
755 N.W.2d 420 (2008)
16 Neb. App. 905
Wendy Jo DREW, on behalf of Madison Joy REED et al., minor children, appellee and cross-appellant,
v.
Patrick William REED, appellant and cross-appellee.
No. A-07-251.
Court of Appeals of Nebraska.
August 12, 2008.
*423 Mark J. Krieger, of Bowman & Krieger, Lincoln, for appellant.
Charles W. Campbell, of Angle, Murphy, Valentino & Campbell, P.C., York, for appellee.
INBODY, Chief Judge, and SIEVERS and CARLSON, Judges.
SIEVERS, Judge.
Patrick William Reed (Patrick) appeals and Wendy Jo Drew (Wendy) cross-appeals from the decision of the district court for Seward County that awarded sole custody of the parties' minor children to Wendy but ordered Patrick to pay child support based on a joint custody calculation, ordered Patrick to pay $21,658 in retroactive child support, ordered Patrick to pay $13,916.31 toward Wendy's attorney fees, and ordered Patrick to pay $27,592 in expert witness fees.

FACTUAL AND PROCEDURAL BACKGROUND
Though never married, Patrick and Wendy engaged in a long-term relationship that produced three children: Madison Joy Reed, born February 8, 1998; Jack William Patrick Reed, born April 3, 2000; and Claire Estelle Reed, born February 12, 2003. Patrick and Wendy ended their relationship in February 2004.
On August 9, 2004, Wendy filed a petition on behalf of Madison, Jack, and Claire to establish paternity and child support. On August 27, the parties filed a stipulation and agreement for a temporary order establishing that Wendy would have custody of the children, subject to Patrick's reasonable rights of visitation, and that Patrick would pay $2,700 per month in child support beginning September 1. The district court approved the parties' stipulation and entered the temporary order.
Trial was held on August 10 and 11, 2006. The court's opinion and judgment was filed on December 8. The district court adjudged that Patrick is the father of the three minor children. The district court incorporated the parties' agreement *424 on parenting time into its judgment, thereby establishing that Wendy would "have sole legal and physical custody" of the parties' three minor children. The judgment provided that Patrick would have parenting time (1) on alternating weekends; (2) every Wednesday from 5:30 p.m. until 8 a.m. Thursday; (3) during the weeks immediately preceding each weekend that he does not have parenting time, for a period commencing at 8 a.m. on Thursday and concluding at 8 a.m. on Friday; (4) every school spring break, without interfering with Wendy's Easter parenting time; (5) two 2-week periods in the summer; and (6) alternating holidays.
The district court stated that even though Wendy was awarded sole legal and physical custody, based on the parties' parenting time agreement, Patrick will have the children 43 percent of every year, which "comes close enough to a factual joint custody" for purposes of setting child support. Thus, the district court used a joint child support calculation and ordered Patrick to pay child support in the amount of $3,337 per month for the three children beginning January 1, 2007. Wendy filed a motion to alter or amend judgment regarding the child support calculation, which was denied by the district court. However, in an addendum to its December 8, 2006, judgment, the district court ordered Patrick to pay $21,658 in retroactive child support. Additionally, taking into account both the December 8 judgment by the district court and the court's later addendum, Patrick was ordered to pay $13,916.31 toward Wendy's attorney fees and $27,592 in expert witness fees. Patrick appeals and Wendy cross-appeals the district court's judgment.

ASSIGNMENTS OF ERROR
Patrick alleges that the district court erred when it ordered him to pay an additional $13,916.31 in attorney fees, $27,592 in expert witness fees, and $21,658 in retroactive child support.
On cross-appeal, Wendy alleges that the district court erred in (1) applying a joint custody calculation for child support, (2) calculating child support using worksheet 3 of the child support guidelines, (3) failing to award child support as determined by worksheet 1, and (4) failing to award retroactive child support based on the calculation of child support pursuant to worksheet 1.

STANDARD OF REVIEW
[1, 2] While a paternity action is one at law, the award of child support in such an action is equitable in nature. State on behalf of Kayla T. v. Risinger, 273 Neb. 694, 731 N.W.2d 892 (2007). A trial court's award of child support in a paternity case will not be disturbed on appeal in the absence of an abuse of discretion by the trial court. Id.
[3] In a paternity action, attorney fees are reviewed de novo on the record to determine whether there has been an abuse of discretion by the trial judge. Henke v. Guerrero, 13 Neb.App. 337, 692 N.W.2d 762 (2005). Absent such an abuse, the award will be affirmed. Id.
[4] In a dissolution action, an appellate court reviews an award of expert witness fees de novo on the record to determine whether there has been an abuse of discretion by the trial judge. See Davidson v. Davidson, 254 Neb. 656, 578 N.W.2d 848 (1998). We find that the same standard applies to an award of expert witness fees in a paternity action.

ANALYSIS

Child Support Calculation.
Wendy argues that the district court erred in applying a joint custody calculation *425 to determine child support. She argues that the district court should have used worksheet 1 to determine the child support award.
The version of the child support guidelines in effect at the time of the trial and judgment in this case stated:
C. Rebuttable Presumption. The child support guidelines shall be applied as a rebuttable presumption. All orders for child support obligations shall be established in accordance with the provisions of the guidelines unless the court finds that one or both parties have produced sufficient evidence to rebut the presumption that the guidelines should be applied.... In the event of a deviation, the reason for the deviation shall be contained in the findings portion of the decree or order, or worksheet 5 should be completed by the court and filed in the court file. Deviations from the guidelines are permissible under the following circumstances:
....
3. if total net income exceeds $10,000 monthly, child support for amounts in excess of $10,000 monthly may be more but shall not be less than the amount which would be computed using the $10,000 monthly income unless other permissible deviations exist[.]
The version of the child support guidelines in effect at the time of the trial and judgment in this case further stated: "L. Joint Physical Custody. When a specific provision for joint physical custody is ordered, support may be calculated using worksheet 3." In its order, the district court specifically stated: "The parties agreed, and I accepted their agreement, that sole legal and physical custody would be placed with Wendy." However, the district court also stated that based on the parties' parenting-time agreement, Patrick will have the children 43 percent of every year, which "comes close enough to a factual joint custody" for purposes of child support.
Using its finding of "factual joint custody," the district court did a joint custody child support calculation citing Elsome v. Elsome, 257 Neb. 889, 601 N.W.2d 537 (1999). Elsome holds that if the evidence establishes a joint physical custody arrangement, courts will so construe it, regardless of how prior decrees or court orders have characterized the arrangement. According to the trial court in this case, the threshold time division is 40 percent, although the Elsome opinion did not explicitly adopt such as a determinative percentage.
[5, 6] In our view, Elsome is distinguishable from the present case. In Elsome, the decree provided that the parties would have "joint legal custody" of their children, but neither party was designated primary physical custodian of the children. The parties had a detailed shared physical custody arrangement based upon 14-day cycles, which generally provided that the children spend 4 days every week with the mother and the following 3 days every week with the father. Thus, in Elsome, the parties had a true joint physical custody arrangement, regardless of whether it was labeled as such. The Elsome opinion favorably cites Pascale v. Pascale, 140 N.J. 583, 660 A.2d 485 (1995), where the New Jersey court said that although there is no established norm for joint physical custody, experts cite schedules within a joint physical custody framework as spending 3 entire days with one parent and 4 entire days with another parent or alternating weeks or even years with each parent. Thus, joint physical custody means the child lives day in and day out with both parents on a rotating basis. Pascale, supra. Numerous "`parenting times'" with a child do not constitute joint physical custody. Pascale, 140 N.J. at 597, 660 *426 A.2d at 492. See, also, Heesacker v. Heesacker, 262 Neb. 179, 629 N.W.2d 558 (2001) (recognizing Pascale, supra).
Alternately living with divorced parents is to be distinguished from cases in which the noncustodial parent has liberal parenting time. In Pool v. Pool, 9 Neb.App. 453, 613 N.W.2d 819 (2000), the father had custody of his children every other weekend, plus an additional weekend day per month; weekday visitation two times a week from 4 to 8 p.m.; visitation on alternating holidays; and extended summer visitation continuously from June 1 to July 31 each year. The arrangements for parenting time for the father in Pool are nearly identical to those involved in the present case. The district court in Pool found that the father had physical custody for 33 percent of the time and determined his support obligation based on the joint physical custody worksheet. On appeal, the mother argued that the court should have used the sole custody worksheet, rather than the joint custody worksheet. This court reversed the decision of the trial court, concluding that the mother was the primary caregiver, that she had sole physical custody of the children, and that the father had been granted "`typical' weekend, holiday, and summer visitation rights." Id. at 458, 613 N.W.2d at 824. Thus, we found that the trial court abused its discretion in basing the child support calculation on the joint custody worksheet, rather than the sole custody worksheet.
Another "liberal parenting time" case is Heesacker, supra, which cited with approval our decision in Pool, supra. In Heesacker, through mediation, the parents agreed to joint custody. The father was to have custody of the parties' child on alternating weekends from Friday after school until Monday morning and from Wednesday after school until Thursday morning. In addition, the father was to have custody an additional Friday night each month and 1 other day each month to be agreed upon by the parties. The visitation plan also called for each parent to have custody of the child for one-half of her summer and winter breaks from school, with holidays and birthdays split on an alternating yearly schedule. The trial court found that the time division was split 65 percent with the mother and 35 percent with the father, and the court used the sole custody worksheet to determine the father's child support obligation.
On appeal, in Heesacker, supra, the father argued that the trial court erred in using a sole custody worksheet, rather than a joint custody worksheet. The Nebraska Supreme Court concluded that the visitation plan adopted by the trial court was "a grant of physical custody to [the mother], with a liberal visitation schedule for [the father]." Heesacker v. Heesacker, 262 Neb. 179, 185, 629 N.W.2d 558, 562 (2001). Accordingly, the Supreme Court determined that the trial court was correct in calculating the father's child support based on the sole custody worksheet.
[7, 8] The present case is more in line with Heesacker, supra, and Pool v. Pool, 9 Neb.App. 453, 613 N.W.2d 819 (2000), where there was clearly liberal parenting time, but such did not justify a joint custody child support calculation. Patrick's children do not live with him day in and day out on a rotating or alternating basis. And while he has extensive and varied parenting times, it is best described as liberal visitation as in Heesacker, supra, and Pool, supra. Thus, the trial court abused its discretion in basing the child support calculation on the joint custody worksheet, rather than the sole custody worksheet which we shall use in making the necessary adjustments to Patrick's child support obligation.


*427 Calculation of Child Support When Income Is Above Guidelines.

[9] Patrick's financial situation is complex. He wholly owns several corporations and is part owner in other corporations that are involved in the cellular telephone industry. Two different accountants did indepth analyses of Patrick's finances and testified regarding his income. We digress to point out that his reply brief attempts to challenge the district court's findings regarding his income. However, errors not assigned in an appellant's initial brief are waived and may not be asserted for the first time in a reply brief. Genetti v. Caterpillar, Inc., 261 Neb. 98, 621 N.W.2d 529 (2001). Because Patrick did not contest the trial court's conclusions regarding his income in his original brief, we need not further detail his finances.
In contrast to the complexities of Patrick's finances, Wendy's financial situation is quite simple. She has been a stay-at-home mother. However, she agreed that she has an earning capacity of $11 per hour, and the district court accepted such. Wendy does not dispute the trial court's conclusion regarding her income capacity.
[10, 11] Thus, we use the district court's findings that Wendy has a gross monthly earning capacity of $1,907 and Patrick has a gross monthly income of $49,195. The district court determined that based on their gross monthly incomes, Wendy's net monthly income is $1,754.03 and Patrick's net monthly income is $29,436.68. Again, neither party challenges this finding. Thus, the parties' combined net monthly income is $31,190.71, making this an "above guidelines" case. The version of the child support guidelines in effect at the time of trial and judgment in this case permitted a deviation from the child support guidelines when the total net income exceeded $10,000 monthly, and given that Patrick's net income was three times that provided for in the guidelines, deviation was clearly justified.
[12] The obvious difficulty in an above guidelines case is determining what deviation from the guidelines is appropriate, and it must be recognized that trial courts have considerable discretion in this regard. In this case, the guidelines in effect at the time of this trial only account for approximately one-third of Patrick's net income. In short, the guidelines do not tell us how to handle child support from the remaining two-thirds of Patrick's net income. Thus, of necessity, the determination of support over the guidelines amount is largely a matter of the trial court's discretion, guided by the evidence before the trial court.
[13] The version of the child support guidelines in effect at the time of trial and judgment in this case simply told us that "child support for amounts in excess of $10,000 may be more but shall not be less than the amount which would be computed using the $10,000 monthly income unless other permissible deviations exist." The district court heard testimony from two accountants and an economics professor on how child support for monthly incomes above $10,000 can be extrapolated from the child support chart for monthly incomes below $10,000. Each witness used a different mathematical approach to, in effect, expand the child support chart beyond a monthly income of $10,000 using extrapolation.
Accountant David Ellingson determined that to avoid a "cap" on child support, for every $50 increase in income above $10,000, the child support payment should be increased by $8Ellingson said that he used $8 for consistency purposes. Thus, Ellingson determined that the child support obligation for three children in this case would be $6,677 (this calculation attributed *428 a net monthly income of $35,233 to Patrick and no monthly income to Wendy). The other two approaches resulted in the child support amount being "capped" with no more increases in child support, although both approaches capped at different monthly incomes. Accountant Leonard Sommer determined that child support "capped" at a monthly income of $43,000, although he did not specify whether this was net or gross income, thereby limiting the utility of his testimony. However, we will assume that he was using net income throughout his calculation. Sommer calculated that in this case, child support for three children would be $4,006.
[14] The approach used by an economics professor, Dr. David Rosenbaum, resulted in a child support figure between that of the two accountants. Rosenbaum determined that for every $500 increase in net monthly income from $5,000 to $10,000, the increase in child support decreased by approximately $2 from the previous increase. We reproduce Rosenbaum's chart for child support for three children when the net income is above $10,000, although we leave out the "middle" of the chart in the interest of saving space, plus that portion is not terribly pertinent. Rosenbaum developed his chart for child support for net income above $10,000 by extrapolation from the data for child support amounts for three children when the net income is between $5,000 and $10,000but his exhibit only goes to net income of $24,500.


      Estimated Child Support Payments
                                 Child
  Net Income    Increase    Support Payment
   $10,000                       $2,645
   $10,500        $78            $2,723
   $11,000        $76            $2,799
   $11,500        $74            $2,873
   $12,000        $72            $2,945
   $12,500        $70            $3,015
    ....
   $21,000        $36            $3,899
   $21,500        $34            $3,933
   $22,000        $32            $3,965
   $22,500        $30            $3,995
   $23,000        $28            $4,023
   $23,500        $26            $4,049
   $24,000        $24            $4,073
   $24,500        $22            $4,095

Because Rosenbaum's chart only went up to $24,500 net income, the court inquired of him what would be the amount of support for three children under his method for net income of $30,000, and his response was $4,205. Rosenbaum further testified that under the extrapolation method he used, child support "caps" at $4,205in other words, that it would not increase any further from net earnings above $30,000 per month, even if a parent netted $100,000 per month. The district court said it "adopt[s] the Rosenbaum approach." Neither parent claims that the trial court's adoption of the Rosenbaum method was error. In addition to the trial court's error in using a joint custody calculation, we also find error in the trial court's application of the Rosenbaum methodology.
The trial court used $31,190 for net joint income, a figure that is unchallenged. The court then sought to extend the Rosenbaum calculation to this figure. It did this by taking the child support for income of $10,000 ($2,645) and adding to that figure the amount for child support for net income of $21,190 ($3,899) to produce a worksheet 1 figure of $6,544. A number of things are wrong with what the district court did. First, it fundamentally ignores Rosenbaum's methodology, as well as his testimony that under his methodology, adopted by the trial court, child support does not increase when net income is above $30,000. This should not be taken as our endorsement of Rosenbaum's finding or concept that support "caps" at $30,000 under his mathematical extrapolation from the guidelines, but merely an *429 observation that, whether intentional or not, the trial court did not really "adopt" Rosenbaum's methodwhich was simply a mathematical extrapolation. The trial court's method, when considered in conjunction with Rosenbaum's testimony that child support is $4,205 for $30,000 of net income, attributes $2,339 in child support to the "last" $1,190 of earningsa clearly illogical proposition. Finally, we bear in mind that we do not, for example, determine the amount of child support for net joint earnings of $9,000 by adding together the amounts on the chart for earnings of $4,000 and $5,000rather, we just look at what the chart says the amount is for $9,000. And while the chart does not go as high as net income of $31,190, Rosenbaum calculated the amount for that level of income while on the witness stand in direct response to the judge's request that he do soand the answer was $4,205, not $6,544. In short, the trial judge's method is contrary to his express adoption of Rosenbaum's calculationwhich was simply an expansion of the child support chart by extrapolation from the known data for child support at $500 increments between $5,000 and $10,000. Therefore, the addition of two child support amounts for two income levels within Rosenbaum's chart is clearly not supported by the evidence provided by Rosenbaum, nor is it a proper use of his methodology.
[15, 16] If Rosenbaum's method is used, child support associated with the parties' combined monthly net income of $31,190 would be $4,205. This is not to say that it could not be higher (or lower) under the trial court's considerable discretion in an above guidelines case, or that Rosenbaum's method of extrapolation is the only acceptable methodology. Rather, although the trial court misapplied Rosenbaum's method, neither party quarrels with the adoption of it as a methodology, and given its logical basis in mathematics, we certainly cannot conclude that it was plain error to use Rosenbaum's method. That said, we emphasize that we have no intent to establish a single methodology or approach to setting child support in an above guidelines situation. Rather, the question of proper support in an above guidelines case is determined by the evidence which encompasses not only methodology but a variety of other matters relating to the circumstances of the parties and the children. In this regard, we note that Rosenbaum freely conceded the illogic of "capping" child support when net income reached $30,000, a result which his method produced purely by mathematics.
The district court found that Patrick is responsible for 94 percent of the support amount. Therefore, Patrick's child support obligation, based on a sole custody calculation, is $3,952.70 per month, and we modify the district court's order to provide for monthly child support for three children in such amount, retroactive to January 1, 2007, the date when the trial court's child support award was to begin. Rosenbaum's extrapolation does not encompass child support for one or two children, and there will be support for the three children for a good number of years. Therefore, when the number of children requiring support changes, the parties, and the trial court if need be, can deal with that matter.

Retroactive Temporary Child Support.
Both parties appeal from the district court's award of retroactive child support. Patrick argues that the district court erred in ordering him to pay $21,658 in retroactive child support when Wendy stipulated to the amount of temporary child support. The record is clear that on August 27, 2004, Wendy, who was represented by counsel, stipulated to temporary child support of $2,700 per month. This stipulation was approved and adopted by the district *430 court on September 2. Wendy argues that the district court erred in failing to award retroactive child support based on the calculation of child support pursuant to worksheet 1. We have handled this contention above, so we turn to Patrick's complaint.
At trial, Wendy testified that at the time the temporary child support agreement was reached, she did not have any information about Patrick's earnings, and that Patrick had never told her anything about his earnings. She also explained that she did not have any money to hire an accountant at that time to review his financial records for the purpose of recommending an appropriate child support amount. Wendy testified that she was not in a position to decline Patrick's offer of $2,700 per month as temporary child support.
[17, 18] We are not persuaded by Wendy's argument for a number of reasons. On cross-examination, Wendy discussed an earlier separation in 2002 which involved mediation to determine the child support for the two children they had at the time, and she acknowledged that she was then advised that Patrick's net income was above $10,000 per month. Moreover, living with Patrick and the children would obviously acquaint Wendy with the sort of lifestyle he was able to afford. Thus, she can hardly be seen as completely in the dark about his finances. Additionally, there were a number of options open to Wendy via discovery to rather quickly gain financial informationtax returns for examplebefore agreeing to a child support amount. And she could have entrusted the decision to the court, negotiated a different stipulation, or made it contingent upon the acquisition of more information. In the final analysis, Wendy voluntarily, with the assistance of counsel, stipulated to $2,700 per month in temporary support, and stipulations voluntarily entered into will be respected and enforced by the courts. See Walters v. Walters, 12 Neb. App. 340, 673 N.W.2d 585 (2004). No showing has been made that the stipulation was not voluntaryWendy's argument being essentially premised on hindsight. And there is no showing that the children lacked for anything during the pendency of this case. In fact, the record clearly shows that anytime Wendy needed additional money during the pendency of this case for the children (or herself), Patrick provided such. Having stipulated to temporary child support, and given the considerations set forth above, we cannot find a basis to void that stipulation by an award of retroactive temporary support. The district court abused its discretion in ordering Patrick to pay $21,658 in retroactive child support. We vacate and set aside that portion of the district court's order.

Attorney Fees.
[19, 20] Patrick argues that the district court erred in ordering him to pay an additional $13,916.31 toward Wendy's attorney fees.
The award of attorney fees depends on multiple factors that include the nature of the case, the services performed and results obtained, the earning capacity of the parties, the length of time required for preparation and presentation of the case, customary charges of the bar, and the general equities of the case.
Gress v. Gress, 271 Neb. 122, 132, 710 N.W.2d 318, 328 (2006). Prior to trial, Patrick had already paid Wendy $2,500 for temporary attorney fees. Wendy submitted a detailed accounting of her attorney fees showing an outstanding balance of $15,184.79, and Patrick was ordered to pay $13,416.31 of such amount. Patrick was later ordered to pay an additional $500 to *431 Wendy's attorney for services rendered in connection with the motion to alter or amend judgment. After a thorough review of the record, and considering the significant difference in the parties' earning capacities, we find that the district court did not abuse its discretion in ordering Patrick to pay the additional $13,916.31 toward Wendy's attorney fees, and we affirm this aspect of the district court's order.

Expert Witness Fees.
[21] Patrick argues that the district court erred in ordering him to pay $27,592 in expert witness fees. We note that the appointment of the expert is not challenged. The district court appointed Ellingson to (1) analyze Patrick's personal and corporate tax returns for the years 2000 through 2004; (2) analyze income earned by and cashflow provided for Patrick from all disclosed sources for the years 2001 through 2004; (3) compute child support based upon the child support guidelines; and (4) prepare a comprehensive report summarizing the analysis and computation and recommending child support, based on the procedures performed. Patrick's financial circumstances are complex, and there is no evidence that such fees were not reasonable and necessary. The district court did not abuse its discretion in ordering Patrick to pay such fees, given the great disparity in the parties' income and the necessity to have expertise to assess Patrick's complex financial situation. We affirm this aspect of the district court's order.

CONCLUSION
We find that although Patrick has numerous and varied parenting times, it is liberal visitation and not joint custody. As a result, the trial court erred in basing the child support calculation on the joint custody worksheet, rather than the sole custody worksheet. Patrick's child support obligation, based on a sole custody calculation, shall be $3,952.70 per month for three children, and we modify the district court's order in this regard. This monthly child support amount shall be paid retroactive to January 1, 2007, the date provided for in the trial court's order.
We find that the district court abused its discretion in ordering Patrick to pay $21,658 in retroactive temporary child support. We vacate and set aside that portion of the district court's order.
The district court did not abuse its discretion in ordering Patrick to pay the additional $13,916.31 toward Wendy's attorney fees, and we affirm this aspect of the district court's order. Nor did the district court abuse its discretion in ordering Patrick to pay attorney fees for Wendy, given the great disparity in the parties' income, and we affirm this aspect of the district court's order.
The district court did not abuse its discretion in ordering Patrick to pay $27,592 in expert witness fees, given the great disparity in the parties' income and the necessity to have expertise to assess his complex financial situation. We affirm this aspect of the district court's order.
AFFIRMED IN PART, AFFIRMED IN PART AS MODIFIED, AND IN PART VACATED AND SET ASIDE.
