + \frac{{A_{\text{C}} }}{{A_{\text{E}} }}\alpha_{\text{C}}$$

, and for 2019 the WAASU model is

$$\alpha_{2019} = \frac{0.33}{{A_{\mathrm{E}}^{\prime } }}\sum {\hat{A}_{i\,} \alpha_{i} } + \frac{0.33}{{A_{\mathrm{E}}^{\prime } }}A_{\mathrm{UHI}} {\mathrm{AF}}_{\mathrm{UHI}} \,\alpha_{\mathrm{UHI}} + \frac{{A_{\mathrm{C}} }}{{A_{\mathrm{E}}^{\prime } }}\alpha_{\mathrm{C}}$$

Here α is the Earth’s Albedo, αi is the albedo of each Earth component with the associated surface area \(\hat{A}{}_{i}\) (the hat indicating all areas excluding the UHI area), similarly αUHI is the UHI albedo associated with its area AUHI, AF is the UHI amplification factor (Sects. 2.1 and 2.2), and AC is the cloud coverage area with average cloud albedo αC (Appendix 5). As explained in Appendix 1, the 0.33 factor arises from the fact that 67% of the Earth is approximately covered by clouds [37].

As well, AE Earth’s surface area in 1950 and AE′ is the Earth’s area in 2019 due to the EAAUHI effective solar area increase, given by

$$A_{\text{E}}^{\prime } = \hat{A}_{\text{E}} + {\text{EAA}}_{\text{UHI}}$$

Here EAA is defined in Eq. 4. Therefore, this increase requires renormalization that is discussed in Sect. 2.5.1. For example, if water covers 56% of the Earth, now it will be slightly less since the Earth’s solar area has increased due to the buildup of cities since 1950 from the number of tall buildings that have increased the Earth’s solar surface area along with other UHI amplification effects. This is captured in the solar effective amplified area.

It is important to note in the WAASU model (Eq. 7) that AF is combined with the UHI area and its albedo value

$$\left( {{\text{A}}_{\text{UHI}} } \right) \, \left( {{\text{AF}}_{\text{UHI}} } \right) \, \left( {\alpha_{\text{UHI}} } \right)$$

This shows the combined effect of the factor in the model and its possible influence on each factor. However, an assumption of the model is αUHI = 0.12 and stays generally constant from 1950 to 2019. Average UHI albedo does not appear to vary much over time in the literature [38]. Therefore, consistent with Eq. 9 we find the amplification effect is mainly related to area growth as described in Sect. 2.1. This allows us to use the term as an effective amplified area (EAA) for the part AUHI × AFUHI.

Note that all the effective surface areas are influenced by the solar irradiance

$${\text{Effective}}\,{\text{Surface}}\;{\text{Area}} = {\text{Surface}}\,{\text{Area}}\, \times \% {\text{Solar}}\,{\text{Irradiance}}.$$

where the surface area includes all areas including EAA. However, we note that the change in the Earth Albedo over time (from 1950 to 2019) is just a function of the UHI area variation, (when holding all unrelated UHI components constant), that is

$$\left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{{{\text{EA}}^{\prime } }} \approx \sum {\left( {{\text{Albedo}}_{\text{UHI}} \, \times \,\% {\text{Solar}}\,{\text{Irradiance}}\, \times \,\frac{{{\text{d}}\,{\text{Surface}}\;{\text{Area}}_{\text{UHI}} }}{{{\text{d}}t}}} \right)}_{i} ,$$

Here EA′ is all other Earth components (held constant). That is the main effect is the UHI surface area change from 1950 to 2019, the albedo and solar irradiance are considered constant.

2.5.1 Model constraints

Because of Eq. 8, this model is subject to the constraint

$${\text{Total}}\,{\text{Area}} = \sum\nolimits_{i} {\{ \% \,{\text{Normalized}}\,{\text{Effective}}\,{\text{Amplified}}\,{\text{Surface}}\;{\text{Areas}}_{i} \} } + \% {\text{Cloud}}\,{\text{Area}}\, = 100\%$$

the small change in area EAAUHI will increase AE slightly as described by Eq. 8. This requires renormalization to meet the requirements of Eq. 12. All areas change slightly including EAAUHI. The UHI change is termed the normalization effective amplified area (NEAA). A full renormalization example is provided in Appendix 6.

To simplify things as much as possible, only five Earth constituents are used: water, sea ice, land, UHI coverage, and clouds (where land is its area minus the UHI coverage). These components are fairly easy to estimate, and references for their values are provided in Appendix 5. Furthermore, we use consistent values found in the IPCC AR5 report (Hartmann et al. [8]) assessment of the Earth’s energy budget for solar irradiance. Table 4 summarizes the constraints from these IPCC values.

Table 4 IPCC Earth energy budget values (Hartmann et al. [8])

The fixed components of our model maintain relative consistency from 1950 to 2019. The non-fixed value is the urban coverage as indicated by Eq. 11. The only unknown value is the land albedo (minus the UHI coverage), and this value is adjusted to obtain the IPCC global albedo, 29.412%, and its Earth surface value of incident/reflected value of 7.059 (see Table 5a).

These values are used as a 1950 starting point, and then, the 2019 increase for the UHI coverage area is inserted. This increases the Earth’s area to greater than 100%. Therefore, renormalization is done per the constraint of Eq. 12. Renormalization is detailed in Appendix 6.

3 Results

Using the extrapolated area coverage in Table 3 with the 3.1 amplification factor applied to the urbanized growth, the resulting global albedo change occurred of 29.399% in 2019 (Table 5b) compared to the earlier 1950 albedo value of 29.412% (Table 5a) for the Schneider nominal case. As well, for the GRUMP worst-case, the albedo changed from 29.412% (Table 6a) to 29.352% (Table 6b) due to the urbanized growth. Dome global albedo values are also provided in Appendix 6.

Table 5 (a) Schneider 1950 effective estimate. (b) Schneider 2019 effective estimate (AF = 3.1)
Table 6 (a) GRUMP 1950 effective estimate. (b) Grump 2019 effective estimate (AF = 3.1)

As we mentioned earlier, the increases in the solar surface area of the Earth, which will occur with city growth of tall buildings and their solar areas, however comparatively small, require renormalization of the Earth’s surface components in the WAASU model (detailed in Appendix 6). This information is displayed in Column 3 in Tables 5b and 6b. While the model is sensitive to urban coverage changes, it works well with renormalization showing a high level of consistency to urban coverage proportionality changes. This consistency is indicated in Table 7 where we find the GRUMP and Schneider long wavelength radiation (LWR) forcing per %EAA averages about 0.096% (W/m2)/%NEAA in the last column.

Table 7 Albedo and radiative increase model results with UHI effective area

Table 7 provides a summary of albedo changes found in the WASSU model along with the expected solar longwave radiation increase. From the above global WAASU model, the estimates of the Earth’s LWR emissions are obtained from the fundamental expression

$$P_{\alpha } = {\mathbf{340}}\;{\text{W}}/{\text{m}}^{ 2} \left( { 1- {\text{Albedo}}} \right).$$

Then, the albedo change from 1950 to 2019 represents the equivalent increase in LWR is given by

$$\Delta P_{\alpha } = {\mathbf{340}}\;{\text{W}}/{\text{m}}^{ 2} \left\{ {\left( { 1- {\text{Albedo}}} \right)_{ 20 1 9} - \left( { 1- {\text{Albedo}}} \right)_{ 1 9 50} } \right\}.$$

The results are compiled in Table 7. The table also includes “What if” estimates, if we could change urbanization to be more reflective with cool roofs to reverse the effect.

The overall results are summarized:

  • Schneider nominal case from 1950 to 2019, the increase in LWR forcing (Row 7) is 0.042 W/m2 and 0.11 W/m2 due to urban area and dome amplification coverage, respectively. These values do not include the addition of GHG re-radiation (see Table 8).

  • GRUMP worst-case from 1950 to 2019 the increase in LWR (Row 7) is 0.204 W/m2 and 0.537 W/m2 due to urban area and dome amplification coverage, respectively. These values do not include the addition of GHG re-radiation (see Table 8).

  • The forcing per unit %NEAA or %EAA has consistency with small variability and averaging about 0.096 W/m2/%NEAA. We also note in Column 8 the consistent value of 1.0 W/m2/%Δalbedo. This is the percent change from the initial albedo value of 29.413%. This value is a useful constant and can be derived [39]. Note these values do not include GHG re-radiation (see Sect. 4).

  • “What if” corrective action results of cool roofs indicate that changing city albedos in both the Schneider and the GRUMP case from 0.12 to an average value of 0.205 would reverse the increase forcing back to 1950 levels. By comparison, He et al. [40] found the average albedo varies from 0.1 to 0.4, averaging 0.25. Note our model found the average land albedo slightly higher at 0.31 (Tables 5 and 6).

4 Discussion on the relative contribution to global warming forcing due to UHIs

In this section, the LWR results in Table 7 are adjusted by including GHG re-radiation forcing that will additionally occur. As well, the total global warming forcing contributions are described.

4.1 Full UHI radiation forcing and associated temperature rise

Estimates in Table 7 provide the LWR forcing, but the anticipated average GHG additional re-radiation forcing increase expected is not included. This average re-radiation GHG factor is roughly estimated as 1.62 [39] (this 62% factor is approximately equal to β4, the effective emissivity of the planetary system) and is exemplified in Table 4. Table 8, Column 4 provides the forcing when the 1.62 factor GHG re-radiation is included and Column 5 shows the associated temperature increase. Appendix 7 provides an overview with a detailed estimate of the forcing and temperature rise assessments found in Table 8.

4.2 IPCC/NOAA radiation forcing comparison

To make relative comparisons with UHI forcing, we compare the forcing results in Table 8 to the IPCC estimate for GHG forcing from the period 1950 to 2019, and GHG warming associated temperature rise. The GHG forcing estimate by IPCC/NOAA [41] is 2.38 W/m2 during this period.

One should note that this value does not include “feedback” (i.e., arctic snow and ice melting) discussed in our conclusions. Column 6 in Table 8 shows the relative forcing ratio to compare it to the UHI strength. For example, the LWR found in the Schneider case for the albedo of 29.3994 was 0.044 W/m2 in Table 7. Then, we estimate with GHG re-radiation as 0.044 W/m2 × 1.62 = 0.071 W/m2 in Column 4 and relative to the IPCC GHG forcing estimate is about 3% (= 0.071/2.38) in Column 6, Table 8. One can also obtain the same percentages in Column 6 by dividing the temperature increase in Column 5 by 0.44 °C. Here 0.44 °C is the temperature rise one obtains from IPCC/NOAA 2.38 W/m2 of forcing without feedback (see Appendix 7) which is a little less than half of the total warming observed since 1950. Note that only Column 6 uses IPCC/NOAA estimates in our results. In the conclusion, we discuss another method using a feedback approach that increases these estimates somewhat as shown in the last column.

Table 8 WAASU Model full forcing and global warming estimate due to UHI in 2019

Finally, the forcing estimate in Column 8, Table 7 is updated in Table 8 from Column 4 divided by Column 3 as

$$\alpha_{{{\text{Global}}\_{\text{Forcing}}}} = 1.62 \times 1.0\;{\text{W}}/{\text{m}}^{2} /\% \Delta {\text{Albedo}} = 1.62\,{\text{W}}/{\text{m}}^{2} /\% \Delta {\text{Albedo}}$$

and from Table 8, Column 4 divided by Column 1, the consistent forcing per %EAA estimate is

$${\text{UHI}}_{{{\text{EAA}}\_{\text{Forcing}}}} \approx 1.62\, \times \,0.096\,{\text{W}}/{\text{m}}^{2} /\% {\text{EAA}} = 0.16\,{\text{W}}/{\text{m}}^{2} /\% {\text{EAA}}$$

where the %EAA is given by Eq. 4 and exemplified in Eq. 5. Examples of how these might be used are provided in Appendix 7. Lastly, as a check, one may note that UHI global warming estimates roughly scales with UHI size as might be expected. For example, in Table 1 the ratio of Schneider to Grump UHI area extent is 2.7/0.51 = 5.3. We note the values in the last column in Table 8 scale close to this factor (i.e., 3% × 5.3 = 15.9% which is close to 13.8% and 7.6% × 5.3 = 40% close to 36%) between Schneider and GRUMP, respectively. Inexact scaling is due to Eq. 4.

5 Conclusions

In this paper, we derived a versatile WAASU model and applied it to provide estimates of the UHI effect (with urban areas) on global warming. This calculation was done with the aid of assumptions for UHI solar amplification factors. These estimates inserted into our WAASU model found that between 0.071 and 0.87 W/m2 of radiative forcing (Table 8) may be possible. This forcing result indicates that about 3% to 36% of global warming may be due to the UHI effect by comparisons to anticipated IPCC/NOAA GHG forcing values with median and mean GW relative percentages of 10.7% to 15.2%, respectively. In this conclusion, we provide an alternate method that includes feedback and which finds slightly larger but comparable warming estimates due to UHIs. This method includes ice loss results in Appendix 4, which found warming feedback using a related WAS model yielded a 0.15 °C rise. This ice loss feedback represents about 16% of GW in 2019 relative to the 0.95 °C estimated increase since 1950.

The wide variations we found on forcing values are due to both the amplification and urban area uncertainties which we now provide guidance for in this conclusion. However, the model found that the forcing per effective amplified UHI area and albedo estimates were consistent showing 0.16 W/m2/%EAA and 1.62 W/m2/%Δalbedo, respectively (see Eqs. 15 and 16). Note that if better estimates are known for the %EAA, (see Eq. 4), then one can quickly assess the impact of the UHI GW effect using the 0.16 W/m2/%EAA estimate.

The WAASU model is versatile. We can quickly look at UHI albedo changes required to offset the estimated forcing. For example, “What if” corrective action results of cool roofs indicated that an average UHI albedos change from 0.12 to 0.21 would reverse the UHI forcing back to 1950 levels. This value was found to be close to the average global land surface albedo of 0.25 [40]. This suggests that the cooling potential of UHIs is very high. For example, if cool roofs and other worldwide changes can be made to raise the UHI albedo to 0.48, (fourfold higher in reflectivity), then this reverse forcing could likely reduce global warming by about 30% or more (estimated from median albedo values).

Therefore, the model can provide albedo-area estimates for reverse forcing similar to the “What if” corrective actions for mitigation/adaptation strategies. As a follow-up study, the author has proposed a similar modeling strategy to estimate select areas changes necessary for surface albedo type global warming solutions [39].

To provide an alternate estimate of the influence that UHIs play in global warming, consider the following temperature breakdown:

$$\Delta T = T_{\text{GHG}} + T_{\text{U - A}} + T_{i} + T(\lambda_{\text{WV}} ,\lambda_{\text{I}} ) + T(\lambda_{\text{A}} ,\lambda_{\text{LR}} ,\lambda_{\text{C}} ,\lambda_{\text{U}} )\;{\text{and}}\;T_{\text{U - A}} = T_{\text{UHI}} + T_{\text{O - U}}$$

Here ΔT = 0.95 °C is the temperature rise from 1950 to 2019, with contributions from GHG (TGHG), urbanization-albedo land-use/land-cover issues (TU-A), UHIs (TUHI), other urbanization-albedo land-use/land-cover effects (rural roads, rooftops, etc.) (TO-U), and other possible smaller temperature rise effects (aerosols, soot on snow, etc.) (Ti). These are all temperature rises related to direct forcing.

T(λ) represents warming due to feedbacks. They are a function of feedbacks responses (water vapor (WV), (I) ice loss, other albedo (A) issues, lapse rate (LR), clouds (C), and (U) urbanization related feedback).

In an article by Liu et al. [42], urbanization area is categorized in an attempt to clarify uncertainties due to the various definitions of what is urban and the wide assessments among authors on how much of the Earth has been urbanized. According to Liu et al., the GRUMP area estimate fits the definition of “urban area,” where the Schneider estimate fits the definition of “built-up area”. Therefore, considering their recommendation, we favor the dome amplification estimate for Schneider as it provides a measure of horizontal and vertical warming area extent from land-use/land-cover albedo UHI effects. Although TA-U is dominated by TUHI, the GRUMP footprint provides a horizontal extent that can be used as a weighting factor in estimation to account for TO_U as it incorporates a larger area. Therefore, this leads to our recommendation for using the median value in Table 8 in Eq. 17 for TU-A.

Note that the median TU-A value is weighted 30% (0.0475 °C/0.16 °C) GRUMP and 70% Schneider (where 0.16 °C is the upper GRUMP estimate in Table 8). Then from rows 4 and 5 in Table 8, we recommended the following:

$$T_{\text{U - A}} = \left\{ {0.0475\,^\circ {\text{C}}\;({\text{Median}}\;{\text{value}})} \right.\;{\text{and}}\;F_{\text{U - A}} = \left\{ {0.256\;{\text{W}}/{\text{m}}^{2} \;({\text{Median}}\;{\text{value}})} \right.$$

Then let

$$T(\lambda_{\text{I}} ,\lambda_{\text{WV}} ) \approx 0.15\;^\circ {\text{C}} + \left[ {(T_{\text{GHG}} + T_{\text{U - A}} )} \right]_{\text{WV}} \;{\text{and}}\;T_{i} + T(\lambda_{\text{A}} ,\lambda_{\text{LR}} ,\lambda_{\text{C}} ,\lambda_{\text{U}} ) \ge 0$$

The inclusion of the TA-U term provides a measurable way to take into account this work. This result can be compared to other authors who have contributed to the understanding of global warming influences from UHIs [1,2,3, 6, 7, 9,10,11,12,13,14,15,16,17]. Equation 19 is based on the results of Appendix 4 for ice loss (0.15 °C). The water vapor feedback factor in Eq. 19 of one-time forcing estimates is based on the findings of other authors [27,28,29].

The feedback term, λU, is difficult to quantify with high confidence, but it is important to account for UHI effects. Here, we point out several key UHI feedbacks effects that have been studied and play a role in global climate change (see also Appendix 2):

  1. 1.

    In a study of wetland reduction in China and its correlation to drought, Cao et al. [43] looked at the wetland distributions and areas for five provinces due to urbanization. These areas showed a total reduction in southwestern China from 1970 to 2008 of 17% ground area, with the highest reduction rate occurring from 2000 to 2008. They found these changes to the wetland area showed a negative correlation with temperature (i.e., wetland decrease, increase in temperature), and a positive correlation with precipitation (i.e., wetland decrease, precipitation decrease). One can conclude that albedo management of urbanization would help increase the loss in condensation. Although some cities find increases in precipitation due to complex warming turbulence, the larger picture indicates that UHIs are a cause of drought. Of course, drought is also the result of other global warming forcing issues in Eq. 17.

  2. 2.

    Drought feedback leads to forest fire feedbacks that not only damage forests that would otherwise remove CO2 from the air, but that also releases CO2 and other GHGs into the atmosphere. Therefore, this is a major offset in CO2 worldwide reduction efforts. This suggests the urgent need for supplementary albedo reverse forcing efforts. Albedo reverse forcing efforts can also be offset by a lack of albedo controls in urbanization development. Therefore, if progress is to be made in climate change mitigation, it is imperative to work on both CO2 and albedo UHI management.

  3. 3.

    Zhao et al. [44] observed that UHI temperatures increase in daytime ΔT by 3.0 °C in humid climates but decrease ΔT by 1.5 °C in dry climates. They found a strong correlation between ΔT increase and daytime precipitation. Their results concluded that albedo management would be a viable means of reducing ΔT on large scales.

  4. 4.

    This effect is often attributed to greenspace decrease of surface roughness due to UHI impermeable smooth surfaces which reduces convection cooling efficiency (Zhao et al. [44], Gunawardena et al. [45])

  5. 5.

    In general, UHIs lessens the possibilities for the evapotranspiration process and thereby reduces the natural cooling effect since vegetation is scarce. UHI cause higher rates of movement of water from the soil, plants, and pavement precipitation evaporation into the atmosphere. Since hotter air can hold more water, it exacerbates dryness. This can promote a local GHG effect and be partly responsible for the observed warming. These effects may to a lesser extend occur on all smooth hot evaporating surfaces (during precipitation periods) including roads and highways.

  6. 6.

    Another problem is due to the large heat capacity of cities that increase the length of warming time after sunset. Nighttime warming creates longer dry periods and also contributes to drought conditions and the potential for forest fires.

  7. 7.

    Lastly, any additional global warming due to TA-U can contribute to other positive feedbacks in Eq. 17 as indicated by Eq. 19.

  8. 8.

    The primary mitigating factor in all these cases would be albedo UHI management of impermeable surfaces.

We now refine our GW UHI estimate by including a feedback method with Eq. 19. We first consider the quantity \(T_{i} + T(\lambda_{\text{A}} ,\lambda_{\text{LR}} ,\lambda_{\text{C}} ,\lambda_{\text{U}} )\) to be small (as we do not have knowledgeable estimates) and approximated it as zero to simplify. Then solving Eqs. 1719, with ΔT = 0.95 °C, the following results are obtained

$$T_{\text{U - A}} + T_{\text{GHG}} = 0.0475\,^\circ {\text{C}} + 0.3525\,^\circ C = 0.4\,^\circ {\text{C}}\;({\text{Median}}\,{\text{value}})$$

This gives the following global warming root cause estimates due to urbanization for TU-A as

$${\text{GW}}\% (T_{\text{U - A}} ) = {{0.0475\,^\circ {\text{C}}} \mathord{\left/ {\vphantom {{0.0475\,^\circ {\text{C}}} {0.4\,^\circ {\text{C}}}}} \right. \kern-0pt} {0.4\,^\circ {\text{C}}}} = 11.9\% \;({\text{Median}}\,{\text{value}})$$

Other values are provided in Table 8, Col. 7. We have now provided two different types of assessments; the IPCC/NOAA and feedback methods that have yielded similar median assessments of 10.7% and 11.9%, respectively. These two estimates are in reasonable agreement and provide an average of 11.3%. This agreement also suggests that our ice loss temperature feedback increase estimate of 0.15 °C in Appendix 4 (using the WAS model) is reasonable. Equation 21 suggests that a little forcing (0.4 °C) can create a lot of feedback and is a primary problem in global warming. In the introduction, it was noted that Zhao and Huang et al. [14, 16] found that about 30% of GW could be due to UHIs. Therefore, our median estimate is a little more than one-third in comparison providing some support. (Full support would require upper bound estimates in Table 8.) It should also be noted in the WAASU model, we were very conservative with cloud coverage, allowing only 33% of solar radiation to reach the Earth through clouds (see Appendix 1), as some attenuated sunlight normally radiates through. The results actually ratios with cloud coverage. For example, if the cloud/atmosphere coverage used was less conservative and closer to an IPCC estimate of 47% (Table 4, 161/340=0.47), then the increase in global warming due mainly by UHIs would go from 11% to about 16% (16%=0.47/0.33x11%).On the other hand, feedbacks are difficult to quantify, and we noted in Appendix 4 that our temperature rise due to ice loss is a rough estimate due to numerous uncertainties encountered by climatologists in fully quantifying the seasonal variations in ice changes. Furthermore, water vapor feedback estimates may be lower than the factor of one-time forcing suggested by some authors [27,28,29].

We conclude that about 11%–16% of global warming is due primarily to the UHI effect based on our median estimates in Table 8. This estimate would likely increase if other feedbacks were included. However, this provides strong engineering judgment that land-use/land-cover albedo effects (i.e., UHI and land-use changes) are responsible for an important portion of global warming and that albedo management of urbanization is an urgent matter. Left unattended along with urbanization expected growth, the influence of UHIs will continue to offset CO2 reduction efforts and increasingly contribute to warming and feedback issues. Furthermore, proper worldwide albedo urbanization management would lead to a substantial reversal of global warming trends [39].

Below, we provide suggestions and corrective actions, which include:

  • Modification of the Paris Climate Agreement to include albedo controls and solutions

  • Albedo guidelines for UHI impermeable surfaces, cool roofs, and roads similar to ongoing CO2 efforts

  • UHI albedo goals: we suggest an albedo increase by a factor of 4 (from 0.12 to 0.48), which could reduce GW by about 30% or more, assessed from our median ranges and other studies [39, 46, 47].

  • Government funding for geoengineering and implementation of albedo solutions

  • Centralize albedo solution efforts in a single government agency (possibly NASA)

  • Guidelines for future albedo design considerations of urbanization areas

  • Further development of white solar cells and their use for cooler panels

  • Requires cars to be more reflective. Although worldwide vehicles do not comprise much of the Earth’s solar area, recommending the preferential manufacturing of cars that are higher in reflectivity (e.g., silver or white) would raise awareness of this issue similar to electric automobiles that help improve CO2 emissions.